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Penyebaran spreadsheet penyesuaian musiman dan eksponensial smoothing Sangat mudah dilakukan penyesuaian musiman dan model pemulusan eksponensial yang sesuai dengan Excel. Gambar layar dan grafik di bawah diambil dari spreadsheet yang telah disiapkan untuk menggambarkan penyesuaian musiman multiplikatif dan pemulusan eksponensial linier pada data penjualan kuartalan berikut dari Outboard Marine: Untuk mendapatkan salinan file spreadsheet itu sendiri, klik di sini. Versi pemulusan eksponensial linier yang akan digunakan di sini untuk tujuan demonstrasi adalah versi Brown8217s, hanya karena dapat diimplementasikan dengan satu kolom formula dan hanya ada satu smoothing constant yang bisa dioptimalkan. Biasanya lebih baik menggunakan versi Holt8217 yang memiliki konstanta pemulusan terpisah untuk tingkat dan tren. Proses peramalan berjalan sebagai berikut: (i) pertama data disesuaikan secara musiman (ii) maka prakiraan dihasilkan untuk data penyesuaian musiman melalui pemulusan eksponensial linier dan (iii) perkiraan musim yang disesuaikan secara musiman adalah kuotasi untuk mendapatkan perkiraan untuk rangkaian aslinya. . Proses penyesuaian musiman dilakukan di kolom D sampai G. Langkah pertama dalam penyesuaian musiman adalah menghitung rata-rata pergerakan terpusat (dilakukan di kolom D). Hal ini dapat dilakukan dengan menghitung rata-rata dua rata-rata satu tahun yang diimbangi dengan satu periode relatif terhadap satu sama lain. (Kombinasi dua rata-rata offset daripada rata-rata tunggal diperlukan untuk tujuan penataan saat jumlah musim genap.) Langkah selanjutnya adalah menghitung rasio terhadap rata-rata pergerakan - i. Data asli dibagi dengan rata-rata bergerak pada setiap periode - yang dilakukan di sini di kolom E. (Ini juga disebut komponen siklus-trenwot dari pola, sejauh kecenderungan dan efek siklus bisnis dapat dianggap sebagai semua hal Tetap setelah rata-rata selama satu tahun penuh data.Tentu saja, perubahan bulan ke bulan yang bukan karena musiman dapat ditentukan oleh banyak faktor lainnya, namun rata-rata 12 bulan rata-rata di atasnya untuk sebagian besar.) Indeks musiman diperkirakan untuk setiap musim dihitung dengan menghitung rata-rata pertama semua rasio untuk musim tertentu, yang dilakukan di sel G3-G6 menggunakan formula AVERAGEIF. Rasio rata-rata kemudian dikompres sehingga jumlahnya mencapai 100 kali jumlah periode dalam satu musim, atau 400 dalam kasus ini, yang dilakukan pada sel H3-H6. Di bawah kolom F, formula VLOOKUP digunakan untuk memasukkan nilai indeks musiman yang sesuai di setiap baris tabel data, sesuai dengan kuartal tahun yang diwakilinya. Rata-rata pergerakan terpusat dan data yang disesuaikan musiman akhirnya terlihat seperti ini: Perhatikan bahwa rata-rata bergerak biasanya terlihat seperti versi yang lebih halus dari rangkaian yang disesuaikan secara musiman, dan ini lebih pendek pada kedua ujungnya. Lembar kerja lain dalam file Excel yang sama menunjukkan penerapan model smoothing eksponensial linier ke data yang disesuaikan secara musiman, dimulai pada kolom G. Nilai untuk konstanta pemulusan (alpha) dimasukkan di atas kolom perkiraan (di sini, di sel H9) dan Untuk kenyamanan itu diberi nama jarak jauh quotAlpha.quot (Nama tersebut diberikan dengan menggunakan perintah quotInsertNameCreatequot.) Model LES diinisialisasi dengan menetapkan dua prakiraan pertama yang sama dengan nilai sebenarnya dari seri yang disesuaikan secara musiman. Rumus yang digunakan di sini untuk perkiraan LES adalah bentuk rekursif tunggal model Brown8217s: Formula ini dimasukkan ke dalam sel yang sesuai dengan periode ketiga (di sini, sel H15) dan disalin dari sana. Perhatikan bahwa perkiraan LES untuk periode saat ini mengacu pada dua observasi sebelumnya dan dua kesalahan perkiraan sebelumnya, serta nilai alpha. Dengan demikian, rumus peramalan pada baris 15 hanya mengacu pada data yang tersedia pada baris 14 dan sebelumnya. (Tentu saja, jika kita ingin menggunakan yang sederhana daripada pemulusan eksponensial linier, kita bisa mengganti formula SES di sini sebagai gantinya. Kita juga bisa menggunakan model LES Holt8217s daripada Brown8217s, yang memerlukan dua kolom formula untuk menghitung tingkat dan tren. Yang digunakan dalam ramalan.) Kesalahan dihitung pada kolom berikutnya (di sini, kolom J) dengan mengurangkan perkiraan dari nilai sebenarnya. Kesalahan kuadrat rata-rata akar dihitung sebagai akar kuadrat dari varians kesalahan ditambah kuadrat rata-rata. (Berikut ini dari identitas matematis: MSE VARIANCE (error) (AVERAGE (error)) 2.) Dalam menghitung mean dan varians dari kesalahan dalam formula ini, dua periode pertama dikecualikan karena model tidak benar-benar mulai meramalkan sampai Periode ketiga (baris 15 di spreadsheet). Nilai alfa yang optimal dapat ditemukan dengan mengubah alpha secara manual sampai RMSE minimum ditemukan, jika tidak, Anda dapat menggunakan quotSolverquot untuk melakukan minimisasi yang tepat. Nilai alfa yang ditemukan Solver ditunjukkan di sini (alpha0.471). Biasanya ide bagus untuk merencanakan kesalahan model (dalam unit yang diubah) dan juga untuk menghitung dan merencanakan autokorelasi mereka pada kelambatan hingga satu musim. Berikut adalah rangkaian rangkaian waktu dari kesalahan (yang disesuaikan secara musiman): Autokorelasi kesalahan dihitung dengan menggunakan fungsi CORREL () untuk menghitung korelasi kesalahan dengan sendirinya yang tertinggal oleh satu atau beberapa periode - rincian ditampilkan dalam model spreadsheet . Berikut adalah sebidang autocorrelations dari kesalahan pada lima kelambatan pertama: Autokorelasi pada lags 1 sampai 3 sangat mendekati nol, namun lonjakan pada lag 4 (yang nilainya 0,35) sedikit merepotkan - ini menunjukkan bahwa Proses penyesuaian musiman belum sepenuhnya berhasil. Namun, sebenarnya hanya sedikit signifikan. 95 band signifikansi untuk menguji apakah autokorelasi berbeda secara signifikan dari nol kira-kira plus-atau-minus 2SQRT (n-k), di mana n adalah ukuran sampel dan k adalah lag. Disini n adalah 38 dan k bervariasi dari 1 sampai 5, jadi kuadrat-akar-of-n-minus-k adalah sekitar 6 untuk semua itu, dan karenanya batas untuk menguji signifikansi statistik penyimpangan dari nol kira-kira plus- Atau-minus 26, atau 0,33. Jika Anda memvariasikan nilai alpha dengan tangan dalam model Excel ini, Anda dapat mengamati pengaruhnya pada deret waktu dan plot autokorelasi dari kesalahan, serta pada kesalahan akar-mean-kuadrat, yang akan digambarkan di bawah ini. Di bagian bawah spreadsheet, rumus peramalan adalah quotbootstrappedquot ke masa depan dengan hanya mengganti perkiraan untuk nilai aktual pada titik di mana data aktual habis - yaitu. Dimana quotthe futurequot dimulai. (Dengan kata lain, di setiap sel di mana nilai data masa depan akan terjadi, referensi sel dimasukkan yang mengarah ke perkiraan yang dibuat untuk periode itu.) Semua formula lainnya hanya disalin dari atas: Perhatikan bahwa kesalahan untuk perkiraan Masa depan semuanya dihitung menjadi nol. Ini tidak berarti kesalahan sebenarnya akan menjadi nol, melainkan hanya mencerminkan fakta bahwa untuk tujuan prediksi, kita mengasumsikan bahwa data masa depan akan sama dengan perkiraan rata-rata. Perkiraan LES yang dihasilkan untuk data penyesuaian musiman terlihat seperti ini: Dengan nilai alpha tertentu ini, yang optimal untuk prediksi satu periode di depan, tren yang diproyeksikan sedikit ke atas, yang mencerminkan tren lokal yang diamati selama 2 tahun terakhir. Atau lebih. Untuk nilai alpha lain, proyeksi tren yang sangat berbeda dapat diperoleh. Biasanya ide bagus untuk melihat apa yang terjadi pada proyeksi tren jangka panjang ketika alfa bervariasi, karena nilai yang terbaik untuk peramalan jangka pendek tidak akan menjadi nilai terbaik untuk memprediksi masa depan yang lebih jauh. Sebagai contoh, berikut ini adalah hasil yang diperoleh jika nilai alpha diatur secara manual menjadi 0,25: Tren jangka panjang yang diproyeksikan sekarang negatif dan bukan positif Dengan nilai alpha yang lebih kecil, model ini menempatkan bobot lebih pada data lama di Perkiraan tingkat dan tren saat ini, dan perkiraan jangka panjangnya mencerminkan tren penurunan yang diamati selama 5 tahun terakhir daripada tren kenaikan yang lebih baru. Bagan ini juga secara jelas mengilustrasikan bagaimana model dengan nilai alpha yang lebih kecil lebih lambat untuk merespons quotturning pointsquot dalam data dan karena itu cenderung membuat kesalahan dari tanda yang sama untuk banyak periode berturut-turut. Kesalahan perkiraan 1 langkah lebih besar rata-rata dibandingkan yang diperoleh sebelumnya (RMSE 34,4 bukan 27,4) dan autokorelasi positif sangat positif. Autokorelasi lag-1 sebesar 0,56 sangat melebihi nilai 0,33 yang dihitung di atas untuk penyimpangan signifikan secara statistik dari nol. Sebagai alternatif untuk menurunkan nilai alpha dalam rangka memperkenalkan lebih banyak konservatisme ke dalam ramalan jangka panjang, faktor penurunan harga perambatan tren kadang ditambahkan ke model untuk membuat tren yang diproyeksikan merata setelah beberapa periode. Langkah terakhir dalam membangun model peramalan adalah untuk memperkirakan tingkat perkiraan LES dengan mengalikannya dengan indeks musiman yang sesuai. Dengan demikian, ramalan yang direvisi di kolom I hanyalah produk dari indeks musiman di kolom F dan perkiraan LES musiman yang disesuaikan di kolom H. Hal ini relatif mudah untuk menghitung interval kepercayaan untuk perkiraan satu langkah yang dibuat oleh model ini: pertama Menghitung RMSE (kesalahan akar-mean-kuadrat, yang merupakan akar kuadrat dari MSE) dan kemudian menghitung interval kepercayaan untuk perkiraan penyesuaian musiman dengan menambahkan dan mengurangkan dua kali RMSE. (Secara umum, interval kepercayaan 95 untuk perkiraan satu periode di depan kira-kira sama dengan perkiraan titik ditambah atau minus dua kali perkiraan deviasi standar dari kesalahan perkiraan, dengan asumsi distribusi kesalahan kira-kira normal dan ukuran sampel Cukup besar, katakanlah 20 atau lebih. Di sini, RMSE dan bukan standar deviasi standar dari kesalahan adalah perkiraan terbaik dari standar deviasi perkiraan kesalahan masa depan karena juga mempertimbangkan variasi yang bias dan juga acak). Batas kepercayaan Untuk perkiraan musiman disesuaikan kemudian direvisi. Bersama dengan perkiraan, dengan mengalikannya dengan indeks musiman yang sesuai. Dalam hal ini RMSE sama dengan 27,4 dan perkiraan penyesuaian musiman untuk periode depan pertama (Des-93) adalah 273,2. Jadi interval kepercayaan 95 yang disesuaikan musiman adalah dari 273,2-227,4 218,4 sampai 273,2227,4 328,0. Mengalikan batas ini dengan indeks musiman Decembers sebesar 68,61. Kita memperoleh batas kepercayaan bawah dan atas 149,8 dan 225,0 sekitar perkiraan titik 93 Desember 187,4. Batas keyakinan untuk prakiraan lebih dari satu periode ke depan biasanya akan melebar seiring perkiraan horizon meningkat, karena ketidakpastian tentang tingkat dan kecenderungan serta faktor musiman, namun sulit untuk menghitungnya secara umum dengan metode analitik. (Cara yang tepat untuk menghitung batas keyakinan untuk perkiraan LES adalah dengan menggunakan teori ARIMA, namun ketidakpastian dalam indeks musiman adalah masalah lain.) Jika Anda menginginkan interval kepercayaan yang realistis untuk perkiraan lebih dari satu periode di depan, ambil semua sumber Dengan kesalahan, taruhan terbaik Anda adalah menggunakan metode empiris: misalnya, untuk mendapatkan interval keyakinan untuk perkiraan 2 langkah di depan, Anda bisa membuat kolom lain di spreadsheet untuk menghitung perkiraan 2 langkah untuk setiap periode ( Dengan melakukan bootstrap perkiraan satu langkah di depan). Kemudian hitung RMSE dari perkiraan kesalahan 2 langkah di depan dan gunakan ini sebagai dasar untuk interval keyakinan 2 langkah di depan.Excel For Statistical Data Analysis Ini adalah situs pendamping webtext dari Business Statistics USA Site Para mis visitantes del mundo De habla hispana, este sitio se encuentra disponible en espaol en: Sitio Espejo para Amrica Latina Sitio de los EEUU Excel adalah paket statistik yang banyak digunakan, yang berfungsi sebagai alat untuk memahami konsep statistik dan perhitungan untuk memeriksa perhitungan pekerjaan tangan Anda dalam memecahkan masalah pekerjaan rumah Anda. Situs ini memberikan pengantar untuk memahami dasar-dasar dan bekerja dengan Excel. Mengulangi contoh numerik bergambar di situs ini akan membantu meningkatkan keakraban Anda dan sebagai hasilnya meningkatkan efektivitas dan efisiensi proses Anda dalam statistik. Untuk mencari situs. Coba E dit F ind di halaman Ctrl f. Masukkan kata atau frasa di kotak dialog, mis. Quot variansquot atau quot meanquot Jika tampilan pertama dari kata kunci bukan apa yang Anda cari, cobalah F ind Next. Pendahuluan Situs ini memberikan pengalaman ilustrasi dalam penggunaan Excel untuk ringkasan data, presentasi, dan untuk analisis statistik dasar lainnya. Saya percaya penggunaan Excel yang populer ada di area di mana Excel benar-benar bisa unggul. Ini termasuk mengatur data, yaitu pengelolaan data dasar, tabulasi dan grafik. Untuk analisis statistik sesungguhnya harus belajar menggunakan paket statistik komersial profesional seperti SAS, dan SPSS. Microsoft Excel 2000 (versi 9) menyediakan seperangkat alat analisis data yang disebut Analysis ToolPak yang dapat Anda gunakan untuk menyimpan langkah-langkah saat Anda mengembangkan analisis statistik yang kompleks. Anda menyediakan data dan parameter untuk setiap analisis alat ini menggunakan fungsi makro statistik yang sesuai dan kemudian menampilkan hasilnya dalam tabel output. Beberapa alat menghasilkan grafik selain tabel output. Jika perintah Analisis Data dapat dipilih pada menu Tools, maka Analysis ToolPak diinstal pada sistem Anda. Namun, jika perintah Analisis Data tidak ada pada menu Tools, Anda perlu menginstal Analysis ToolPak dengan melakukan hal berikut: Langkah 1: Pada menu Tools, klik Add-Ins. Jika Analysis ToolPak tidak tercantum dalam kotak dialog Add-Ins, klik Browse dan cari drive, nama folder, dan nama file untuk Analysis ToolPak Add-in Analys32.xll yang biasanya ada di folder FilesMicrosoft OfficeOfficeLibraryAnalisis Program. Setelah Anda menemukan file tersebut, pilih dan klik OK. Langkah 2: Jika Anda tidak menemukan file Analys32.xll, maka Anda harus menginstalnya. Masukkan Disk Microsoft Office 2000 Anda ke dalam drive CD ROM. Pilih Run dari menu Start Windows. Jelajahi dan pilih drive untuk CD Anda. Pilih Setup.exe, klik Open, dan klik OK. Klik tombol Add or Remove Features. Klik next ke Microsoft Excel untuk Windows. Klik next to Add-in. Klik panah bawah di samping Analysis ToolPak. Pilih Run dari My Computer. Pilih tombol Update Now. Excel sekarang akan memperbarui sistem Anda untuk menyertakan Analysis ToolPak. Luncurkan Excel. Pada menu Tools, klik Add-Ins. - dan pilih kotak centang Analysis ToolPak. Langkah 3: Alat AnalisisPak Add-In sekarang terinstal dan Analisis Data. Sekarang akan dipilih pada menu Tools.Microsoft Excel adalah paket spreadsheet yang kuat yang tersedia untuk Microsoft Windows dan Apple Macintosh. Perangkat lunak Spreadsheet digunakan untuk menyimpan informasi dalam kolom dan baris yang kemudian dapat diatur dan diproses. Spreadsheets dirancang untuk bekerja dengan baik dengan angka tapi sering menyertakan teks. Excel mengatur pekerjaan Anda ke dalam buku kerja setiap buku kerja dapat berisi banyak lembar kerja lembar kerja yang digunakan untuk membuat daftar dan menganalisis data. Excel tersedia di semua PC akses publik (yaitu, di Lab Perpustakaan dan PC). Hal ini dapat dibuka baik dengan memilih Start - Programs - Microsoft Excel atau dengan mengklik Short Cut Excel yang ada di desktop Anda, atau pada PC manapun, atau pada bilah Alat Kantor. Membuka Dokumen: Klik File-Open (CtrlO) untuk membuka kembali buku kerja yang ada untuk mengubah area direktori atau drive untuk mencari file di lokasi lain Untuk membuat workbook baru, klik File-New-Blank Document. Menyimpan dan Menutup Dokumen: Untuk menyimpan dokumen Anda dengan nama file, lokasi dan format filenya saat ini, klik File - Save. Jika Anda menabung untuk pertama kalinya, klik File-Save choosetype nama untuk dokumen Anda lalu klik OK. Juga gunakan File-Save jika Anda ingin menyimpan ke filenamelocation yang berbeda. Setelah selesai mengerjakan dokumen, Anda harus menutupnya. Buka menu File dan klik Close. Jika Anda telah membuat perubahan sejak file terakhir disimpan, Anda akan ditanya apakah Anda ingin menyimpannya. Layar Excel Workbook dan lembar kerja: Saat Anda memulai Excel, lembar kerja kosong ditampilkan yang terdiri dari beberapa grid sel dengan baris bernomor di halaman dan kolom dengan judul berdasarkan abjad di seluruh halaman. Setiap sel direferensikan oleh koordinatnya (misalnya A3 digunakan untuk merujuk ke sel di kolom A dan baris 3 B10: B20 digunakan untuk merujuk pada kisaran sel pada kolom B dan baris 10 sampai 20). Pekerjaan Anda disimpan dalam file Excel yang disebut workbook. Setiap buku kerja mungkin berisi beberapa lembar kerja dan grafik - lembar kerja saat ini disebut lembar aktif. Untuk melihat lembar kerja yang berbeda dalam buku kerja, klik Lembar Tab yang sesuai. Anda dapat mengakses dan menjalankan perintah langsung dari menu utama atau Anda dapat menunjuk ke salah satu tombol toolbar (kotak tampilan yang muncul di bawah tombol, saat Anda meletakkan kursor di atasnya, menunjukkan nama tombol) dan klik sekali. Bergerak di Sekitar Lembar Kerja: Penting untuk dapat bergerak di sekitar lembar kerja secara efektif karena Anda hanya dapat memasukkan atau mengubah data pada posisi kursor. Anda dapat memindahkan kursor dengan menggunakan tombol panah atau dengan mengarahkan mouse ke sel yang diperlukan dan mengkliknya. Setelah dipilih sel menjadi sel aktif dan dikenali oleh batas tebal hanya satu sel yang bisa aktif sekaligus. Untuk beralih dari satu lembar kerja ke lembar kerja lainnya, klik tab lembar. (Jika buku kerja Anda berisi banyak lembar, klik kanan tombol gulir tab lalu klik lembar yang Anda inginkan.) Nama lembar aktif ditampilkan dalam huruf tebal. Bergerak Antara Sel: Berikut adalah cara pintas keyboard untuk memindahkan sel aktif: Home - bergerak ke kolom pertama di baris saat ini CtrlHome - bergerak ke sudut kiri atas dokumen Akhir kemudian Home - bergerak ke sel terakhir dalam dokumen To Bergerak di antara sel pada lembar kerja, klik sel atau gunakan tombol panah. Untuk melihat area sheet yang berbeda, gunakan scroll bar dan klik pada tanda panah atau area di bawah kotak gulir di scroll bar vertikal atau horizontal. Perhatikan bahwa ukuran kotak gulir menunjukkan jumlah proporsional area yang digunakan dari lembar yang terlihat di jendela. Posisi kotak gulir menunjukkan lokasi relatif area yang terlihat dalam lembar kerja. Memasukkan Data Lembar kerja baru adalah kotak baris dan kolom. Baris diberi label dengan angka, dan kolom diberi label dengan huruf. Setiap persimpangan baris dan kolom adalah sel. Setiap sel memiliki alamat. Yaitu kolom huruf dan nomor baris. Panah pada lembar kerja ke titik kanan ke sel A1, yang saat ini disorot. Menunjukkan bahwa itu adalah sel aktif. Sel harus aktif memasukkan informasi ke dalamnya. Untuk menyorot (pilih) sel, klik di atasnya. Untuk memilih lebih dari satu sel: Klik sel (misalnya A1), kemudian tahan tombol shift saat Anda mengklik yang lain (misalnya D4) untuk memilih semua sel antara dan termasuk A1 dan D4. Klik pada sel (misalnya A1) dan seret mouse ke kisaran yang diinginkan, lepaskan pada sel lain (misalnya D4) untuk memilih semua sel antara dan termasuk A1 dan D4. Untuk memilih beberapa sel yang tidak bersebelahan, tekan kontrol dan klik Sel yang ingin Anda pilih. Klik nomor atau huruf yang melabeli baris atau kolom untuk memilih keseluruhan baris atau kolom. Satu lembar kerja bisa memiliki hingga 256 kolom dan 65.536 baris, jadi akan lama sebelum Anda kehabisan ruang. Setiap sel bisa mengandung label. Nilai. Nilai logis Atau formula. Label dapat berisi kombinasi huruf, angka, atau simbol. Nilai adalah angka. Hanya nilai (angka) yang bisa digunakan dalam perhitungan. Nilai juga bisa berupa tanggal atau waktu. Nilai logis benar atau salah. Formula secara otomatis melakukan perhitungan terhadap nilai pada sel tertentu lainnya dan menampilkan hasilnya di sel tempat formula dimasukkan (misalnya, Anda dapat menentukan sel D3 itu Adalah berisi jumlah angka di B3 dan C3 nomor yang ditampilkan di D3 kemudian akan menjadi funtion dari angka yang masuk ke B3 dan C3). Untuk memasukkan informasi ke dalam sel, pilih sel dan mulai mengetik. Perhatikan bahwa saat Anda mengetik informasi ke dalam sel, informasi yang Anda masukkan juga ditampilkan di formula bar. Anda juga bisa memasukkan informasi ke dalam formula bar, dan informasinya akan muncul di sel yang dipilih. Bila Anda telah selesai memasukkan label atau nilai: Tekan Enter untuk berpindah ke sel berikutnya di bawah ini (dalam hal ini, A2) Tekan Tab untuk berpindah ke sel berikutnya ke kanan (dalam kasus ini, B1) Klik sel untuk memilih Itu Memasuki Label Kecuali informasi yang Anda masukkan diformat sebagai nilai atau formula, Excel akan menafsirkannya sebagai label, dan default menyelaraskan teks di sisi kiri sel. Jika Anda membuat lembar kerja yang panjang dan Anda akan mengulangi informasi label yang sama di banyak sel yang berbeda, Anda dapat menggunakan fungsi AutoComplete. Fungsi ini akan melihat entri lain di kolom yang sama dan mencoba mencocokkan entri sebelumnya dengan entri Anda saat ini. Misalnya, jika Anda sudah mengetikkan Wesleyan di sel lain dan Anda mengetik W di sel baru, Excel secara otomatis akan masuk Wesleyan. Jika Anda bermaksud memasukkan Wesleyan ke dalam sel, tugas Anda selesai, dan Anda dapat beralih ke sel berikutnya. Jika Anda ingin mengetikkan sesuatu yang lain, mis. Williams, masuk ke sel, terus mengetik untuk memasukkan istilah tersebut. Untuk mengaktifkan funtion AutoComplete, klik Tools pada menu bar, lalu pilih Options, lalu pilih Edit, dan klik untuk memberi tanda centang pada kotak di sebelah Enable AutoComplete untuk nilai sel. Cara lain untuk cepat memasukkan label berulang adalah dengan menggunakan fitur Pick List. Klik kanan pada sel, lalu pilih Pick From List. Ini akan memberi Anda daftar semua entri lainnya di sel di kolom itu. Klik pada item di menu untuk memasukkannya ke sel yang sekarang dipilih. Nilai adalah angka, tanggal, atau waktu, ditambah beberapa simbol jika perlu untuk menentukan angka lebih jauh seperti pada. - () 93. Angka diasumsikan positif untuk memasukkan angka negatif, gunakan tanda minus - atau lampirkan nomor dalam tanda kurung (). Tanggal disimpan sebagai MMDDYYYY, namun Anda tidak perlu memasukkannya persis dalam format itu. Jika Anda masuk ke jan 9 atau jan-9, Excel akan mengetahuinya pada tanggal 9 Januari tahun ini, dan menyimpannya pada 192002. Masukkan tahun empat digit selama satu tahun selain tahun berjalan (misalnya jan 9, 1999). Untuk memasukkan tanggal hari ini, tekan kontrol dan pada saat bersamaan. Kali default menjadi 24 jam. Gunakan a atau p untuk menunjukkan am atau pm jika Anda menggunakan jam 12 jam (mis. 8:30 p ditafsirkan pada pukul 20.30). Untuk memasukkan waktu saat ini, tekan kontrol dan: (shift-titik koma) pada saat bersamaan. Entri yang diartikan sebagai nilai (angka, tanggal, atau waktu) diselaraskan ke sisi kanan sel, untuk memformat ulang sebuah nilai. Angka Pembulatan yang Memenuhi Kriteria Tertentu: Untuk menerapkan warna ke nilai andor dan minimum maksimum: Pilih sel di wilayah ini, dan tekan CtrlShift (pada Excel 2003, tekan ini atau CtrlA) untuk memilih Current Region. Dari menu Format, pilih Conditional Formatting. Dalam Kondisi 1, pilih Formula Is, dan ketik MAX (F: F) F1. Klik Format, pilih Font tab, pilih warna, dan kemudian klik Oke. Dalam Kondisi 2, pilih Formula Is, dan ketik MIN (F: F) F1. Ulangi langkah 4, pilih warna yang berbeda dari yang Anda pilih untuk Kondisi 1, lalu klik OK. Catatan: Pastikan untuk membedakan antara referensi absolut dan referensi relatif saat memasukkan rumus. Angka Pengambilan yang Memenuhi Kriteria Spesifik Masalah: Membulatkan semua angka di kolom A ke angka nol, kecuali yang ada di desimal pertama. Solusi: Gunakan fungsi JIKA, MOD, dan ROUND dengan rumus sebagai berikut: IF (MOD (A2,1) 0,5, A2, ROUND (A2,0)) Untuk Menyalin dan Menyisipkan Semua Sel dalam Lembar Pilih sel di lembar Dengan menekan CtrlA (di Excel 2003, pilih sel di area kosong sebelum menekan CtrlA, atau dari sel yang dipilih dalam rentang Current RegionList, tekan CtrlAA). ATAU Klik Pilih Semua di persimpangan kiri atas baris dan kolom. Tekan CtrlC. Tekan CtrlPage Down untuk memilih lembar lain, lalu pilih sel A1. Tekan enter. Untuk Menyalin Seluruh Lembar Menyalin seluruh lembar berarti menyalin sel, parameter pengaturan halaman, dan Nama-nama yang ditentukan. Opsi 1: Gerakkan penunjuk mouse ke tab lembar. Tekan Ctrl, dan tahan mouse untuk menyeret lembar ke lokasi lain. Lepaskan tombol mouse dan tombol Ctrl. Opsi 2: Klik kanan tab sheet yang sesuai. Dari menu jalan pintas, pilih Pindah atau Salin. Pindahkan atau Salin kotak dialog memungkinkan seseorang untuk menyalin lembar baik ke lokasi yang berbeda dalam buku kerja saat ini atau ke buku kerja yang berbeda. Pastikan untuk menandai kotak centang Create a copy. Opsi 3: Dari menu Window, pilih Arrange. Pilih Tiled to tile all open workbooks di jendela. Gunakan Opsi 1 (menyeret lembar sambil menekan Ctrl) untuk menyalin atau memindahkan lembar. Sortasi menurut Kolom Pengaturan default untuk sortir dalam urutan Ascending atau Descending adalah dengan baris. Untuk mengurutkan menurut kolom: Dari menu Data, pilih Sortir, lalu pilih. Pilih tombol pilihan Sortir ke kanan dan klik OK. Di dalam Sort by option dari kotak dialog Sort, pilih nomor baris dimana kolom akan diurutkan dan klik OK. Statistik Deskriptif Alat Analisis DataPak memiliki alat Statistik Deskriptif yang memberi Anda kemudahan untuk menghitung statistik ringkasan untuk sekumpulan data sampel. Ringkasan statistik meliputi Mean, Standard Error, Median, Mode, Standard Deviation, Varians, Kurtosis, Skewness, Range, Minimum, Maximum, Sum, and Count. Alat ini menghilangkan kebutuhan untuk mengetikkan fungsi indivividual untuk menemukan masing-masing hasil ini. Excel mencakup bilah alat yang rumit dan dapat disesuaikan, misalnya bilah alat standar yang ditunjukkan di sini: Beberapa ikon berguna untuk perhitungan matematis: adalah ikon Autosum, yang memasukkan jumlah rumus () untuk menambahkan rentang sel. Adalah ikon FunctionWizard, yang memberi Anda akses ke semua fungsi yang tersedia. Adalah ikon GraphWizard, memberikan akses ke semua jenis grafik yang tersedia, seperti yang ditunjukkan pada tampilan ini: Excel dapat digunakan untuk menghasilkan ukuran lokasi dan variabilitas variabel. Misalkan kita ingin menemukan statistik deskriptif untuk data sampel: 2, 4, 6, dan 8. Langkah 1. Pilih menu pull-down Tools, jika Anda melihat analisis data, klik opsi ini, jika tidak, klik add-in . Pilihan untuk menginstal alat analisis pak. Langkah 2. Klik pada pilihan analisis data. Langkah 3. Pilih Deskriptif Statistik dari Daftar Alat Analisis. Langkah 4. Saat kotak dialog muncul: Masukkan A1: A4 di kotak input range, A1 adalah nilai pada kolom A dan baris 1. Dalam hal ini nilai ini adalah 2. Menggunakan teknik yang sama masukkan NILAI lain sampai Anda mencapai yang terakhir. Jika sampel terdiri dari 20 nomor, Anda dapat memilih misalnya A1, A2, A3, dll sebagai kisaran masukan. Langkah 5. Pilih rentang output. Dalam hal ini B1. Klik ringkasan statistik untuk melihat hasilnya. Bila Anda mengklik OK. Anda akan melihat hasilnya di kisaran yang dipilih. Seperti yang akan Anda lihat, rata-rata sampel adalah 5, mediannya adalah 5, standar deviasi adalah 2.581989, varians sampelnya adalah 6.666667, kisarannya adalah 6 dan seterusnya. Masing-masing faktor ini mungkin penting dalam perhitungan prosedur statistik yang berbeda. Distribusi Normal Pertimbangkan masalah untuk menemukan probabilitas mendapatkan nilai kurang dari nilai tertentu di bawah distribusi probabilitas normal. Sebagai contoh ilustratif, mari kita anggap nilai SAT nasional secara normal didistribusikan dengan rata-rata dan standar deviasi 500 dan 100. Jawablah pertanyaan berikut berdasarkan informasi yang diberikan: A: Berapakah probabilitas skor siswa yang dipilih secara acak akan kurang dari 600 poin B: Berapakah probabilitas skor siswa yang dipilih secara acak akan melebihi 600 poin C: Berapakah probabilitasnya Bahwa nilai siswa yang dipilih secara acak antara 400 dan 600 Petunjuk: Dengan menggunakan Excel Anda dapat menemukan probabilitas mendapatkan nilai kira-kira kurang dari atau sama dengan nilai tertentu. Dalam masalah, bila mean dan standar deviasi populasi diberikan, Anda harus menggunakan akal sehat untuk menemukan probabilitas yang berbeda berdasarkan pertanyaan karena Anda tahu area di bawah kurva normal adalah 1. Pada lembar kerja, pilih Sel dimana Anda ingin jawaban muncul. Misalkan, Anda memilih nomor satu, A1. Dari menu, pilih quotinsert pull-downquot. Langkah 2-3 Dari menu, pilih insert, lalu klik pada Function. Langkah 4. Setelah mengklik pilihan Function, dialog Paste Function muncul dari Function Category. Pilih Statistik kemudian NORMDIST dari kotak Function Name Klik OK Langkah 5. Setelah mengklik OK, kotak distribusi NORMDIST akan muncul: i. Masukkan 600 di X (kotak nilai) ii. Masukkan 500 di kotak Mean iii. Masukkan 100 di kotak deviasi Standar iv. Ketik quottruequot di kotak kumulatif, lalu klik OK. Seperti yang Anda lihat, nilai 0.84134474 muncul di A1, menunjukkan probabilitas bahwa skor siswa yang dipilih secara acak di bawah 600 poin. Dengan menggunakan akal sehat, kita bisa menjawab kuququot bagian dengan mengurangkan 0.84134474 dari 1. Jadi, jawaban kuom kuadrat adalah 1- 0,8413474 atau 0.158653. Ini adalah probabilitas bahwa skor siswa yang dipilih secara acak lebih besar dari 600 poin. Untuk menjawab bagian kuotot, gunakan teknik yang sama untuk menemukan probabilitas atau area di sisi kiri nilai 600 dan 400. Karena area atau probabilitas ini saling tumpang tindih untuk menjawab pertanyaan, Anda harus mengurangi probabilitas yang lebih kecil dari probabilitas yang lebih besar. Jawabannya sama dengan 0.84134474 - 0.15865526 ​​yaitu 0.68269. Screen shot seharusnya terlihat seperti berikut: Menghitung nilai variabel acak yang sering disebut nilai quotxquot Anda dapat menggunakan NORMINV dari kotak fungsi untuk menghitung nilai variabel acak - jika probabilitas ke sisi kiri variabel ini diberikan. Sebenarnya, Anda harus menggunakan fungsi ini untuk menghitung persentil yang berbeda. Dalam masalah ini seseorang bisa bertanya berapa skor seorang siswa yang persentilnya 90 Ini berarti kira-kira 90 nilai siswa kurang dari angka ini. Di sisi lain jika kita diminta untuk melakukan masalah ini dengan tangan, kita harus menghitung nilai x dengan menggunakan rumus distribusi normal x m zd. Sekarang mari kita gunakan Excel untuk menghitung P90. Pada fungsi Paste, klik dialog statistik, lalu klik NORMINV. Tangkapan layar akan terlihat seperti berikut: Bila Anda melihat NORMINV kotak dialog muncul. saya. Masukkan 0,90 untuk probabilitas (ini berarti bahwa kira-kira 90 nilai siswa kurang dari nilai yang kita cari) ii. Masukkan 500 untuk mean (ini adalah mean dari distribusi normal dalam kasus kami) iii. Masukkan 100 untuk standar deviasi (ini adalah standar deviasi dari distribusi normal dalam kasus kami) Pada akhir layar ini Anda akan melihat hasil rumus yang kira-kira 628 poin. Ini berarti 10 besar siswa mendapat nilai lebih baik dari 628. Interval Kepercayaan untuk Mean Misalkan kita berharap untuk memperkirakan interval kepercayaan untuk rata-rata populasi. Depending on the size of your sample size you may use one of the following cases: Large Sample Size (n is larger than, say 30): The general formula for developing a confidence interval for a population means is: In this formula is the mean of the sample Z is the interval coefficient, which can be found from the normal distribution table (for example the interval coefficient for a 95 confidence level is 1.96). S is the standard deviation of the sample and n is the sample size. Now we would like to show how Excel is used to develop a certain confidence interval of a population mean based on a sample information. As you see in order to evaluate this formula you need quotthe mean of the samplequot and the margin of error Excel will automatically calculate these quantities for you. The only things you have to do are: add the margin of error to the mean of the sample, Find the upper limit of the interval and subtract the margin of error from the mean to the lower limit of the interval. To demonstrate how Excel finds these quantities we will use the data set, which contains the hourly income of 36 work-study students here, at the University of Baltimore. These numbers appear in cells A1 to A36 on an Excel work sheet. After entering the data, we followed the descriptive statistic procedure to calculate the unknown quantities. The only additional step is to click on the confidence interval in the descriptive statistics dialog box and enter the given confidence level, in this case 95. Here is, the above procedures in step-by-step: Step 1. Enter data in cells A1 to A36 (on the spreadsheet) Step 2. From the menus select Tools Step 3. Click on Data Analysis then choose the Descriptive Statistics option then click OK . On the descriptive statistics dialog, click on Summary Statistic. After you have done that, click on the confidence interval level and type 95 - or in other problems whatever confidence interval you desire. In the Output Range box enter B1 or what ever location you desire. Now click on OK . The screen shot would look like the following: As you see, the spreadsheet shows that the mean of the sample is 6.902777778 and the absolute value of the margin of error 0.231678109. This mean is based on this sample information. A 95 confidence interval for the hourly income of the UB work-study students has an upper limit of 6.902777778 0.231678109 and a lower limit of 6.902777778 - 0.231678109. On the other hand, we can say that of all the intervals formed this way 95 contains the mean of the population. Or, for practical purposes, we can be 95 confident that the mean of the population is between 6.902777778 - 0.231678109 and 6.902777778 0.231678109. We can be at least 95 confident that interval 6.68 and 7.13 contains the average hourly income of a work-study student. Smal Sample Size (say less than 30) If the sample n is less than 30 or we must use the small sample procedure to develop a confidence interval for the mean of a population. The general formula for developing confidence intervals for the population mean based on small a sample is: In this formula is the mean of the sample. is the interval coefficient providing an area of in the upper tail of a t distribution with n-1 degrees of freedom which can be found from a t distribution table (for example the interval coefficient for a 90 confidence level is 1.833 if the sample is 10). S is the standard deviation of the sample and n is the sample size. Now you would like to see how Excel is used to develop a certain confidence interval of a population mean based on this small sample information. As you see, to evaluate this formula you need quotthe mean of the samplequot and the margin of error Excel will automatically calculate these quantities the way it did for large samples. Again, the only things you have to do are: add the margin of error to the mean of the sample, , find the upper limit of the interval and to subtract the margin of error from the mean to find the lower limit of the interval. To demonstrate how Excel finds these quantities we will use the data set, which contains the hourly incomes of 10 work-study students here, at the University of Baltimore. These numbers appear in cells A1 to A10 on an Excel work sheet. After entering the data we follow the descriptive statistic procedure to calculate the unknown quantities (exactly the way we found quantities for large sample). Here you are with the procedures in step-by-step form: Step 1. Enter data in cells A1 to A10 on the spreadsheet Step 2. From the menus select Tools Step 3. Click on Data Analysis then choose the Descriptive Statistics option. Click OK on the descriptive statistics dialog, click on Summary Statistic, click on the confidence interval level and type in 90 or in other problems whichever confidence interval you desire. In the Output Range box, enter B1 or whatever location you desire. Now click on OK . The screen shot will look like the following: Now, like the calculation of the confidence interval for the large sample, calculate the confidence interval of the population based on this small sample information. The confidence interval is: 6.8 0.414426102 or 6.39 7.21. We can be at least 90 confidant that the interval 6.39 and 7.21 contains the true mean of the population. Test of Hypothesis Concerning the Population Mean Again, we must distinguish two cases with respect to the size of your sample Large Sample Size (say, over 30): In this section you wish to know how Excel can be used to conduct a hypothesis test about a population mean. We will use the hourly incomes of different work-study students than those introduced earlier in the confidence interval section. Data are entered in cells A1 to A36. The objective is to test the following Null and Alternative hypothesis: The null hypothesis indicates that the average hourly income of a work-study student is equal to 7 per hour however, the alternative hypothesis indicates that the average hourly income is not equal to 7 per hour. I will repeat the steps taken in descriptive statistics and at the very end will show how to find the value of the test statistics in this case, z, using a cell formula. Step 1. Enter data in cells A1 to A36 (on the spreadsheet) Step 2. From the menus select Tools Step 3. Click on Data Analysis then choose the Descriptive Statistics option, click OK . On the descriptive statistics dialog, click on Summary Statistic. Select the Output Range box, enter B1 or whichever location you desire. Sekarang klik OK. (To calculate the value of the test statistics search for the mean of the sample then the standard error. In this output, these values are in cells C3 and C4.) Step 4. Select cell D1 and enter the cell formula (C3 - 7)C4. The screen shot should look like the following: The value in cell D1 is the value of the test statistics. Since this value falls in acceptance range of -1.96 to 1.96 (from the normal distribution table), we fail to reject the null hypothesis. Small Sample Size (say, less than 30): Using steps taken the large sample size case, Excel can be used to conduct a hypothesis for small-sample case. Lets use the hourly income of 10 work-study students at UB to conduct the following hypothesis. The null hypothesis indicates that average hourly income of a work-study student is equal to 7 per hour .The alternative hypothesis indicates that average hourly income is not equal to 7 per hour. I will repeat the steps taken in descriptive statistics and at the very end will show how to find the value of the test statistics in this case quottquot using a cell formula. Step 1. Enter data in cells A1 to A10 (on the spreadsheet) Step 2. From the menus select Tools Step 3. Click on Data Analysis then choose the Descriptive Statistics option. Klik OK On the descriptive statistics dialog, click on Summary Statistic. Select the Output Range boxes, enter B1 or whatever location you chose. Again, click on OK . (To calculate the value of the test statistics search for the mean of the sample then the standard error, in this output these values are in cells C3 and C4.) Step 4. Select cell D1 and enter the cell formula (C3 - 7)C4. The screen shot would look like the following: Since the value of test statistic t -0.66896 falls in acceptance range -2.262 to 2.262 (from t table, where 0.025 and the degrees of freedom is 9), we fail to reject the null hypothesis. Difference Between Mean of Two Populations In this section we will show how Excel is used to conduct a hypothesis test about the difference between two population means assuming that populations have equal variances. The data in this case are taken from various offices here at the University of Baltimore. I collected the hourly income data of 36 randomly selected work-study students and 36 student assistants. The hourly income range for work-study students was 6 - 8 while the hourly income range for student assistants was 6-9. The main objective in this hypothesis testing is to see whether there is a significant difference between the means of the two populations. The NULL and the ALTERNATIVE hypothesis is that the means are equal and the means are not equal, respectively. Referring to the spreadsheet, I chose A1 and A2 as label centers. The work-study students hourly income for a sample size 36 are shown in cells A2:A37 . and the student assistants hourly income for a sample size 36 is shown in cells B2:B37 Data for Work Study Student: 6, 6, 6, 6, 6, 6, 6, 6.5, 6.5, 6.5, 6.5, 6.5, 6.5, 7, 7, 7, 7, 7, 7, 7, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 8, 8, 8, 8, 8, 8, 8, 8, 8. Data for Student Assistant: 6, 6, 6, 6, 6, 6.5, 6.5, 6.5, 6.5, 6.5, 7, 7, 7, 7, 7, 7.5, 7.5, 7.5, 7.5, 7.5, 7.5, 8, 8, 8, 8, 8, 8, 8, 8.5, 8.5, 8.5, 8.5, 8.5, 9, 9, 9, 9. Use the Descriptive Statistics procedure to calculate the variances of the two samples. The Excel procedure for testing the difference between the two population means will require information on the variances of the two populations. Since the variances of the two populations are unknowns they should be replaced with sample variances. The descriptive for both samples show that the variance of first sample is s 1 2 0.55546218 . while the variance of the second sample s 2 2 0.969748 . To conduct the desired test hypothesis with Excel the following steps can be taken: Step 1. From the menus select Tools then click on the Data Analysis option. Step 2. When the Data Analysis dialog box appears: Choose z-Test: Two Sample for means then click OK Step 3. When the z-Test: Two Sample for means dialog box appears: Enter A1:A36 in the variable 1 range box (work-study students hourly income) Enter B1:B36 in the variable 2 range box (student assistants hourly income) Enter 0 in the Hypothesis Mean Difference box (if you desire to test a mean difference other than 0, enter that value) Enter the variance of the first sample in the Variable 1 Variance box Enter the variance of the second sample in the Variable 2 Variance box and select Labels Enter 0.05 or, whatever level of significance you desire, in the Alpha box Select a suitable Output Range for the results, I chose C19 . then click OK. The value of test statistic z-1.9845824 appears in our case in cell D24. The rejection rule for this test is z 1.96 from the normal distribution table. In the Excel output these values for a two-tail test are z 1.959961082. Since the value of the test statistic z-1.9845824 is less than -1.959961082 we reject the null hypothesis. We can also draw this conclusion by comparing the p-value for a two tail -test and the alpha value. Since p-value 0.047190813 is less than a0.05 we reject the null hypothesis. Overall we can say, based on the sample results, the two populations means are different. Small Samples: n 1 OR n 2 are less than 30 In this section we will show how Excel is used to conduct a hypothesis test about the difference between two population means. - Given that the populations have equal variances when two small independent samples are taken from both populations. Similar to the above case, the data in this case are taken from various offices here at the University of Baltimore. I collected hourly income data of 11 randomly selected work-study students and 11 randomly selected student assistants. The hourly income range for both groups was similar range, 6 - 8 and 6-9. The main objective in this hypothesis testing is similar too, to see whether there is a significant difference between the means of the two populations. The NULL and the ALTERNATIVE hypothesis are that the means are equal and they are not equal, respectively. Referring to the spreadsheet, we chose A1 and A2 as label centers. The work-study students hourly income for a sample size 11 are shown in cells A2:A12 . and the student assistants hourly income for a sample size 11 is shown in cells B2:B12 . Unlike previous case, you do not have to calculate the variances of the two samples, Excel will automatically calculate these quantities and use them in the calculation of the value of the test statistic. Similar to the previous case, but a bit different in step 2, to conduct the desired test hypothesis with Excel the following steps can be taken: Step 1. From the menus select Tools then click on the Data Analysis option. Step 2. When the Data Analysis dialog box appears: Choose t-Test: Two Sample Assuming Equal Variances then click OK Step 3 When the t-Test: Two Sample Assuming Equal Variances dialog box appears : Enter A1:A12 in the variable 1 range box (work-study student hourly income) Enter B1:B12 in the variable 2 range box (student assistant hourly income) Enter 0 in the Hypothesis Mean Difference box(if you desire to test a mean difference other than zero, enter that value) then select Labels Enter 0.05 or, whatever level of significance you desire, in the Alpha box Select a suitable Output Range for the results, I chose C1, then click OK. The value of the test statistic t-1.362229828 appears, in our case, in cell D10. The rejection rule for this test is t 2.086 from the t distribution table where the t value is based on a t distribution with n 1 -n 2 -2 degrees of freedom and where the area of the upper one tail is 0.025 ( that is equal to alpha2). In the Excel output the values for a two-tail test are t 2.085962478. Since the value of the test statistic t-1.362229828, is in an acceptance range of t 2.085962478, we fail to reject the null hypothesis. We can also draw this conclusion by comparing the p-value for a two-tail test and the alpha value. Since the p-value 0.188271278 is greater than a0.05 again . we fail to reject the null hypothesis. Overall we can say, based on sample results, the two populations means are equal. Enter data in an Excel work sheet starting with cell A2 and ending with cell C8. The following steps should be taken to find the proper output for interpretation. Step 1. From the menus select Tools and click on Data Analysis option. Step 2. When data analysis dialog appears, choose Anova single-factor option enter A2:C8 in the input range box. Select labels in first row. Step3. Select any cell as output(in here we selected A11). Click OK. The general form of Anova table looks like following: Source of Variation Suppose the test is done at level of significance a 0.05, we reject the null hypothesis. This means there is a significant difference between means of hourly incomes of student assistants in these departments. The Two-way ANOVA Without Replication In this section, the study involves six students who were offered different hourly wages in three different department services here at the University of Baltimore. The objective is to see whether the hourly incomes are the same. Therefore, we can consider the following: Treatment: Hourly payments in the three departments Blocks: Each student is a block since each student has worked in the three different departments The general form of Anova table would look like: Source of Variation Degrees of freedom To find the Excel output for the above data the following steps can be taken: Step 1. From the menus select Tools and click on Data Analysis option. Step2. When data analysis box appears: select Anova two-factor without replication then Enter A2: D8 in the input range. Select labels in first row. Step3. Select an output range (in here we selected A11) then OK. Source of Variation NOTE: FMSTMSE 0.9805560.497222 1.972067 F 3.33 from table (5 numerator DF and 10 denominator DF) Since 1.972067 Goodness-of-Fit Test for Discrete Random Variables The CHI-SQUARE distribution can be used in a hypothesis test involving a population variance. However, in this section we would like to test and see how close a sample results are to the expected results. Example: The Multinomial Random Variable In this example the objective is to see whether or not based on a randomly selected sample information the standards set for a population is met. There are so many practical examples that can be used in this situation. For example it is assumed the guidelines for hiring people with different ethnic background for the US government is set at 70(WHITE), 20(African American) and 10(others), respectively. A randomly selected sample of 1000 US employees shows the following results that is summarized in a table. EXPECTED NUMBER OF EMPLOYEES OBSERVED FROM SAMPLE As you see the observed sample numbers for groups two and three are lower than their expected values unlike group one which has a higher expected value. Is this a clear sign of discrimination with respect to ethnic background Well depends on how much lower the expected values are. The lower amount might not statistically be significant. To see whether these differences are significant we can use Excel and find the value of the CHI-SQUARE. If this value falls within the acceptance region we can assume that the guidelines are met otherwise they are not. Now lets enter these numbers into Excel spread- sheet. We used cells B7-B9 for the expected proportions, C7-C9 for the observed values and D7-D9 for the expected frequency. To calculate the expected frequency for a category, you can multiply the proportion of that category by the sample size (in here 1000). The formula for the first cell of the expected value column, D7 is 1000B7. To find other entries in the expected value column, use the copy and the paste menu as shown in the following picture. These are important values for the chi-square test. The observed range in this case is C7: C9 while the expected range is D7: D9. The null and the alternative hypothesis for this test are as follows: H A . The population proportions are not P W 0.70, P A 0.20 and P O 0.10 Now lets use Excel to calculate the p-value in a CHI-SQUARE test. Step 1. Select a cell in the work sheet, the location which you like the p value of the CHI-SQUARE to appear. We chose cell D12. Step 2. From the menus, select insert then click on the Function option, Paste Function dialog box appears. Step 3. Refer to function category box and choose statistical . from function name box select CHITEST and click on OK . Step 4. When the CHITEST dialog appears: Enter C7: C9 in the actual-range box then enter D7: D9 in the expected-range box, and finally click on OK . The p-value will appear in the selected cell, D12. As you see the p value is 0.002392 which is less than the value of the level of significance (in this case the level of significance, a 0.10). Hence the null hypothesis should be rejected. This means based on the sample information the guidelines are not met. Notice if you type CHITEST(C7:C9,D7:D9) in the formula bar the p-value will show up in the designated cell. NOTE: Excel can actually find the value of the CHI-SQUARE. To find this value first select an empty cell on the spread sheet then in the formula bar type CHIINV(D12,2). D12 designates the p-Value found previously and 2 is the degrees of freedom (number of rows minus one). The CHI-SQUARE value in this case is 12.07121. If we refer to the CHI-SQUARE table we will see that the cut off is 4.60517 since 12.071214.60517 we reject the null. The following screen shot shows you how to the CHI-SQUARE value. Test of Independence: Contingency Tables The CHI-SQUARE distribution is also used to test and see whether two variables are independent or not. For example based on sample data you might want to see whether smoking and gender are independent events for a certain population. The variables of interest in this case are smoking and the gender of an individual. Another example in this situation could involve the age range of an individual and his or her smoking habit. Similar to case one data may appear in a table but unlike the case one this table may contains several columns in addition to rows. The initial table contains the observed values. To find expected values for this table we set up another table similar to this one. To find the value of each cell in the new table we should multiply the sum of the cell column by the sum of the cell row and divide the results by the grand total. The grand total is the total number of observations in a study. Now based on the following table test whether or not the smoking habit and gender of the population that the following sample taken from are independent. On the other hand is that true that males in this population smoke more than females You could use formula bar to calculate the expected values for the expected range. For example to find the expected value for the cell C5 which is replaced in c11 you could click on the formula bar and enter C6D5D6 then enter in cell C11. Step 1. Observed Range b4:c5 Smoking and gender So the observed range is b4:c5 and the expected range is b10:c11. Step 3. Click on fx (paste function) Step 4. When Paste Function dialog box appears, click on Statistical in function category and CHITEST in the function name then click OK. When the CHITEST box appears, enter b4:c5 for the actual range, then b10:c11 for the expected range. Step 5. Click on OK (the p-value appears). 0.477395 Conclusion: Since p-value is greater than the level of significance (0.05), fails to reject the null. This means smoking and gender are independent events. Based on sample information one can not assure females smoke more than males or the other way around. Step 6. To find the chi-square value, use CHINV function, when Chinv box appears enter 0.477395 for probability part, then 1 for the degrees of freedom. Degrees of freedom(number of columns-1)X(number of rows-1) Test Hypothesis Concerning the Variance of Two Populations In this section we would like to examine whether or not the variances of two populations are equal. Whenever independent simple random samples of equal or different sizes such as n 1 and n 2 are taken from two normal distributions with equal variances, the sampling distribution of s 1 2 s 2 2 has F distribution with n 1 - 1 degrees of freedom for the numerator and n 2 - 1 degrees of freedom for the denominator. In the ratio s 1 2 s 2 2 the numerator s 1 2 and the denominator s 2 2 are variances of the first and the second sample, respectively. The following figure shows the graph of an F distribution with 10 degrees of freedom for both the numerator and the denominator. Unlike the normal distribution as you see the F distribution is not symmetric. The shape of an F distribution is positively skewed and depends on the degrees of freedom for the numerator and the denominator. The value of F is always positive. Now let see whether or not the variances of hourly income of student-assistant and work-study students based on samples taken from populations previously are equal. Assume that the hypothesis test in this case is conducted at a 0.10. The null and the alternative are: Rejection Rule: Reject the null hypothesis if Flt F 0.095 or Fgt F 0.05 where F, the value of the test statistic is equal to s 1 2 s 2 2. with 10 degrees of freedom for both the numerator and the denominator. We can find the value of F .05 from the F distribution table. If s 1 2 s 2 2. we do not need to know the value of F 0.095 otherwise, F 0.95 1 F 0.05 for equal sample sizes. A survey of eleven student-assistant and eleven work-study students shows the following descriptive statistics. Our objective is to find the value of s 1 2 s 2 2. where s 1 2 is the value of the variance of student assistant sample and s 2 2 is the value of the variance of the work study students sample. As you see these values are in cells F8 and D8 of the descriptive statistic output. To calculate the value of s 1 2 s 2 2. select a cell such as A16 and enter cell formula F8D8 and enter. This is the value of F in our problem. Since this value, F1.984615385, falls in acceptance area we fail to reject the null hypothesis. Hence, the sample results do support the conclusion that student assistants hourly income variance is equal to the work study students hourly income variance. The following screen shoot shows how to find the F value. We can follow the same format for one tail test(s). Linear Correlation and Regression Analysis In this section the objective is to see whether there is a correlation between two variables and to find a model that predicts one variable in terms of the other variable. There are so many examples that we could mention but we will mention the popular ones in the world of business. Usually independent variable is presented by the letter x and the dependent variable is presented by the letter y. A business man would like to see whether there is a relationship between the number of cases of sold and the temperature in a hot summer day based on information taken from the past. He also would like to estimate the number cases of soda which will be sold in a particular hot summer day in a ball game. He clearly recorded temperatures and number of cases of soda sold on those particular days. The following table shows the recorded data from June 1 through June 13. The weatherman predicts a 94F degree temperature for June 14. The businessman would like to meet all demands for the cases of sodas ordered by customers on June 14. Now lets use Excel to find the linear correlation coefficient and the regression line equation. The linear correlation coefficient is a quantity between -1 and 1. This quantity is denoted by R . The closer R to 1 the stronger positive (direct) correlation and similarly the closer R to -1 the stronger negative (inverse) correlation exists between the two variables. The general form of the regression line is y mx b. In this formula, m is the slope of the line and b is the y-intercept. You can find these quantities from the Excel output. In this situation the variable y (the dependent variable) is the number of cases of soda and the x (independent variable) is the temperature. To find the Excel output the following steps can be taken: Step 1. From the menus choose Tools and click on Data Analysis. Step 2. When Data Analysis dialog box appears, click on correlation. Step 3. When correlation dialog box appears, enter B1:C14 in the input range box. Click on Labels in first row and enter a16 in the output range box. Click on OK. As you see the correlation between the number of cases of soda demanded and the temperature is a very strong positive correlation. This means as the temperature increases the demand for cases of soda is also increasing. The linear correlation coefficient is 0.966598577 which is very close to 1. Now lets follow same steps but a bit different to find the regression equation. Step 1. From the menus choose Tools and click on Data Analysis Step 2 . When Data Analysis dialog box appears, click on regression . Step 3. When Regression dialog box appears, enter b1:b14 in the y-range box and c1:c14 in the x-range box. Click on labels . Step 4. Enter a19 in the output range box . Note: The regression equation in general should look like Ym X b. In this equation m is the slope of the regression line and b is its y-intercept. Adjusted R Square The relationship between the number of cans of soda and the temperature is: Y 0.879202711 X 9.17800767 The number of cans of soda 0.879202711(Temperature) 9.17800767. Referring to this expression we can approximately predict the number of cases of soda needed on June 14. The weather forecast for this is 94 degrees, hence the number of cans of soda needed is equal to The number of cases of soda0.879202711(94) 9.17800767 91.82 or about 92 cases. Moving Average and Exponential Smoothing Moving Average Models: Use the Add Trendline option to analyze a moving average forecasting model in Excel. You must first create a graph of the time series you want to analyze. Select the range that contains your data and make a scatter plot of the data. Once the chart is created, follow these steps: Click on the chart to select it, and click on any point on the line to select the data series. When you click on the chart to select it, a new option, Chart, s added to the menu bar. From the Chart menu, select Add Trendline. The following is the moving average of order 4 for weekly sales: Exponential Smoothing Models: The simplest way to analyze a timer series using an Exponential Smoothing model in Excel is to use the data analysis tool. This tool works almost exactly like the one for Moving Average, except that you will need to input the value of a instead of the number of periods, k. Once you have entered the data range and the damping factor, 1- a. and indicated what output you want and a location, the analysis is the same as the one for the Moving Average model. Applications and Numerical Examples Descriptive Statistics: Suppose you have the following, n 10, data: 1.2, 1.5, 2.6, 3.8, 2.4, 1.9, 3.5, 2.5, 2.4, 3.0 Type your n data points into the cells A1 through An. Click on the Tools menu. (At the bottom of the Tools menu will be a submenu Data Analysis. , if the Analysis Tool Pack has been properly installed.) Clicking on Data Analysis. will lead to a menu from which Descriptive Statistics is to be selected. Select Descriptive Statistics by pointing at it and clicking twice, or by highlighting it and clicking on the Okay button. Within the Descriptive Statistics submenu, a. for the input range enter A1:Dn, assuming you typed the data into cells A1 to An.b. click on the output range button and enter the output range C1:C16. C. click on the Summary Statistics box d. finally, click on Okay. The Central Tendency: The data can be sorted in ascending order: 1.2, 1.5, 1.9, 2.4, 2.4, 2.5, 2.6, 3.0, 3.5, 3.8 The mean, median and mode are computed as follows: (1.2 1.5 2.6 3.8 2.4 1.9 3.5 2.5 2.4 3.0) 10 2.48 The mode is 2.4, since it is the only value that occurs twice. The midrange is (1.2 3.8) 2 2.5. Note that the mean, median and mode of this set of data are very close to each other. This suggests that the data is very symmetrically distributed. Variance: The variance of a set of data is the average of the cumulative measure of the squares of the difference of all the data values from the mean. The sample variance-based estimation for the population variance are computed differently. The sample variance is simply the arithmetic mean of the squares of the difference between each data value in the sample and the mean of the sample. On the other hand, the formula for an estimate for the variance in the population is similar to the formula for the sample variance, except that the denominator in the fraction is (n-1) instead of n. However, you should not worry about this difference if the sample size is large, say over 30. Compute an estimate for the variance of the population . given the following sorted data: 1.2, 1.5, 1.9, 2.4, 2.4, 2.5, 2.6, 3.0, 3.5, 3.8 mean 2.48 as computed earlier. An estimate for the population variance is: s 2 1 (10-1) (1.2 - 2.48) 2 (1.5 - 2.48) 2 (1.9 - 2.48) 2 (2.4 -2.48) 2 (2.4 - 2.48) 2 (2.5 - 2.48) 2 (2.6 - 2.48) 2 (3.0 - 2.48) 2 (3.5 -2.48) 2 (3.8 - 2.48) 2 (1 9) (1.6384 0.9604 0.3364 0.0064 0.0064 0.0004 0.0144 0.2704 1.0404 1.7424) 0.6684 Therefore, the standard deviation is s ( 0.6684 ) 12 0.8176 Probability and Expected Values: Newsweek reported that average take for bank robberies was 3,244 but 85 percent of the robbers were caught. Assuming 60 percent of those caught lose their entire take and 40 percent lose half, graph the probability mass function using EXCEL. Calculate the expected take from a bank robbery. Does it pay to be a bank robber To construct the probability function for bank robberies, first define the random variable x, bank robbery take. If the robber is not caught, x 3,244. If the robber is caught and manages to keep half, x 1,622. If the robber is caught and loses it all, then x 0. The associated probabilities for these x values are 0.15 (1 - 0.85), 0.34 (0.85)(0.4), and 0.51 (0.85)(0.6). After entering the x values in cells A1, A2 and A3 and after entering the associated probabilities in B1, B2, and B3, the following steps lead to the probability mass function: Click on ChartWizard. The ChartWizard Step 1 of 4 screen will appear. Highlight Column at ChartWizard Step 1 of 4 and click Next. At ChartWizard Step 2 of 4 Chart Source Data, enter B1:B3 for Data range, and click column button for Series in. A graph will appear. Click on series toward the top of the screen to get a new page. At the bottom of the Series page, is a rectangle for Category (X) axis labels: Click on this rectangle and then highlight A1:A3. At Step 3 of 4 move on by clicking on Next, and at Step 4 of 4, click on Finish. The expected value of a robbery is 1,038.08. E(X) (0)(0.51)(1622)(0.34) (3244)(0.15) 0 551.48 486.60 1038.08 The expected return on a bank robbery is positive. On average, bank robbers get 1,038.08 per heist. If criminals make their decisions strictly on this expected value, then it pays to rob banks. A decision rule based only on an expected value, however, ignores the risks or variability in the returns. In addition, our expected value calculations do not include the cost of jail time, which could be viewed by criminals as substantial. Discrete Continuous Random Variables: Binomial Distribution Application: A multiple choice test has four unrelated questions. Each question has five possible choices but only one is correct. Thus, a person who guesses randomly has a probability of 0.2 of guessing correctly. Draw a tree diagram showing the different ways in which a test taker could get 0, 1, 2, 3 and 4 correct answers. Sketch the probability mass function for this test. What is the probability a person who guesses will get two or more correct Solution: Letting Y stand for a correct answer and N a wrong answer, where the probability of Y is 0.2 and the probability of N is 0.8 for each of the four questions, the probability tree diagram is shown in the textbook on page 182. This probability tree diagram shows the branches that must be followed to show the calculations captured in the binomial mass function for n 4 and 0.2. For example, the tree diagram shows the six different branch systems that yield two correct and two wrong answers (which corresponds to 4(22) 6. The binomial mass function shows the probability of two correct answers as P(x 2 n 4, p 0.2) 6(.2)2(.8)2 6(0.0256) 0.1536 P(2) Which is obtained from excel by using the BINOMDIST Command, where the first entry is x, the second is n, and the third is mass (0) or cumulative (1) that is, entering BINOMDIST(2,4,0.2,0) IN ANY EXCEL CELL YIELDS 0.1536 AND BINOMDIST(3,4,0.2,0) YIELDS P(x3n4, p 0.2) 0.0256 BINOMDIST(4,4,0.2,0) YIELDS P(x4n4, p 0.2) 0.0016 1-BINOMDIST(1,4,0.2,1) YIELDS P(x 179 2 n 4, p 0.2) 0.1808 Normal Example: If the time required to complete an examination by those with a certain learning disability is believed to be distributed normally, with mean of 65 minutes and a standard deviation of 15 minutes, then when can the exam be terminated so that 99 percent of those with the disability can finish Solution: Because t he average and standard deviation are known, what needs to be established is the amount of time, above the mean time, such that 99 percent of the distribution is lower. This is a distance that is measured in standard deviations as given by the Z value corresponding to the 0.99 probability found in the body of Appendix B, Table 5,as shown in the textbook OR the commands entered into any cell of Excel to find this Z value is NORMINV(0.99,0,1) for 2.326342. The closest cumulative probability that can be found is 0.9901, in the row labeled 2.3 and column headed by .03, Z 2.33, which is only an approximation for the more exact 2.326342 found in Excel. Using this more exact value the calculation with mean m and standard deviation s in the following formula would be Z ( X - m ) s That is, Z ( x - 65)15 Thus, x 65 15(2.32634) 99.9 minutes. Alternatively, instead of standardizing with the Z distribution using Excel we can simply work directly with the normal distribution with a mean of 65 and standard deviation of 15 and enter NORMINV(0.99,65,15). In general to obtain the x value for which alpha percent of a normal random variables values are lower, the following NORMINV command may be used, where the first entry is a. the second is m. and the third is s. Another Example: In the early 1980s, the Toro Company of Minneapolis, Minnesota, advertised that it would refund the purchase price of a snow blower if the following winters snowfall was less than 21 percent of the local average. If the average snowfall is 45.25 inches, with a standard deviation of 12.2 inches, what is the likelihood that Toro will have to make refunds Solution: Within limits, snowfall is a continuous random variable that can be expected to vary symmetrically around its mean, with values closer to the mean occurring most often. Thus, it seems reasonable to assume that snowfall (x) is approximately normally distributed with a mean of 45.25 inches and standard deviation of 12.2 inches. Nine and one half inches is 21 percent of the mean snowfall of 45.25 inches and, with a standard deviation of 12.2 inches, the number of standard deviations between 45.25 inches and 9.5 inches is Z: Z ( x - m ) s (9.50 - 45.25)12.2 -2.93 Using Appendix B, Table 5, the textbook demonstrates the determination of P(x 163 9.50) P(z 163 -2.93) 0.17, the probability of snowfall less than 9.5 inches. Using Excel, this normal probability is obtained with the NORMDIST command, where the first entry is x, the second is mean m. the third is standard deviation s, and the fourth is CUMULATIVE (1). Entering NORMDIST(9.5,45.25,12.2,1), Gives P( x 163 9.50) 0.001693. Sampling Distribution and the Central Limit Theorem : A bakery sells an average of 24 loaves of bread per day. Sales (x) are normally distributed with a standard deviation of 4. If a random sample of size n 1 (day) is selected, what is the probability this x value will exceed 28 If a random sample of size n 4 (days) is selected, what is theprobability that xbar 179 28 Why does the answer in part 1 differ from that in part 2 1. The sampling distribution of the sample mean xbar is normal with a mean of 24 and a standard error of the mean of 4. Thus, using Excel, 0.15866 1-NORMDIST(28,24,4,1). 2. The sampling distribution of the sample mean xbar is normal with a mean of 24 and a standard error of the mean of 2 using Excel, 0.02275 1-NORMDIST(28,24,2,1). Regression Analysis: The highway deaths per 100 million vehicle miles and highway speed limits for 10 countries, are given below: (Death, Speed) (3.0, 55), (3.3, 55), (3.4, 55), (3.5, 70), (4.1, 55), (4.3, 60), (4.7, 55), (4.9, 60), (5.1, 60), and (6.1, 75). From this we can see that five countries with the same speed limit have very different positions on the safety list. For example, Britain. with a speed limit of 70 is demonstrably safer than Japan, at 55. Can we argue that, speed has little to do with safety. Use regression analysis to answer this question. Solution: Enter the ten paired y and x data into cells A2 to A11 and B2 to B11, with the death rate label in A1 and speed limits label in B1, the following steps produce the regression output. Choose Regression from Data Analysis in the Tools menu. The Regression dialog box will will appear. Note: Use the mouse to move between the boxes and buttons. Click on the desired box or button. The large rectangular boxes require a range from the worksheet. A range may be typed in or selected by highlighting the cells with the mouse after clicking on the box. If the dialog box blocks the data, it can be moved on the screen by clicking on the title bar and dragging. For the Input Y Range, enter A1 to A11, and for the Input X Range enter B1 to B11. Because the Y and X ranges include the Death and Speed labels in A1 and B1, select the Labels box with a click. Click the Output Range button and type reference cell, which in this demonstration is A13. To get the predicted values of Y (Death rates) and residuals select the Residuals box with a click. Your screen display should show a Table, clicking OK will give the SUMMARY OUTPUT, ANOVA AND RESIDUAL OUTPUT The first section of the EXCEL printout gives SUMMARY OUTPUT. The Multiple R is the square root of the R Square the computation and interpretation of which we have already discussed. The Standard Error of estimate (which will be discussed in the next chapter) is s 0.86423, which is the square root of Residual SS 5.97511 divided by its degrees of freedom, df 8, as given in the ANOVA section. We will also discuss the adjusted R-square of 0.21325 in the following chapters. Under the ANOVA section are the estimated regression coefficients and related statistics that will be discussed in detail in the next chapter. For now it is sufficient to recognize that the calculated coefficient values for the slope and y intercept are provided (b 0.07556 and a -0.29333). Next to these coefficient estimates is information on the variability in the distribution of the least-squares estimators from which these specific estimates were drawn: the column titled Std. Error contains the standard deviations (standard errors) of the intercept and slope distributions the t-ratio and p columns give the calculated values of the t statistics and associated p-values. As shown in Chapter 13, the t statistic of 1.85458 and p-value of 0.10077, for example, indicates that the sample slope (0.07556) is sufficiently different from zero, at even the 0.10 two-tail Type I error level, to conclude that there is a significant relationship between deaths and speed limits in the population. This conclusion is contrary to assertion that speed has little to do with safety. SUMMARY OUTPUT: Multiple R 0.54833, R Square 0.30067, Adjusted R Square 0.21325, Standard Error 0.86423, Observations 10 ANOVA df SS MS F P-value Regression 1 2.56889 2.56889 3.43945 0.10077 Residual 8 5.97511 0.74689 Total 9 8.54400 Coeffs. Estimate Std. Error T Stat P-value Lower 95 Upper 95 Intercept -0.29333 2.45963 -0.11926 0.90801 -5.96526 5.37860 Speed 0.07556 0.04074 1.85458 0.10077 -0.01839 0.16950 Predicted Residuals 3.86222 -0.86222 3.86222 -0.56222 3.86222 -0.46222 4.99556 -1.49556 3.86222 0.23778 4.24000 0.06000 3.86222 0.83778 4.24000 0.66000 4.24000 0.86000 5.37333 0.72667 Microsoft Excel Add-Ins Forecasting with regression requires the Excel add-in called Analysis ToolPak , and linear programming requires the Excel add-in called Solver . How you check to see if these are activated on your computer, and how to activate them if they are not active, varies with Excel version. Here are instructions for the most common versions. If Excel will not let you activate Data Analysis and Solver, you must use a different computer. Excel 20022003: Start Excel, then click Tools and look for Data Analysis and for Solver. If both are there, press Esc (escape) and continue with the respective assignment. Otherwise click Tools, Add-Ins, and check the boxes for Analysis ToolPak and for Solver, then click OK. Click Tools again, and both tools should be there. Excel 2007: Start Excel 2007 and click the Data tab at the top. Look to see if Data Analysis and Solver show in the Analysis section at the far right. If both are there, continue with the respective assignment. Otherwise, do the following steps exactly as indicated: -click the 8220Office Button8221 at top left -click the Excel Options button near the bottom of the resulting window -click the Add-ins button on the left of the next screen -near the bottom at Manage Excel Add-ins, click Go -check the boxes for Analysis ToolPak and Solver Add-in if they are not already checked, then click OK -click the Data tab as above and verify that the add-ins show. Excel 2010: Start Excel 2010 and click the Data tab at the top. Look to see if Data Analysis and Solver show in the Analysis section at the far right. If both are there, continue with the respective assignment. Otherwise, do the following steps exactly as indicated: -click the File tab at top left -click the Options button near the bottom of the left side -click the Add-ins button near the bottom left of the next screen -near the bottom at Manage Excel Add-ins, click Go -check the boxes for Analysis ToolPak and Solver Add-in if they are not already checked, then click OK -click the Data tab as above and verify that the add-ins show. Solving Linear Programs by Excel Some of these examples can be modified for other types problems Computer-assisted Learning: E-Labs and Computational Tools My teaching style deprecates the plug the numbers into the software and let the magic box work it out approach. Personal computers, spreadsheets, e.g. Excel. professional statistical packages (e.g. such as SPSS), and other information technologies are now ubiquitous in statistical data analysis. Without using these tools, one cannot perform any realistic statistical data analysis on large data sets. The appearance of other computer software, JavaScript Applets. Statistical Demonstrations Applets. and Online Computation are the most important events in the process of teaching and learning concepts in model-based statistical decision making courses. These tools allow you to construct numerical examples to understand the concepts, and to find their significance for yourself. Use any or online interactive tools available on the WWW to perform statistical experiments (with the same purpose, as you used to do experiments in physics labs to learn physics) to understand statistical concepts such as Central Limit Theorem are entertaining and educating. Computer-assisted learning is similar to the experiential model of learning. The adherents of experiential learning are fairly adamant about how we learn. Learning seldom takes place by rote. Learning occurs because we immerse ourselves in a situation in which we are forced to perform and think. You get feedback from the computer output and then adjust your thinking-process if needed. A SPSS-Example . SPSS-Examples . SPSS-More Examples . (Statistical Package for the Social Sciences) is a data management and analysis product. It can perform a variety of data analysis and presentation functions, including statistical analyses and graphical presentation of data. SAS (Statistical Analysis System) is a system of software packages some of its basic functions and uses are: database management inputting, cleaning and manipulating data, statistical analysis, calculating simple statistics such as means, variances, correlations running standard routines such as regressions. Available at: SPSSSAS Packages on Citrix (Installing and Accessing ) Use your email ID and Password: Technical Difficulties OTS Call Center (401) 837-6262 Excel Examples. Excel More Examples It is Excellent for Descriptive Statistics, and getting acceptance is improving, as computational tool for Inferential Statistics. The Value of Performing Experiment: If the learning environment is focused on background information, knowledge of terms and new concepts, the learner is likely to learn that basic information successfully. However, this basic knowledge may not be sufficient to enable the learner to carry out successfully the on-the-job tasks that require more than basic knowledge. Thus, the probability of making real errors in the business environment is high. On the other hand, if the learning environment allows the learner to experience and learn from failures within a variety of situations similar to what they would experience in the real world of their job, the probability of having similar failures in their business environment is low. This is the realm of simulations-a safe place to fail. The appearance of statistical software is one of the most important events in the process of decision making under uncertainty. Statistical software systems are used to construct examples, to understand the existing concepts, and to find new statistical properties. On the other hand, new developments in the process of decision making under uncertainty often motivate developments of new approaches and revision of the existing software systems. Statistical software systems rely on a cooperation of statisticians, and software developers. Beside the professional statistical software Online statistical computation . and the use of a scientific calculator is required for the course. A Scientific Calculator is the one, which has capability to give you, say, the result of square root of 5. Any calculator that goes beyond the 4 operations is fine for this course. These calculators allow you to perform simple calculations you need in this course, for example, enabling you to take square root, to raise e to the power of say, 0.36. dan seterusnya. These types of calculators are called general Scientific Calculators. There are also more specific and advanced calculators for mathematical computations in other areas such as Finance, Accounting, and even Statistics. The last one, for example, computes mean, variance, skewness, and kurtosis of a sample by simply entering all data one-by-one and then pressing any of the mean, variance, skewness, and kurtosis keys. Without a computer one cannot perform any realistic statistical data analysis. Students who are signing up for the course are expected to know the basics of Excel. As a starting point, you need visiting the Excel Web site created for this course. If you are challenged by or unfamiliar with Excel, you may seek tutorial help from the Academic Resource Center at 410-837-5385, E-mail. What and How to Hand-in My Computer Assignment For the computer assignment I do recommend in checking your hand computation homework, and checking some of the numerical examples from your textbook. As part of your homework assignment you don not have to hand in the printout of the computer assisted learning, however, you must include within your handing homework a paragraph entitled Computer Implementation describing your (positive or negative) experience. Interesting and Useful Sites The Copyright Statement: The fair use, according to the 1996 Fair Use Guidelines for Educational Multimedia. of materials presented on this Web site is permitted for non-commercial and classroom purposes only. This site may be mirrored intact (including these notices), on any server with public access. All files are available at home.ubalt.eduntsbarshBusiness-stat for mirroring. Kindly e-mail me your comments, suggestions, and concerns. Terima kasih. EOF: CopyRights 1994-2015.Smoothing and filtering are two of the most commonly used time series techniques for removing noise from the underlying data to help reveal the important features and components (e.g. trend, seasonality, etc.). However, we can also use smoothing to fill in missing values andor conduct a forecast. In this issue, we will discuss five (5) different smoothing methods: weighted moving average (WMA i ), simple exponential smoothing, double exponential smoothing, linear exponential smoothing, and triple exponential smoothing. Why should we care Smoothing is very often used (and abused) in the industry to make a quick visual examination of the data properties (e.g. trend, seasonality, etc.), fit in missing values, and conduct a quick out-of-sample forecast. Why do we have so many smoothing functions As we will see in this paper, each function works for a different assumption about the underlying data. For instance, simple exponential smoothing assumes the data has a stable mean (or at least a slow moving mean), so simple exponential smoothing will do poorly in forecasting data exhibiting seasonality or a trend. In this paper, we will go over each smoothing function, highlight its assumptions and parameters, and demonstrate its application through examples. Weighted Moving Average (WMA) A moving average is commonly used with time series data to smooth out short-term fluctuations and highlight longer-term trends or cycles. A weighted moving average has multiplying factors to give different weights to data at different positions in the sample window. The weighted moving average has a fixed window (i.e. N) and the factors are typically chosen to given more weight to recent observations. The window size (N) determines the number of points averaged at each time, so a larger windows size is less responsive to new changes in the original time series and a small window size can cause the smoothed output to be noisy. For out of sample forecasting purposes: Example 1: Lets consider monthly sales for Company X, using a 4-month (equal-weighted) moving average. Note that the moving average is always lagging behind the data and the out-of-sample forecast converges to a constant value. Lets try to use a weighting scheme (see below) which gives more emphasis to the latest observation. We plotted the equal-weighted moving average and WMA on the same graph. The WMA seems more responsive to recent changes and the out-of sample forecast converges to the same value as the moving average. Example 2: Lets examine the WMA in the presence of trend and seasonality. For this example, well use the international passenger airline data. The moving average window is 12 months. The MA and the WMA keep pace with the trend, but the out-of-sample forecast flattens. Furthermore, although the WMA exhibits some seasonality, it is always lagging behind the original data. (Browns) Simple Exponential Smoothing Simple exponential smoothing is similar to the WMA with the exception that the window size if infinite and the weighting factors decrease exponentially. As we have seen in the WMA, the simple exponential is suited for time series with a stable mean, or at least a very slow moving mean. Example 1: Lets use the monthly sales data (as we did in the WMA example). In the example above, we chose the smoothing factor to be 0.8, which begs the question: What is the best value for the smoothing factor Estimating the best value from the data Using the TSSUB function (to compute the error), SUMSQ, and Excel data tables, we computed the sum of the squared errors (SSE) and plotted the results: The SSE reaches its minimum value around 0.8, so we picked this value for our smoothing. (Holt-Winters) Double Exponential Smoothing Simple exponential smoothing does not do well in the presence of a trend, so several method devised under the double exponential umbrella are proposed to handle this type of data. NumXL supports Holt-Winters double exponential smoothing, which take the following formulation: Example 1: Lets examine the international passengers airline data We chose an Alpha value of 0.9 and a Beta of 0.1. Please note that although double smoothing traces the original data well, the out-of-sample forecast is inferior to the simple moving average. How do we find the best smoothing factors We take a similar approach to our simple exponential smoothing example, but modified for two variables. We compute the sum of the squared errors construct a two-variable data table, and pick the alpha and beta values that minimize the overall SSE. (Browns) Linear Exponential Smoothing This is another method of double exponential smoothing function, but it has one smoothing factor: Browns double exponential smoothing takes one parameter less than Holt-Winters function, but it may not offer as good a fit as that function. Example 1: Lets use the same example in Holt-Winters double exponential and compare the optimal sum of the squared error. The Browns double exponential does not fit the sample data as well as the Holt-Winters method, but the out-of sample (in this particular case) is better. How do we find the best smoothing factor ( ) We use the same method to select the alpha value that minimizes the sum of the squared error. For the example sample data, the alpha is found to be 0.8. (Winters) Triple Exponential Smoothing The triple exponential smoothing takes into account seasonal changes as well as trends. This method requires 4 parameters: The formulation for triple exponential smoothing is more involved than any of the earlier ones. Please, check our online reference manual for the exact formulation. Using the international passengers airline data, we can apply winters triple exponential smoothing, find optimal parameters, and conduct an out-of sample forecast. Obviously, the Winters triple exponential smoothing is best applied for this data sample, as it tracks the values well and the out-of sample forecast exhibits seasonality (L12). How do we find the best smoothing factor ( ) Again, we need to pick the values that minimize the overall sum of the squared errors (SSE), but the data tables can be used for more than two variables, so we resort to the Excel solver: (1) Setup the minimization problem, with the SSE as the utility function (2) The constraints for this problem Conclusion support Files
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