{"id":53,"date":"2025-08-14T00:57:15","date_gmt":"2025-08-14T00:57:15","guid":{"rendered":"https:\/\/openbooks.spmvv.ac.in\/businessopsanalytics\/chapter\/introduction-to-statistical-analysis-for-operations\/"},"modified":"2026-07-10T23:59:30","modified_gmt":"2026-07-10T23:59:30","slug":"introduction-to-statistical-analysis-for-operations","status":"publish","type":"chapter","link":"https:\/\/openbooks.spmvv.ac.in\/businessopsanalytics\/chapter\/introduction-to-statistical-analysis-for-operations\/","title":{"raw":"Introduction to Statistical Analysis for Operations","rendered":"Introduction to Statistical Analysis for Operations"},"content":{"raw":"<a href=\"http:\/\/pressbooks.usnh.edu\/businessopsanalytics\/files\/2025\/08\/Chapter-9-Header.png\"><img src=\"https:\/\/openbooks.spmvv.ac.in\/app\/uploads\/sites\/3\/2025\/08\/Chapter-9-Header.png\"><\/a>\n<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Objectives<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<p data-start=\"36\" data-end=\"89\">By the end of this chapter, students will be able to:<\/p>\n\n<ol data-start=\"91\" data-end=\"624\" data-is-last-node=\"\" data-is-only-node=\"\">\n \t<li data-start=\"91\" data-end=\"181\">\n<p data-start=\"94\" data-end=\"181\">Summarize operational data (mean, median, standard deviation) in Excel and R.<\/p>\n<\/li>\n \t<li data-start=\"182\" data-end=\"274\">\n<p data-start=\"185\" data-end=\"274\">Choose and apply Normal, Binomial, and Poisson models; compute probabilities\/quantiles.<\/p>\n<\/li>\n \t<li data-start=\"275\" data-end=\"370\">\n<p data-start=\"278\" data-end=\"370\">Run a two-sample t-test; report the difference in means, confidence interval, and p-value.<\/p>\n<\/li>\n \t<li data-start=\"371\" data-end=\"466\">\n<p data-start=\"374\" data-end=\"466\">Analyze association with correlation and simple linear regression; interpret slope and R\u00b2.<\/p>\n<\/li>\n \t<li data-start=\"467\" data-end=\"563\">\n<p data-start=\"470\" data-end=\"563\">Perform quick checks for outliers, variance, and independence; note limits of the analysis.<\/p>\n<\/li>\n \t<li data-start=\"564\" data-end=\"624\" data-is-last-node=\"\">\n<p data-start=\"567\" data-end=\"624\" data-is-last-node=\"\">Convert results into a concise operations recommendation.<\/p>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<h2 data-start=\"279\" data-end=\"330\">9.1 Basic Statistical Terms (Operations Context)<\/h2>\n<p data-start=\"332\" data-end=\"794\">In earlier chapters you organized data, improved quality, and visualized performance. Here we formalize those summaries with a few core statistics you already know from your prerequisite course, focusing on how to apply and interpret them in operations. We\u2019ll compute each measure in Excel and mirror it in R, using two <em data-start=\"664\" data-end=\"686\">Pemi Coffee Roasters<\/em> datasets: Dataset A (fulfillment times, days) and Dataset B (items per order and fulfillment time).<\/p>\n\n<h3 data-start=\"796\" data-end=\"815\">Mean and Median<\/h3>\n<p data-start=\"816\" data-end=\"939\">The mean summarizes average performance; the median is the middle value and is less sensitive to a few slow orders.<\/p>\n\n<ul data-start=\"941\" data-end=\"1214\">\n \t<li data-start=\"941\" data-end=\"1038\">\n<p data-start=\"943\" data-end=\"1038\"><strong data-start=\"943\" data-end=\"952\">Excel<\/strong> (Dataset A in <code data-start=\"967\" data-end=\"975\">B2:B25<\/code>)<br data-start=\"976\" data-end=\"979\"><span style=\"font-size: 12pt\"><code data-start=\"981\" data-end=\"999\">=AVERAGE(B2:B25)<\/code> \u2192 <code data-start=\"1002\" data-end=\"1009\">5.875<\/code><\/span><br data-start=\"1009\" data-end=\"1012\"><span style=\"font-size: 12pt\"><code data-start=\"1014\" data-end=\"1031\">=MEDIAN(B2:B25)<\/code> \u2192 <code data-start=\"1035\" data-end=\"1038\">5<\/code><\/span><\/p>\n<\/li>\n \t<li data-start=\"1039\" data-end=\"1214\">\n<p data-start=\"1041\" data-end=\"1048\"><strong data-start=\"1041\" data-end=\"1046\">R<\/strong><\/p>\n\n<div class=\"contain-inline-size rounded-2xl relative bg-token-sidebar-surface-primary\">\n<div class=\"sticky top-9\"><\/div>\n<div class=\"overflow-y-auto p-4\" dir=\"ltr\"><span style=\"font-size: 12pt\"><code class=\"whitespace-pre! language-r\">dfA <span class=\"hljs-operator\">&lt;-<\/span> read.csv<span class=\"hljs-punctuation\">(<\/span><span class=\"hljs-string\">\"Chapter9_Dataset_A_FulfillmentTimes.csv\"<\/span><span class=\"hljs-punctuation\">)<\/span>\n<\/code><\/span><\/div>\n<div class=\"overflow-y-auto p-4\" dir=\"ltr\"><span style=\"font-size: 12pt\"><code class=\"whitespace-pre! language-r\">mean<span class=\"hljs-punctuation\">(<\/span>dfA<span class=\"hljs-operator\">$<\/span>FulfillmentTimeDays<span class=\"hljs-punctuation\">)<\/span>     <span class=\"hljs-comment\"># [1] 5.875<\/span>\n<\/code><\/span><\/div>\n<div class=\"overflow-y-auto p-4\" dir=\"ltr\"><span style=\"font-size: 12pt\"><code class=\"whitespace-pre! language-r\">median<span class=\"hljs-punctuation\">(<\/span>dfA<span class=\"hljs-operator\">$<\/span>FulfillmentTimeDays<span class=\"hljs-punctuation\">)<\/span>   <span class=\"hljs-comment\"># [1] 5<\/span>\n<\/code><\/span><\/div>\n<\/div><\/li>\n<\/ul>\n<h3 data-start=\"1216\" data-end=\"1268\">Variability: Range, Variance, Standard Deviation<\/h3>\n<p data-start=\"1269\" data-end=\"1390\">Consistency matters. The range is max \u2212 min. Variance and standard deviation quantify spread around the mean.<\/p>\n\n<ul data-start=\"1392\" data-end=\"1774\">\n \t<li data-start=\"1392\" data-end=\"1524\">\n<p data-start=\"1394\" data-end=\"1524\"><span style=\"font-size: 12pt\"><strong data-start=\"1394\" data-end=\"1403\">Excel<\/strong><\/span><br data-start=\"1403\" data-end=\"1406\"><span style=\"font-size: 12pt\"><code data-start=\"1408\" data-end=\"1434\">=MAX(B2:B25)-MIN(B2:B25)<\/code> \u2192 <code data-start=\"1437\" data-end=\"1440\">8<\/code><\/span><br data-start=\"1440\" data-end=\"1443\"><span style=\"font-size: 12pt\"><code data-start=\"1445\" data-end=\"1461\">=VAR.S(B2:B25)<\/code> \u2192 <code data-start=\"1475\" data-end=\"1482\">3.592<\/code><\/span><br data-start=\"1482\" data-end=\"1485\"><span style=\"font-size: 12pt\"><code data-start=\"1487\" data-end=\"1505\">=STDEV.S(B2:B25)<\/code> \u2192 <code data-start=\"1517\" data-end=\"1524\">1.895<\/code><\/span><\/p>\n<\/li>\n \t<li data-start=\"1525\" data-end=\"1774\">\n<p data-start=\"1527\" data-end=\"1534\"><span style=\"font-size: 12pt\"><strong data-start=\"1527\" data-end=\"1532\">R<\/strong><\/span><\/p>\n\n<div class=\"contain-inline-size rounded-2xl relative bg-token-sidebar-surface-primary\">\n<div class=\"sticky top-9\"><\/div>\n<div class=\"overflow-y-auto p-4\" dir=\"ltr\"><span style=\"font-size: 12pt\"><code class=\"whitespace-pre! language-r\"><span class=\"hljs-built_in\">max<\/span><span class=\"hljs-punctuation\">(<\/span>dfA<span class=\"hljs-operator\">$<\/span>FulfillmentTimeDays<span class=\"hljs-punctuation\">)<\/span> <span class=\"hljs-operator\">-<\/span> <span class=\"hljs-built_in\">min<\/span><span class=\"hljs-punctuation\">(<\/span>dfA<span class=\"hljs-operator\">$<\/span>FulfillmentTimeDays<span class=\"hljs-punctuation\">)<\/span>  <span class=\"hljs-comment\"># [1] 8<\/span>\n<\/code><\/span><\/div>\n<div class=\"overflow-y-auto p-4\" dir=\"ltr\"><span style=\"font-size: 12pt\"><code class=\"whitespace-pre! language-r\">var<span class=\"hljs-punctuation\">(<\/span>dfA<span class=\"hljs-operator\">$<\/span>FulfillmentTimeDays<span class=\"hljs-punctuation\">)<\/span>                                 <span class=\"hljs-comment\"># [1] 3.592389<\/span>\n<\/code><\/span><\/div>\n<div class=\"overflow-y-auto p-4\" dir=\"ltr\"><span style=\"font-size: 12pt\"><code class=\"whitespace-pre! language-r\"><span style=\"font-size: 12pt\">sd<span class=\"hljs-punctuation\">(<\/span>dfA<span class=\"hljs-operator\">$<\/span>FulfillmentTimeDays<span class=\"hljs-punctuation\">)<\/span>                                  <span class=\"hljs-comment\"># [1] 1.895360<\/span><\/span>\n<\/code><\/span><\/div>\n<\/div><\/li>\n<\/ul>\n<h3 data-start=\"1776\" data-end=\"1814\">Association (Preview): Correlation<\/h3>\n<p data-start=\"1815\" data-end=\"1943\"><span style=\"font-size: 12pt\">Correlation indicates whether two variables move together\u2014for example, whether larger orders take longer to fulfill.<\/span><\/p>\n\n<ul>\n \t<li><span style=\"font-size: 12pt\"><strong data-start=\"1394\" data-end=\"1403\">Excel <\/strong>(Dataset B: items in <code data-start=\"1978\" data-end=\"1986\">B2:B31<\/code>, time in <code data-start=\"1996\" data-end=\"2004\">C2:C31<\/code>)<\/span><br data-start=\"1403\" data-end=\"1406\"><span style=\"font-size: 12pt\"><code data-start=\"1408\" data-end=\"1434\">=CORREL(B2:B31, C2:C31) \u2192 0.991<\/code><\/span><\/li>\n \t<li data-start=\"2046\" data-end=\"2203\">\n<p data-start=\"2048\" data-end=\"2055\"><span style=\"font-size: 12pt\"><strong data-start=\"2048\" data-end=\"2053\">R<\/strong><\/span><\/p>\n\n<div class=\"contain-inline-size rounded-2xl relative bg-token-sidebar-surface-primary\">\n<div class=\"sticky top-9\">\n<div class=\"absolute end-0 bottom-0 flex h-9 items-center pe-2\">\n<div class=\"bg-token-bg-elevated-secondary text-token-text-secondary flex items-center gap-4 rounded-sm px-2 font-sans text-xs\"><span style=\"font-size: 12pt\"><code class=\"whitespace-pre! language-r\" style=\"background-color: #ffffff\">dfB <span class=\"hljs-operator\">&lt;-<\/span> read.csv<span class=\"hljs-punctuation\">(<\/span><span class=\"hljs-string\">\"Chapter9_Dataset_B_OrderSize_vs_Fulfillment.csv\"<\/span><span class=\"hljs-punctuation\">)<\/span>\n<\/code><\/span><\/div>\n<div class=\"bg-token-bg-elevated-secondary text-token-text-secondary flex items-center gap-4 rounded-sm px-2 font-sans text-xs\"><span style=\"font-size: 12pt\"><code class=\"whitespace-pre! language-r\" style=\"background-color: #ffffff\">cor<span class=\"hljs-punctuation\">(<\/span>dfB<span class=\"hljs-operator\">$<\/span>ItemsInOrder<span class=\"hljs-punctuation\">,<\/span> dfB<span class=\"hljs-operator\">$<\/span>FulfillmentTimeDays<span class=\"hljs-punctuation\">)<\/span>  <span class=\"hljs-comment\"># [1] 0.9911393<\/span><\/code><\/span><span class=\"\" data-state=\"closed\"><\/span><\/div>\n<\/div>\n<\/div>\n<\/div><\/li>\n<\/ul>\n<h3 data-start=\"2205\" data-end=\"2263\">Applied Example: Fulfillment Times at Pemi (Dataset A)<strong style=\"text-align: initial;font-size: 1em\" data-start=\"2264\" data-end=\"2273\">Excel<\/strong><\/h3>\n<div class=\"contain-inline-size rounded-2xl relative bg-token-sidebar-surface-primary\">\n<div class=\"overflow-y-auto p-4\" dir=\"ltr\">\n\n<span style=\"font-size: 12pt\"><span style=\"font-size: 12pt\"><code class=\"whitespace-pre! language-r\"><\/code><\/span><\/span>\n<p data-start=\"2276\" data-end=\"2312\"><span style=\"font-size: 12pt\">Mean: <code data-start=\"2282\" data-end=\"2300\">=AVERAGE(B2:B25)<\/code> \u2192 <code data-start=\"2303\" data-end=\"2310\">5.875<\/code><\/span><\/p>\n<p data-start=\"2276\" data-end=\"2312\"><span style=\"font-size: 12pt\"><code data-start=\"2303\" data-end=\"2310\"><\/code>Median: <code data-start=\"2323\" data-end=\"2340\">=MEDIAN(B2:B25)<\/code> \u2192 <code data-start=\"2343\" data-end=\"2346\">5<\/code><\/span><\/p>\n<p data-start=\"2276\" data-end=\"2312\"><span style=\"font-size: 12pt\"><code data-start=\"2343\" data-end=\"2346\"><\/code>Std. dev.: <code data-start=\"2362\" data-end=\"2380\">=STDEV.S(B2:B25)<\/code> \u2192 <code data-start=\"2383\" data-end=\"2390\">1.895<\/code><\/span><\/p>\n<p data-start=\"2276\" data-end=\"2312\"><code data-start=\"2383\" data-end=\"2390\"><\/code>Optional histogram: Data \u2192 Data Analysis \u2192 Histogram<\/p>\n<span style=\"font-size: 12pt\"><span style=\"font-size: 12pt\"><code class=\"whitespace-pre! language-r\"><\/code><\/span><\/span>\n<p data-start=\"2454\" data-end=\"2459\"><strong data-start=\"2454\" data-end=\"2459\">R<\/strong><\/p>\n<span style=\"font-size: 12pt\"><span style=\"font-size: 12pt\"><code class=\"whitespace-pre! language-r\"><\/code><\/span><\/span>\n<pre class=\"overflow-visible!\" data-start=\"2460\" data-end=\"2688\"><span style=\"font-size: 12pt\">dfA &lt;- read.csv(\"Chapter9_Dataset_A_FulfillmentTimes.csv\")<\/span>\n<span style=\"font-size: 12pt\">mean(dfA$FulfillmentTimeDays) # [1] 5.875<\/span>\n<span style=\"font-size: 12pt\">median(dfA$FulfillmentTimeDays) # [1] 5<\/span>\n<span style=\"font-size: 12pt\">sd(dfA$FulfillmentTimeDays) # [1] 1.895360<\/span>\n<span style=\"font-size: 12pt\">hist(dfA$FulfillmentTimeDays)<\/span><\/pre>\n<span style=\"font-size: 12pt\"><code class=\"whitespace-pre! language-r\"><\/code><\/span>\n\n<\/div>\n<\/div>\n<p data-start=\"2690\" data-end=\"2947\"><strong data-start=\"2690\" data-end=\"2708\">Interpretation<\/strong><br data-start=\"2708\" data-end=\"2711\">The mean exceeds the median, indicating a right tail (a few slow orders). A standard deviation near 1.9 suggests noticeable variability. Next steps: isolate slow orders and investigate causes (capacity, supplier delays, address issues).<\/p>\n\n<h2 data-start=\"199\" data-end=\"272\"><strong data-start=\"202\" data-end=\"272\">9.3 Hypothesis testing: two-sample t-test (small vs. large orders)<\/strong><\/h2>\n<p data-start=\"274\" data-end=\"324\">Question: do larger orders take longer to fulfill?<\/p>\n<p data-start=\"326\" data-end=\"357\">Define two groups in Dataset B:<\/p>\n\n<ul data-start=\"358\" data-end=\"409\">\n \t<li data-start=\"358\" data-end=\"383\">\n<p data-start=\"360\" data-end=\"383\">Small orders: items \u2264 3<\/p>\n<\/li>\n \t<li data-start=\"384\" data-end=\"409\">\n<p data-start=\"386\" data-end=\"409\">Large orders: items &gt; 3<\/p>\n<\/li>\n<\/ul>\n<p data-start=\"411\" data-end=\"500\">Set up in <strong data-start=\"421\" data-end=\"430\">Excel<\/strong>: add a helper column <code data-start=\"452\" data-end=\"464\">size_group<\/code> with<br data-start=\"469\" data-end=\"472\"><code data-start=\"472\" data-end=\"500\">=IF(B2&lt;=3,\"Small\",\"Large\")<\/code><\/p>\n<p data-start=\"502\" data-end=\"593\">Compare mean fulfillment time by group using Welch\u2019s two-sample t-test (unequal variances).<\/p>\n<p data-start=\"595\" data-end=\"604\"><strong data-start=\"595\" data-end=\"604\">Excel<\/strong><\/p>\n\n<ol data-start=\"605\" data-end=\"945\">\n \t<li data-start=\"605\" data-end=\"722\">\n<p data-start=\"608\" data-end=\"722\">Compute group means and standard deviations with <code data-start=\"657\" data-end=\"672\">=AVERAGEIFS()<\/code> and <code data-start=\"677\" data-end=\"696\">=STDEV.S(IF(...))<\/code> or with filters\/pivots.<\/p>\n<\/li>\n \t<li data-start=\"723\" data-end=\"945\">\n<p data-start=\"726\" data-end=\"797\">Data \u2192 Data Analysis \u2192 t-Test: Two-Sample Assuming Unequal Variances.<\/p>\n\n<ul data-start=\"801\" data-end=\"945\">\n \t<li data-start=\"801\" data-end=\"838\">\n<p data-start=\"803\" data-end=\"838\">Variable 1 Range: times for Small<\/p>\n<\/li>\n \t<li data-start=\"842\" data-end=\"879\">\n<p data-start=\"844\" data-end=\"879\">Variable 2 Range: times for Large<\/p>\n<\/li>\n \t<li data-start=\"883\" data-end=\"918\">\n<p data-start=\"885\" data-end=\"918\">Hypothesized mean difference: 0<\/p>\n<\/li>\n \t<li data-start=\"922\" data-end=\"945\">\n<p data-start=\"924\" data-end=\"945\">Output to a new range<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<p data-start=\"947\" data-end=\"952\"><strong data-start=\"947\" data-end=\"952\">R<\/strong><\/p>\n\n<div class=\"contain-inline-size rounded-2xl relative bg-token-sidebar-surface-primary\">\n<div class=\"sticky top-9\">\n<div class=\"absolute end-0 bottom-0 flex h-9 items-center pe-2\">\n<div class=\"bg-token-bg-elevated-secondary text-token-text-secondary flex items-center gap-4 rounded-sm px-2 font-sans text-xs\"><span class=\"\" data-state=\"closed\"><\/span><\/div>\n<\/div>\n<\/div>\n<div class=\"overflow-y-auto p-4\" dir=\"ltr\"><code class=\"whitespace-pre! language-r\"><span>dfB <span class=\"hljs-operator\">&lt;-<\/span> read.csv<span class=\"hljs-punctuation\">(<\/span><span class=\"hljs-string\">\"Chapter9_Dataset_B_OrderSize_vs_Fulfillment.csv\"<\/span><span class=\"hljs-punctuation\">)<\/span>\ndfB<span class=\"hljs-operator\">$<\/span>size_group <span class=\"hljs-operator\">&lt;-<\/span> ifelse<span class=\"hljs-punctuation\">(<\/span>dfB<span class=\"hljs-operator\">$<\/span>ItemsInOrder <span class=\"hljs-operator\">&lt;=<\/span> <span class=\"hljs-number\">3<\/span><span class=\"hljs-punctuation\">,<\/span> <span class=\"hljs-string\">\"Small\"<\/span><span class=\"hljs-punctuation\">,<\/span> <span class=\"hljs-string\">\"Large\"<\/span><span class=\"hljs-punctuation\">)<\/span>\nt.test<span class=\"hljs-punctuation\">(<\/span>FulfillmentTimeDays <span class=\"hljs-operator\">~<\/span> size_group<span class=\"hljs-punctuation\">,<\/span> data <span class=\"hljs-operator\">=<\/span> dfB<span class=\"hljs-punctuation\">)<\/span>  <span class=\"hljs-comment\"># Welch by default<\/span>\n<\/span><\/code><\/div>\n<\/div>\n<p data-start=\"1169\" data-end=\"1212\">What you should see with the provided data:<\/p>\n\n<ul data-start=\"1213\" data-end=\"1298\">\n \t<li data-start=\"1213\" data-end=\"1262\">\n<p data-start=\"1215\" data-end=\"1262\">Difference in means \u2248 1.54 days (Large \u2212 Small)<\/p>\n<\/li>\n \t<li data-start=\"1263\" data-end=\"1286\">\n<p data-start=\"1265\" data-end=\"1286\">95% CI \u2248 [1.22, 1.86]<\/p>\n<\/li>\n \t<li data-start=\"1287\" data-end=\"1298\">\n<p data-start=\"1289\" data-end=\"1298\">p &lt; 0.001<\/p>\n<\/li>\n<\/ul>\n<p data-start=\"1300\" data-end=\"1451\">Interpretation: in this sample, larger orders take longer by about a day and a half on average. The effect is practically and statistically meaningful.<\/p>\n<p data-start=\"1453\" data-end=\"1551\">Transition: correlation and a simple linear model quantify the items \u2192 time relationship directly.<\/p>\n\n<h2 data-start=\"1558\" data-end=\"1609\"><strong data-start=\"1561\" data-end=\"1609\">9.4 Correlation and simple linear regression<\/strong><\/h2>\n<p data-start=\"1611\" data-end=\"1734\">Use correlation to summarize association and a simple linear regression to estimate how much time increases per added item.<\/p>\n<p data-start=\"1736\" data-end=\"1745\"><strong data-start=\"1736\" data-end=\"1745\">Excel<\/strong><\/p>\n\n<ul data-start=\"1746\" data-end=\"1958\">\n \t<li data-start=\"1746\" data-end=\"1788\">\n<p data-start=\"1748\" data-end=\"1788\">Correlation:<span style=\"font-size: 12pt\"> <code data-start=\"1761\" data-end=\"1786\">=CORREL(B2:B31, C2:C31)<\/code><\/span><\/p>\n<\/li>\n \t<li data-start=\"1789\" data-end=\"1866\">\n<p data-start=\"1791\" data-end=\"1866\">Scatterplot: Insert \u2192 Scatter. Add a trendline \u2192 display equation and R\u00b2.<\/p>\n<\/li>\n \t<li data-start=\"1867\" data-end=\"1958\">\n<p data-start=\"1869\" data-end=\"1958\"><span style=\"font-size: 12pt\">Functions: <code data-start=\"1880\" data-end=\"1904\">=SLOPE(C2:C31, B2:B31)<\/code>, <code data-start=\"1906\" data-end=\"1934\">=INTERCEPT(C2:C31, B2:B31)<\/code>, <code data-start=\"1936\" data-end=\"1958\">=RSQ(C2:C31, B2:B31)<\/code><\/span><\/p>\n<\/li>\n<\/ul>\n<p data-start=\"1960\" data-end=\"1965\"><strong data-start=\"1960\" data-end=\"1965\">R<\/strong><\/p>\n\n<div class=\"contain-inline-size rounded-2xl relative bg-token-sidebar-surface-primary\">\n<div class=\"sticky top-9\">\n<div class=\"absolute end-0 bottom-0 flex h-9 items-center pe-2\">\n<div class=\"bg-token-bg-elevated-secondary text-token-text-secondary flex items-center gap-4 rounded-sm px-2 font-sans text-xs\"><span class=\"\" data-state=\"closed\"><\/span><\/div>\n<\/div>\n<\/div>\n<div class=\"overflow-y-auto p-4\" dir=\"ltr\"><code class=\"whitespace-pre! language-r\"><span><span style=\"font-size: 12pt\">cor<span class=\"hljs-punctuation\">(<\/span>dfB<span class=\"hljs-operator\">$<\/span>ItemsInOrder<span class=\"hljs-punctuation\">,<\/span> dfB<span class=\"hljs-operator\">$<\/span>FulfillmentTimeDays<span class=\"hljs-punctuation\">)<\/span>      <span class=\"hljs-comment\"># \u2248 0.991<\/span><\/span>\n<span style=\"font-size: 12pt\">\nm <span class=\"hljs-operator\">&lt;-<\/span> lm<span class=\"hljs-punctuation\">(<\/span>FulfillmentTimeDays <span class=\"hljs-operator\">~<\/span> ItemsInOrder<span class=\"hljs-punctuation\">,<\/span> data <span class=\"hljs-operator\">=<\/span> dfB<span class=\"hljs-punctuation\">)<\/span><\/span>\n<span style=\"font-size: 12pt\">\nsummary<span class=\"hljs-punctuation\">(<\/span>m<span class=\"hljs-punctuation\">)<\/span><\/span>\n<\/span><\/code><\/div>\n<\/div>\n<p data-start=\"2105\" data-end=\"2161\">With the provided data, you should see results close to:<\/p>\n\n<ul data-start=\"2162\" data-end=\"2244\">\n \t<li data-start=\"2162\" data-end=\"2204\">\n<p data-start=\"2164\" data-end=\"2204\">Slope \u2248 0.430 days per additional item<\/p>\n<\/li>\n \t<li data-start=\"2205\" data-end=\"2231\">\n<p data-start=\"2207\" data-end=\"2231\">Intercept \u2248 4.232 days<\/p>\n<\/li>\n \t<li data-start=\"2232\" data-end=\"2244\">\n<p data-start=\"2234\" data-end=\"2244\">R\u00b2 \u2248 0.982<\/p>\n<\/li>\n<\/ul>\n<p data-start=\"2246\" data-end=\"2413\">Interpretation: each additional item adds roughly 0.43 days on average. The fit is strong in this sample. Do not extrapolate beyond the observed range without a check.<\/p>\n<p data-start=\"2415\" data-end=\"2514\">Transition: before relying on these results, confirm basic assumptions and scan for obvious issues.<\/p>\n\n<h2 data-start=\"2521\" data-end=\"2565\"><strong data-start=\"2524\" data-end=\"2565\">9.5 Assumptions and quick diagnostics<\/strong><\/h2>\n<p data-start=\"2567\" data-end=\"2662\">Hypothesis tests and linear models rest on simple conditions. A few quick checks go a long way.<\/p>\n<p data-start=\"2664\" data-end=\"2688\">t-test (Small vs. Large)<\/p>\n\n<ul data-start=\"2689\" data-end=\"3043\">\n \t<li data-start=\"2689\" data-end=\"2781\">\n<p data-start=\"2691\" data-end=\"2781\">Independence: orders should be independent observations. Segment by shift\/promo if needed.<\/p>\n<\/li>\n \t<li data-start=\"2782\" data-end=\"2914\">\n<p data-start=\"2784\" data-end=\"2914\">Shape: with n \u2248 15 per group, the t-test is robust, but heavy skew\/outliers can distort results. Use group histograms or boxplots.<\/p>\n<\/li>\n \t<li data-start=\"2915\" data-end=\"3043\">\n<p data-start=\"2917\" data-end=\"3043\">Variances: Welch\u2019s test (used above) does not assume equal variances. If you suspect very different spreads, keep using Welch.<\/p>\n<\/li>\n<\/ul>\n<p data-start=\"3045\" data-end=\"3073\"><strong data-start=\"3045\" data-end=\"3073\">Excel (practical checks)<\/strong><\/p>\n\n<ul data-start=\"3074\" data-end=\"3371\">\n \t<li data-start=\"3074\" data-end=\"3142\">\n<p data-start=\"3076\" data-end=\"3142\">Build small and large group histograms or boxplots (via charts).<\/p>\n<\/li>\n \t<li data-start=\"3143\" data-end=\"3190\">\n<p data-start=\"3145\" data-end=\"3190\">Compute group variances with <code data-start=\"3174\" data-end=\"3187\">=VAR.S(...)<\/code>.<\/p>\n<\/li>\n \t<li data-start=\"3191\" data-end=\"3371\">\n<p data-start=\"3193\" data-end=\"3371\">If you ran the regression, add predicted values <span style=\"font-size: 12pt\"><code data-start=\"3241\" data-end=\"3267\">=intercept + slope*Items<\/code> and create a residuals column <code data-start=\"3298\" data-end=\"3319\">=Actual - Predicted<\/code>. Plot residuals vs. p<\/span>redicted; look for no pattern.<\/p>\n<\/li>\n<\/ul>\n<p data-start=\"3373\" data-end=\"3395\"><strong data-start=\"3373\" data-end=\"3395\">R (quick commands)<\/strong><\/p>\n\n<div class=\"contain-inline-size rounded-2xl relative bg-token-sidebar-surface-primary\">\n<div class=\"sticky top-9\">\n<div class=\"absolute end-0 bottom-0 flex h-9 items-center pe-2\">\n<div class=\"bg-token-bg-elevated-secondary text-token-text-secondary flex items-center gap-4 rounded-sm px-2 font-sans text-xs\"><span class=\"\" data-state=\"closed\"><\/span><\/div>\n<\/div>\n<\/div>\n<div class=\"overflow-y-auto p-4\" dir=\"ltr\"><code class=\"whitespace-pre! language-r\"><span><span style=\"font-size: 12pt\">boxplot<span class=\"hljs-punctuation\">(<\/span>FulfillmentTimeDays <span class=\"hljs-operator\">~<\/span> size_group<span class=\"hljs-punctuation\">,<\/span> data <span class=\"hljs-operator\">=<\/span> dfB<span class=\"hljs-punctuation\">)<\/span><\/span>\n<span style=\"font-size: 12pt\">\nby<span class=\"hljs-punctuation\">(<\/span>dfB<span class=\"hljs-operator\">$<\/span>FulfillmentTimeDays<span class=\"hljs-punctuation\">,<\/span> dfB<span class=\"hljs-operator\">$<\/span>size_group<span class=\"hljs-punctuation\">,<\/span> sd<span class=\"hljs-punctuation\">)<\/span>  <span class=\"hljs-comment\"># group spreads<\/span><\/span>\n<span class=\"hljs-comment\" style=\"font-size: 12pt\">\n# Regression diagnostics<\/span>\n<span style=\"font-size: 12pt\">\nplot<span class=\"hljs-punctuation\">(<\/span>m<span class=\"hljs-punctuation\">,<\/span> which <span class=\"hljs-operator\">=<\/span> <span class=\"hljs-number\">1<\/span><span class=\"hljs-punctuation\">)<\/span>  <span class=\"hljs-comment\"># Residuals vs Fitted<\/span><\/span>\n<span style=\"font-size: 12pt\">\nplot<span class=\"hljs-punctuation\">(<\/span>m<span class=\"hljs-punctuation\">,<\/span> which <span class=\"hljs-operator\">=<\/span> <span class=\"hljs-number\">2<\/span><span class=\"hljs-punctuation\">)<\/span>  <span class=\"hljs-comment\"># Normal Q-Q<\/span><\/span>\n<\/span><\/code><\/div>\n<\/div>\n<p data-start=\"3625\" data-end=\"3635\">Regression<\/p>\n\n<ul data-start=\"3636\" data-end=\"3923\">\n \t<li data-start=\"3636\" data-end=\"3706\">\n<p data-start=\"3638\" data-end=\"3706\">Linearity: residuals vs. fitted should look like noise, not a curve.<\/p>\n<\/li>\n \t<li data-start=\"3707\" data-end=\"3814\">\n<p data-start=\"3709\" data-end=\"3814\">Constant variance: avoid funnel shapes; consider a variance-stabilizing approach or segmenting if needed.<\/p>\n<\/li>\n \t<li data-start=\"3815\" data-end=\"3923\">\n<p data-start=\"3817\" data-end=\"3923\">Influence: in small samples, a single unusual order can sway the fit; inspect points with large residuals.<\/p>\n<\/li>\n<\/ul>\n<p data-start=\"3925\" data-end=\"4021\">Transition: with checks in place, synthesize findings in a brief memo that a manager can act on.<\/p>\n\n<h2 data-start=\"4028\" data-end=\"4068\"><strong data-start=\"4031\" data-end=\"4068\">9.6 Applied assignment: Pemi memo<\/strong><\/h2>\n<p data-start=\"4070\" data-end=\"4181\">Goal: use the chapter workflow to answer a specific operations question and communicate a clear recommendation.<\/p>\n<p data-start=\"4183\" data-end=\"4191\">Datasets<\/p>\n\n<ul data-start=\"4192\" data-end=\"4285\">\n \t<li data-start=\"4192\" data-end=\"4235\">\n<p data-start=\"4194\" data-end=\"4235\">Chapter9_Dataset_A_FulfillmentTimes.csv<\/p>\n<\/li>\n \t<li data-start=\"4236\" data-end=\"4285\">\n<p data-start=\"4238\" data-end=\"4285\">Chapter9_Dataset_B_OrderSize_vs_Fulfillment.csv<\/p>\n<\/li>\n<\/ul>\n<p data-start=\"4287\" data-end=\"4292\">Tasks<\/p>\n\n<ol data-start=\"4293\" data-end=\"4970\">\n \t<li data-start=\"4293\" data-end=\"4537\">\n<p data-start=\"4296\" data-end=\"4303\">Excel<\/p>\n\n<ul data-start=\"4307\" data-end=\"4537\">\n \t<li data-start=\"4307\" data-end=\"4387\">\n<p data-start=\"4309\" data-end=\"4387\">Dataset A: mean, median, standard deviation; histogram of fulfillment times.<\/p>\n<\/li>\n \t<li data-start=\"4391\" data-end=\"4472\">\n<p data-start=\"4393\" data-end=\"4472\">Dataset B: correlation (items vs. time); two-sample t-test (Small vs. Large).<\/p>\n<\/li>\n \t<li data-start=\"4476\" data-end=\"4537\">\n<p data-start=\"4478\" data-end=\"4537\">Optional: scatterplot with trendline, show equation and R\u00b2.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n \t<li data-start=\"4539\" data-end=\"4707\">\n<p data-start=\"4542\" data-end=\"4545\">R<\/p>\n\n<ul data-start=\"4549\" data-end=\"4707\">\n \t<li data-start=\"4549\" data-end=\"4654\">\n<p data-start=\"4551\" data-end=\"4654\">Replicate Excel results with <span style=\"font-size: 12pt\"><code data-start=\"4580\" data-end=\"4588\">mean()<\/code>, <code data-start=\"4590\" data-end=\"4600\">median()<\/code>, <code data-start=\"4602\" data-end=\"4608\">sd()<\/code>, <code data-start=\"4610\" data-end=\"4618\">hist()<\/code>, <code data-start=\"4620\" data-end=\"4627\">cor()<\/code>, <code data-start=\"4629\" data-end=\"4639\">t.test()<\/code>, and <code data-start=\"4645\" data-end=\"4651\">lm()<\/code>.<\/span><\/p>\n<\/li>\n \t<li data-start=\"4658\" data-end=\"4707\">\n<p data-start=\"4660\" data-end=\"4707\">Confirm that Excel and R agree within rounding.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n \t<li data-start=\"4709\" data-end=\"4970\">\n<p data-start=\"4712\" data-end=\"4746\">Managerial memo (1\u20132 paragraphs)<\/p>\n\n<ul data-start=\"4750\" data-end=\"4970\">\n \t<li data-start=\"4750\" data-end=\"4810\">\n<p data-start=\"4752\" data-end=\"4810\">Is the difference in mean time by size group meaningful?<\/p>\n<\/li>\n \t<li data-start=\"4814\" data-end=\"4868\">\n<p data-start=\"4816\" data-end=\"4868\">About how much time does each additional item add?<\/p>\n<\/li>\n \t<li data-start=\"4872\" data-end=\"4970\">\n<p data-start=\"4874\" data-end=\"4970\">One concrete action you recommend (e.g., staffing during peaks, batching rules, target setting).<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<p data-start=\"4972\" data-end=\"4984\">Deliverables<\/p>\n\n<ul data-start=\"4985\" data-end=\"5167\">\n \t<li data-start=\"4985\" data-end=\"5067\">\n<p data-start=\"4987\" data-end=\"5067\">One Excel file with clearly labeled sheets (A_Summary, B_ttest, B_Regression).<\/p>\n<\/li>\n \t<li data-start=\"5068\" data-end=\"5145\">\n<p data-start=\"5070\" data-end=\"5145\">One R script (.R) or R Markdown (.Rmd) with commands and printed outputs.<\/p>\n<\/li>\n \t<li data-start=\"5146\" data-end=\"5167\">\n<p data-start=\"5148\" data-end=\"5167\">A PDF or DOCX memo.<\/p>\n<\/li>\n<\/ul>\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\n<p class=\"textbox__title\">Key Takeaways<\/p>\n\n<\/header>\n<div class=\"textbox__content\">\n<ul>\n \t<li data-start=\"230\" data-end=\"319\">\n<p data-start=\"232\" data-end=\"319\">Mean and median summarize central tendency; the median is less sensitive to outliers.<\/p>\n<\/li>\n \t<li data-start=\"320\" data-end=\"431\">\n<p data-start=\"322\" data-end=\"431\">Variance and standard deviation measure variability around the mean, indicating consistency in performance.<\/p>\n<\/li>\n \t<li data-start=\"432\" data-end=\"509\">\n<p data-start=\"434\" data-end=\"509\">Correlation quantifies the strength of association between two variables.<\/p>\n<\/li>\n \t<li data-start=\"510\" data-end=\"618\">\n<p data-start=\"512\" data-end=\"618\">A two-sample t-test compares group means and can show whether differences are statistically significant.<\/p>\n<\/li>\n \t<li data-start=\"619\" data-end=\"744\">\n<p data-start=\"621\" data-end=\"744\">Linear regression estimates the relationship between predictors (e.g., order size) and outcomes (e.g., fulfillment time).<\/p>\n<\/li>\n \t<li data-start=\"745\" data-end=\"856\">\n<p data-start=\"747\" data-end=\"856\">Diagnostic checks (shape, variance, residuals) ensure statistical results are reliable for decision-making.<\/p>\n<\/li>\n \t<li data-start=\"857\" data-end=\"978\">\n<p data-start=\"859\" data-end=\"978\">Excel and R can be used together to calculate, visualize, and cross-verify these statistics in an operations context.<\/p>\n<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<h2>Chapter Nine References<\/h2>\n<p data-start=\"207\" data-end=\"317\">Microsoft. (n.d.). <em data-start=\"226\" data-end=\"262\">Statistical functions (reference).<\/em> Microsoft.<a href=\"https:\/\/support.microsoft.com\/en-us\/excel\"> https:\/\/support.microsoft.com\/en-us\/excel<\/a><\/p>\n<p data-start=\"320\" data-end=\"467\">R Core Team. (2023). <em data-start=\"341\" data-end=\"399\">R: A language and environment for statistical computing.<\/em> R Foundation for Statistical Computing. <a href=\"https:\/\/www.r-project.org\">https:\/\/www.r-project.org<\/a><span aria-hidden=\"true\" class=\"ms-0.5 inline-block align-middle leading-none\"><\/span><\/p>\n<p data-start=\"470\" data-end=\"621\">Wickham, H., &amp; Grolemund, G. (2017). <em data-start=\"507\" data-end=\"580\">R for data science: Import, tidy, transform, visualize, and model data.<\/em> O\u2019Reilly Media. <a data-start=\"597\" data-end=\"619\" rel=\"noopener\" target=\"_new\" class=\"decorated-link cursor-pointer\" href=\"https:\/\/r4ds.had.co.nz\">https:\/\/r4ds.had.co.nz<\/a><\/p>","rendered":"<p><a href=\"http:\/\/pressbooks.usnh.edu\/businessopsanalytics\/files\/2025\/08\/Chapter-9-Header.png\"><img decoding=\"async\" src=\"https:\/\/openbooks.spmvv.ac.in\/app\/uploads\/sites\/3\/2025\/08\/Chapter-9-Header.png\" alt=\"image\" \/><\/a><\/p>\n<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Objectives<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p data-start=\"36\" data-end=\"89\">By the end of this chapter, students will be able to:<\/p>\n<ol data-start=\"91\" data-end=\"624\" data-is-last-node=\"\" data-is-only-node=\"\">\n<li data-start=\"91\" data-end=\"181\">\n<p data-start=\"94\" data-end=\"181\">Summarize operational data (mean, median, standard deviation) in Excel and R.<\/p>\n<\/li>\n<li data-start=\"182\" data-end=\"274\">\n<p data-start=\"185\" data-end=\"274\">Choose and apply Normal, Binomial, and Poisson models; compute probabilities\/quantiles.<\/p>\n<\/li>\n<li data-start=\"275\" data-end=\"370\">\n<p data-start=\"278\" data-end=\"370\">Run a two-sample t-test; report the difference in means, confidence interval, and p-value.<\/p>\n<\/li>\n<li data-start=\"371\" data-end=\"466\">\n<p data-start=\"374\" data-end=\"466\">Analyze association with correlation and simple linear regression; interpret slope and R\u00b2.<\/p>\n<\/li>\n<li data-start=\"467\" data-end=\"563\">\n<p data-start=\"470\" data-end=\"563\">Perform quick checks for outliers, variance, and independence; note limits of the analysis.<\/p>\n<\/li>\n<li data-start=\"564\" data-end=\"624\" data-is-last-node=\"\">\n<p data-start=\"567\" data-end=\"624\" data-is-last-node=\"\">Convert results into a concise operations recommendation.<\/p>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<h2 data-start=\"279\" data-end=\"330\">9.1 Basic Statistical Terms (Operations Context)<\/h2>\n<p data-start=\"332\" data-end=\"794\">In earlier chapters you organized data, improved quality, and visualized performance. Here we formalize those summaries with a few core statistics you already know from your prerequisite course, focusing on how to apply and interpret them in operations. We\u2019ll compute each measure in Excel and mirror it in R, using two <em data-start=\"664\" data-end=\"686\">Pemi Coffee Roasters<\/em> datasets: Dataset A (fulfillment times, days) and Dataset B (items per order and fulfillment time).<\/p>\n<h3 data-start=\"796\" data-end=\"815\">Mean and Median<\/h3>\n<p data-start=\"816\" data-end=\"939\">The mean summarizes average performance; the median is the middle value and is less sensitive to a few slow orders.<\/p>\n<ul data-start=\"941\" data-end=\"1214\">\n<li data-start=\"941\" data-end=\"1038\">\n<p data-start=\"943\" data-end=\"1038\"><strong data-start=\"943\" data-end=\"952\">Excel<\/strong> (Dataset A in <code data-start=\"967\" data-end=\"975\">B2:B25<\/code>)<br data-start=\"976\" data-end=\"979\" \/><span style=\"font-size: 12pt\"><code data-start=\"981\" data-end=\"999\">=AVERAGE(B2:B25)<\/code> \u2192 <code data-start=\"1002\" data-end=\"1009\">5.875<\/code><\/span><br data-start=\"1009\" data-end=\"1012\" \/><span style=\"font-size: 12pt\"><code data-start=\"1014\" data-end=\"1031\">=MEDIAN(B2:B25)<\/code> \u2192 <code data-start=\"1035\" data-end=\"1038\">5<\/code><\/span><\/p>\n<\/li>\n<li data-start=\"1039\" data-end=\"1214\">\n<p data-start=\"1041\" data-end=\"1048\"><strong data-start=\"1041\" data-end=\"1046\">R<\/strong><\/p>\n<div class=\"contain-inline-size rounded-2xl relative bg-token-sidebar-surface-primary\">\n<div class=\"sticky top-9\"><\/div>\n<div class=\"overflow-y-auto p-4\" dir=\"ltr\"><span style=\"font-size: 12pt\"><code class=\"whitespace-pre! language-r\">dfA <span class=\"hljs-operator\">&lt;-<\/span> read.csv<span class=\"hljs-punctuation\">(<\/span><span class=\"hljs-string\">\"Chapter9_Dataset_A_FulfillmentTimes.csv\"<\/span><span class=\"hljs-punctuation\">)<\/span><br \/>\n<\/code><\/span><\/div>\n<div class=\"overflow-y-auto p-4\" dir=\"ltr\"><span style=\"font-size: 12pt\"><code class=\"whitespace-pre! language-r\">mean<span class=\"hljs-punctuation\">(<\/span>dfA<span class=\"hljs-operator\">$<\/span>FulfillmentTimeDays<span class=\"hljs-punctuation\">)<\/span>     <span class=\"hljs-comment\"># [1] 5.875<\/span><br \/>\n<\/code><\/span><\/div>\n<div class=\"overflow-y-auto p-4\" dir=\"ltr\"><span style=\"font-size: 12pt\"><code class=\"whitespace-pre! language-r\">median<span class=\"hljs-punctuation\">(<\/span>dfA<span class=\"hljs-operator\">$<\/span>FulfillmentTimeDays<span class=\"hljs-punctuation\">)<\/span>   <span class=\"hljs-comment\"># [1] 5<\/span><br \/>\n<\/code><\/span><\/div>\n<\/div>\n<\/li>\n<\/ul>\n<h3 data-start=\"1216\" data-end=\"1268\">Variability: Range, Variance, Standard Deviation<\/h3>\n<p data-start=\"1269\" data-end=\"1390\">Consistency matters. The range is max \u2212 min. Variance and standard deviation quantify spread around the mean.<\/p>\n<ul data-start=\"1392\" data-end=\"1774\">\n<li data-start=\"1392\" data-end=\"1524\">\n<p data-start=\"1394\" data-end=\"1524\"><span style=\"font-size: 12pt\"><strong data-start=\"1394\" data-end=\"1403\">Excel<\/strong><\/span><br data-start=\"1403\" data-end=\"1406\" \/><span style=\"font-size: 12pt\"><code data-start=\"1408\" data-end=\"1434\">=MAX(B2:B25)-MIN(B2:B25)<\/code> \u2192 <code data-start=\"1437\" data-end=\"1440\">8<\/code><\/span><br data-start=\"1440\" data-end=\"1443\" \/><span style=\"font-size: 12pt\"><code data-start=\"1445\" data-end=\"1461\">=VAR.S(B2:B25)<\/code> \u2192 <code data-start=\"1475\" data-end=\"1482\">3.592<\/code><\/span><br data-start=\"1482\" data-end=\"1485\" \/><span style=\"font-size: 12pt\"><code data-start=\"1487\" data-end=\"1505\">=STDEV.S(B2:B25)<\/code> \u2192 <code data-start=\"1517\" data-end=\"1524\">1.895<\/code><\/span><\/p>\n<\/li>\n<li data-start=\"1525\" data-end=\"1774\">\n<p data-start=\"1527\" data-end=\"1534\"><span style=\"font-size: 12pt\"><strong data-start=\"1527\" data-end=\"1532\">R<\/strong><\/span><\/p>\n<div class=\"contain-inline-size rounded-2xl relative bg-token-sidebar-surface-primary\">\n<div class=\"sticky top-9\"><\/div>\n<div class=\"overflow-y-auto p-4\" dir=\"ltr\"><span style=\"font-size: 12pt\"><code class=\"whitespace-pre! language-r\"><span class=\"hljs-built_in\">max<\/span><span class=\"hljs-punctuation\">(<\/span>dfA<span class=\"hljs-operator\">$<\/span>FulfillmentTimeDays<span class=\"hljs-punctuation\">)<\/span> <span class=\"hljs-operator\">-<\/span> <span class=\"hljs-built_in\">min<\/span><span class=\"hljs-punctuation\">(<\/span>dfA<span class=\"hljs-operator\">$<\/span>FulfillmentTimeDays<span class=\"hljs-punctuation\">)<\/span>  <span class=\"hljs-comment\"># [1] 8<\/span><br \/>\n<\/code><\/span><\/div>\n<div class=\"overflow-y-auto p-4\" dir=\"ltr\"><span style=\"font-size: 12pt\"><code class=\"whitespace-pre! language-r\">var<span class=\"hljs-punctuation\">(<\/span>dfA<span class=\"hljs-operator\">$<\/span>FulfillmentTimeDays<span class=\"hljs-punctuation\">)<\/span>                                 <span class=\"hljs-comment\"># [1] 3.592389<\/span><br \/>\n<\/code><\/span><\/div>\n<div class=\"overflow-y-auto p-4\" dir=\"ltr\"><span style=\"font-size: 12pt\"><code class=\"whitespace-pre! language-r\"><span style=\"font-size: 12pt\">sd<span class=\"hljs-punctuation\">(<\/span>dfA<span class=\"hljs-operator\">$<\/span>FulfillmentTimeDays<span class=\"hljs-punctuation\">)<\/span>                                  <span class=\"hljs-comment\"># [1] 1.895360<\/span><\/span><br \/>\n<\/code><\/span><\/div>\n<\/div>\n<\/li>\n<\/ul>\n<h3 data-start=\"1776\" data-end=\"1814\">Association (Preview): Correlation<\/h3>\n<p data-start=\"1815\" data-end=\"1943\"><span style=\"font-size: 12pt\">Correlation indicates whether two variables move together\u2014for example, whether larger orders take longer to fulfill.<\/span><\/p>\n<ul>\n<li><span style=\"font-size: 12pt\"><strong data-start=\"1394\" data-end=\"1403\">Excel <\/strong>(Dataset B: items in <code data-start=\"1978\" data-end=\"1986\">B2:B31<\/code>, time in <code data-start=\"1996\" data-end=\"2004\">C2:C31<\/code>)<\/span><br data-start=\"1403\" data-end=\"1406\" \/><span style=\"font-size: 12pt\"><code data-start=\"1408\" data-end=\"1434\">=CORREL(B2:B31, C2:C31) \u2192 0.991<\/code><\/span><\/li>\n<li data-start=\"2046\" data-end=\"2203\">\n<p data-start=\"2048\" data-end=\"2055\"><span style=\"font-size: 12pt\"><strong data-start=\"2048\" data-end=\"2053\">R<\/strong><\/span><\/p>\n<div class=\"contain-inline-size rounded-2xl relative bg-token-sidebar-surface-primary\">\n<div class=\"sticky top-9\">\n<div class=\"absolute end-0 bottom-0 flex h-9 items-center pe-2\">\n<div class=\"bg-token-bg-elevated-secondary text-token-text-secondary flex items-center gap-4 rounded-sm px-2 font-sans text-xs\"><span style=\"font-size: 12pt\"><code class=\"whitespace-pre! language-r\" style=\"background-color: #ffffff\">dfB <span class=\"hljs-operator\">&lt;-<\/span> read.csv<span class=\"hljs-punctuation\">(<\/span><span class=\"hljs-string\">\"Chapter9_Dataset_B_OrderSize_vs_Fulfillment.csv\"<\/span><span class=\"hljs-punctuation\">)<\/span><br \/>\n<\/code><\/span><\/div>\n<div class=\"bg-token-bg-elevated-secondary text-token-text-secondary flex items-center gap-4 rounded-sm px-2 font-sans text-xs\"><span style=\"font-size: 12pt\"><code class=\"whitespace-pre! language-r\" style=\"background-color: #ffffff\">cor<span class=\"hljs-punctuation\">(<\/span>dfB<span class=\"hljs-operator\">$<\/span>ItemsInOrder<span class=\"hljs-punctuation\">,<\/span> dfB<span class=\"hljs-operator\">$<\/span>FulfillmentTimeDays<span class=\"hljs-punctuation\">)<\/span>  <span class=\"hljs-comment\"># [1] 0.9911393<\/span><\/code><\/span><span class=\"\" data-state=\"closed\"><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/li>\n<\/ul>\n<h3 data-start=\"2205\" data-end=\"2263\">Applied Example: Fulfillment Times at Pemi (Dataset A)<strong style=\"text-align: initial;font-size: 1em\" data-start=\"2264\" data-end=\"2273\">Excel<\/strong><\/h3>\n<div class=\"contain-inline-size rounded-2xl relative bg-token-sidebar-surface-primary\">\n<div class=\"overflow-y-auto p-4\" dir=\"ltr\">\n<p><span style=\"font-size: 12pt\"><span style=\"font-size: 12pt\"><code class=\"whitespace-pre! language-r\"><\/code><\/span><\/span><\/p>\n<p data-start=\"2276\" data-end=\"2312\"><span style=\"font-size: 12pt\">Mean: <code data-start=\"2282\" data-end=\"2300\">=AVERAGE(B2:B25)<\/code> \u2192 <code data-start=\"2303\" data-end=\"2310\">5.875<\/code><\/span><\/p>\n<p data-start=\"2276\" data-end=\"2312\"><span style=\"font-size: 12pt\"><code data-start=\"2303\" data-end=\"2310\"><\/code>Median: <code data-start=\"2323\" data-end=\"2340\">=MEDIAN(B2:B25)<\/code> \u2192 <code data-start=\"2343\" data-end=\"2346\">5<\/code><\/span><\/p>\n<p data-start=\"2276\" data-end=\"2312\"><span style=\"font-size: 12pt\"><code data-start=\"2343\" data-end=\"2346\"><\/code>Std. dev.: <code data-start=\"2362\" data-end=\"2380\">=STDEV.S(B2:B25)<\/code> \u2192 <code data-start=\"2383\" data-end=\"2390\">1.895<\/code><\/span><\/p>\n<p data-start=\"2276\" data-end=\"2312\"><code data-start=\"2383\" data-end=\"2390\"><\/code>Optional histogram: Data \u2192 Data Analysis \u2192 Histogram<\/p>\n<p><span style=\"font-size: 12pt\"><span style=\"font-size: 12pt\"><code class=\"whitespace-pre! language-r\"><\/code><\/span><\/span><\/p>\n<p data-start=\"2454\" data-end=\"2459\"><strong data-start=\"2454\" data-end=\"2459\">R<\/strong><\/p>\n<p><span style=\"font-size: 12pt\"><span style=\"font-size: 12pt\"><code class=\"whitespace-pre! language-r\"><\/code><\/span><\/span><\/p>\n<pre class=\"overflow-visible!\" data-start=\"2460\" data-end=\"2688\"><span style=\"font-size: 12pt\">dfA &lt;- read.csv(\"Chapter9_Dataset_A_FulfillmentTimes.csv\")<\/span>\n<span style=\"font-size: 12pt\">mean(dfA$FulfillmentTimeDays) # [1] 5.875<\/span>\n<span style=\"font-size: 12pt\">median(dfA$FulfillmentTimeDays) # [1] 5<\/span>\n<span style=\"font-size: 12pt\">sd(dfA$FulfillmentTimeDays) # [1] 1.895360<\/span>\n<span style=\"font-size: 12pt\">hist(dfA$FulfillmentTimeDays)<\/span><\/pre>\n<p><span style=\"font-size: 12pt\"><code class=\"whitespace-pre! language-r\"><\/code><\/span><\/p>\n<\/div>\n<\/div>\n<p data-start=\"2690\" data-end=\"2947\"><strong data-start=\"2690\" data-end=\"2708\">Interpretation<\/strong><br data-start=\"2708\" data-end=\"2711\" \/>The mean exceeds the median, indicating a right tail (a few slow orders). A standard deviation near 1.9 suggests noticeable variability. Next steps: isolate slow orders and investigate causes (capacity, supplier delays, address issues).<\/p>\n<h2 data-start=\"199\" data-end=\"272\"><strong data-start=\"202\" data-end=\"272\">9.3 Hypothesis testing: two-sample t-test (small vs. large orders)<\/strong><\/h2>\n<p data-start=\"274\" data-end=\"324\">Question: do larger orders take longer to fulfill?<\/p>\n<p data-start=\"326\" data-end=\"357\">Define two groups in Dataset B:<\/p>\n<ul data-start=\"358\" data-end=\"409\">\n<li data-start=\"358\" data-end=\"383\">\n<p data-start=\"360\" data-end=\"383\">Small orders: items \u2264 3<\/p>\n<\/li>\n<li data-start=\"384\" data-end=\"409\">\n<p data-start=\"386\" data-end=\"409\">Large orders: items &gt; 3<\/p>\n<\/li>\n<\/ul>\n<p data-start=\"411\" data-end=\"500\">Set up in <strong data-start=\"421\" data-end=\"430\">Excel<\/strong>: add a helper column <code data-start=\"452\" data-end=\"464\">size_group<\/code> with<br data-start=\"469\" data-end=\"472\" \/><code data-start=\"472\" data-end=\"500\">=IF(B2&lt;=3,\"Small\",\"Large\")<\/code><\/p>\n<p data-start=\"502\" data-end=\"593\">Compare mean fulfillment time by group using Welch\u2019s two-sample t-test (unequal variances).<\/p>\n<p data-start=\"595\" data-end=\"604\"><strong data-start=\"595\" data-end=\"604\">Excel<\/strong><\/p>\n<ol data-start=\"605\" data-end=\"945\">\n<li data-start=\"605\" data-end=\"722\">\n<p data-start=\"608\" data-end=\"722\">Compute group means and standard deviations with <code data-start=\"657\" data-end=\"672\">=AVERAGEIFS()<\/code> and <code data-start=\"677\" data-end=\"696\">=STDEV.S(IF(...))<\/code> or with filters\/pivots.<\/p>\n<\/li>\n<li data-start=\"723\" data-end=\"945\">\n<p data-start=\"726\" data-end=\"797\">Data \u2192 Data Analysis \u2192 t-Test: Two-Sample Assuming Unequal Variances.<\/p>\n<ul data-start=\"801\" data-end=\"945\">\n<li data-start=\"801\" data-end=\"838\">\n<p data-start=\"803\" data-end=\"838\">Variable 1 Range: times for Small<\/p>\n<\/li>\n<li data-start=\"842\" data-end=\"879\">\n<p data-start=\"844\" data-end=\"879\">Variable 2 Range: times for Large<\/p>\n<\/li>\n<li data-start=\"883\" data-end=\"918\">\n<p data-start=\"885\" data-end=\"918\">Hypothesized mean difference: 0<\/p>\n<\/li>\n<li data-start=\"922\" data-end=\"945\">\n<p data-start=\"924\" data-end=\"945\">Output to a new range<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<p data-start=\"947\" data-end=\"952\"><strong data-start=\"947\" data-end=\"952\">R<\/strong><\/p>\n<div class=\"contain-inline-size rounded-2xl relative bg-token-sidebar-surface-primary\">\n<div class=\"sticky top-9\">\n<div class=\"absolute end-0 bottom-0 flex h-9 items-center pe-2\">\n<div class=\"bg-token-bg-elevated-secondary text-token-text-secondary flex items-center gap-4 rounded-sm px-2 font-sans text-xs\"><span class=\"\" data-state=\"closed\"><\/span><\/div>\n<\/div>\n<\/div>\n<div class=\"overflow-y-auto p-4\" dir=\"ltr\"><code class=\"whitespace-pre! language-r\"><span>dfB <span class=\"hljs-operator\">&lt;-<\/span> read.csv<span class=\"hljs-punctuation\">(<\/span><span class=\"hljs-string\">\"Chapter9_Dataset_B_OrderSize_vs_Fulfillment.csv\"<\/span><span class=\"hljs-punctuation\">)<\/span><br \/>\ndfB<span class=\"hljs-operator\">$<\/span>size_group <span class=\"hljs-operator\">&lt;-<\/span> ifelse<span class=\"hljs-punctuation\">(<\/span>dfB<span class=\"hljs-operator\">$<\/span>ItemsInOrder <span class=\"hljs-operator\">&lt;=<\/span> <span class=\"hljs-number\">3<\/span><span class=\"hljs-punctuation\">,<\/span> <span class=\"hljs-string\">\"Small\"<\/span><span class=\"hljs-punctuation\">,<\/span> <span class=\"hljs-string\">\"Large\"<\/span><span class=\"hljs-punctuation\">)<\/span><br \/>\nt.test<span class=\"hljs-punctuation\">(<\/span>FulfillmentTimeDays <span class=\"hljs-operator\">~<\/span> size_group<span class=\"hljs-punctuation\">,<\/span> data <span class=\"hljs-operator\">=<\/span> dfB<span class=\"hljs-punctuation\">)<\/span>  <span class=\"hljs-comment\"># Welch by default<\/span><br \/>\n<\/span><\/code><\/div>\n<\/div>\n<p data-start=\"1169\" data-end=\"1212\">What you should see with the provided data:<\/p>\n<ul data-start=\"1213\" data-end=\"1298\">\n<li data-start=\"1213\" data-end=\"1262\">\n<p data-start=\"1215\" data-end=\"1262\">Difference in means \u2248 1.54 days (Large \u2212 Small)<\/p>\n<\/li>\n<li data-start=\"1263\" data-end=\"1286\">\n<p data-start=\"1265\" data-end=\"1286\">95% CI \u2248 [1.22, 1.86]<\/p>\n<\/li>\n<li data-start=\"1287\" data-end=\"1298\">\n<p data-start=\"1289\" data-end=\"1298\">p &lt; 0.001<\/p>\n<\/li>\n<\/ul>\n<p data-start=\"1300\" data-end=\"1451\">Interpretation: in this sample, larger orders take longer by about a day and a half on average. The effect is practically and statistically meaningful.<\/p>\n<p data-start=\"1453\" data-end=\"1551\">Transition: correlation and a simple linear model quantify the items \u2192 time relationship directly.<\/p>\n<h2 data-start=\"1558\" data-end=\"1609\"><strong data-start=\"1561\" data-end=\"1609\">9.4 Correlation and simple linear regression<\/strong><\/h2>\n<p data-start=\"1611\" data-end=\"1734\">Use correlation to summarize association and a simple linear regression to estimate how much time increases per added item.<\/p>\n<p data-start=\"1736\" data-end=\"1745\"><strong data-start=\"1736\" data-end=\"1745\">Excel<\/strong><\/p>\n<ul data-start=\"1746\" data-end=\"1958\">\n<li data-start=\"1746\" data-end=\"1788\">\n<p data-start=\"1748\" data-end=\"1788\">Correlation:<span style=\"font-size: 12pt\"> <code data-start=\"1761\" data-end=\"1786\">=CORREL(B2:B31, C2:C31)<\/code><\/span><\/p>\n<\/li>\n<li data-start=\"1789\" data-end=\"1866\">\n<p data-start=\"1791\" data-end=\"1866\">Scatterplot: Insert \u2192 Scatter. Add a trendline \u2192 display equation and R\u00b2.<\/p>\n<\/li>\n<li data-start=\"1867\" data-end=\"1958\">\n<p data-start=\"1869\" data-end=\"1958\"><span style=\"font-size: 12pt\">Functions: <code data-start=\"1880\" data-end=\"1904\">=SLOPE(C2:C31, B2:B31)<\/code>, <code data-start=\"1906\" data-end=\"1934\">=INTERCEPT(C2:C31, B2:B31)<\/code>, <code data-start=\"1936\" data-end=\"1958\">=RSQ(C2:C31, B2:B31)<\/code><\/span><\/p>\n<\/li>\n<\/ul>\n<p data-start=\"1960\" data-end=\"1965\"><strong data-start=\"1960\" data-end=\"1965\">R<\/strong><\/p>\n<div class=\"contain-inline-size rounded-2xl relative bg-token-sidebar-surface-primary\">\n<div class=\"sticky top-9\">\n<div class=\"absolute end-0 bottom-0 flex h-9 items-center pe-2\">\n<div class=\"bg-token-bg-elevated-secondary text-token-text-secondary flex items-center gap-4 rounded-sm px-2 font-sans text-xs\"><span class=\"\" data-state=\"closed\"><\/span><\/div>\n<\/div>\n<\/div>\n<div class=\"overflow-y-auto p-4\" dir=\"ltr\"><code class=\"whitespace-pre! language-r\"><span><span style=\"font-size: 12pt\">cor<span class=\"hljs-punctuation\">(<\/span>dfB<span class=\"hljs-operator\">$<\/span>ItemsInOrder<span class=\"hljs-punctuation\">,<\/span> dfB<span class=\"hljs-operator\">$<\/span>FulfillmentTimeDays<span class=\"hljs-punctuation\">)<\/span>      <span class=\"hljs-comment\"># \u2248 0.991<\/span><\/span><br \/>\n<span style=\"font-size: 12pt\"><br \/>\nm <span class=\"hljs-operator\">&lt;-<\/span> lm<span class=\"hljs-punctuation\">(<\/span>FulfillmentTimeDays <span class=\"hljs-operator\">~<\/span> ItemsInOrder<span class=\"hljs-punctuation\">,<\/span> data <span class=\"hljs-operator\">=<\/span> dfB<span class=\"hljs-punctuation\">)<\/span><\/span><br \/>\n<span style=\"font-size: 12pt\"><br \/>\nsummary<span class=\"hljs-punctuation\">(<\/span>m<span class=\"hljs-punctuation\">)<\/span><\/span><br \/>\n<\/span><\/code><\/div>\n<\/div>\n<p data-start=\"2105\" data-end=\"2161\">With the provided data, you should see results close to:<\/p>\n<ul data-start=\"2162\" data-end=\"2244\">\n<li data-start=\"2162\" data-end=\"2204\">\n<p data-start=\"2164\" data-end=\"2204\">Slope \u2248 0.430 days per additional item<\/p>\n<\/li>\n<li data-start=\"2205\" data-end=\"2231\">\n<p data-start=\"2207\" data-end=\"2231\">Intercept \u2248 4.232 days<\/p>\n<\/li>\n<li data-start=\"2232\" data-end=\"2244\">\n<p data-start=\"2234\" data-end=\"2244\">R\u00b2 \u2248 0.982<\/p>\n<\/li>\n<\/ul>\n<p data-start=\"2246\" data-end=\"2413\">Interpretation: each additional item adds roughly 0.43 days on average. The fit is strong in this sample. Do not extrapolate beyond the observed range without a check.<\/p>\n<p data-start=\"2415\" data-end=\"2514\">Transition: before relying on these results, confirm basic assumptions and scan for obvious issues.<\/p>\n<h2 data-start=\"2521\" data-end=\"2565\"><strong data-start=\"2524\" data-end=\"2565\">9.5 Assumptions and quick diagnostics<\/strong><\/h2>\n<p data-start=\"2567\" data-end=\"2662\">Hypothesis tests and linear models rest on simple conditions. A few quick checks go a long way.<\/p>\n<p data-start=\"2664\" data-end=\"2688\">t-test (Small vs. Large)<\/p>\n<ul data-start=\"2689\" data-end=\"3043\">\n<li data-start=\"2689\" data-end=\"2781\">\n<p data-start=\"2691\" data-end=\"2781\">Independence: orders should be independent observations. Segment by shift\/promo if needed.<\/p>\n<\/li>\n<li data-start=\"2782\" data-end=\"2914\">\n<p data-start=\"2784\" data-end=\"2914\">Shape: with n \u2248 15 per group, the t-test is robust, but heavy skew\/outliers can distort results. Use group histograms or boxplots.<\/p>\n<\/li>\n<li data-start=\"2915\" data-end=\"3043\">\n<p data-start=\"2917\" data-end=\"3043\">Variances: Welch\u2019s test (used above) does not assume equal variances. If you suspect very different spreads, keep using Welch.<\/p>\n<\/li>\n<\/ul>\n<p data-start=\"3045\" data-end=\"3073\"><strong data-start=\"3045\" data-end=\"3073\">Excel (practical checks)<\/strong><\/p>\n<ul data-start=\"3074\" data-end=\"3371\">\n<li data-start=\"3074\" data-end=\"3142\">\n<p data-start=\"3076\" data-end=\"3142\">Build small and large group histograms or boxplots (via charts).<\/p>\n<\/li>\n<li data-start=\"3143\" data-end=\"3190\">\n<p data-start=\"3145\" data-end=\"3190\">Compute group variances with <code data-start=\"3174\" data-end=\"3187\">=VAR.S(...)<\/code>.<\/p>\n<\/li>\n<li data-start=\"3191\" data-end=\"3371\">\n<p data-start=\"3193\" data-end=\"3371\">If you ran the regression, add predicted values <span style=\"font-size: 12pt\"><code data-start=\"3241\" data-end=\"3267\">=intercept + slope*Items<\/code> and create a residuals column <code data-start=\"3298\" data-end=\"3319\">=Actual - Predicted<\/code>. Plot residuals vs. p<\/span>redicted; look for no pattern.<\/p>\n<\/li>\n<\/ul>\n<p data-start=\"3373\" data-end=\"3395\"><strong data-start=\"3373\" data-end=\"3395\">R (quick commands)<\/strong><\/p>\n<div class=\"contain-inline-size rounded-2xl relative bg-token-sidebar-surface-primary\">\n<div class=\"sticky top-9\">\n<div class=\"absolute end-0 bottom-0 flex h-9 items-center pe-2\">\n<div class=\"bg-token-bg-elevated-secondary text-token-text-secondary flex items-center gap-4 rounded-sm px-2 font-sans text-xs\"><span class=\"\" data-state=\"closed\"><\/span><\/div>\n<\/div>\n<\/div>\n<div class=\"overflow-y-auto p-4\" dir=\"ltr\"><code class=\"whitespace-pre! language-r\"><span><span style=\"font-size: 12pt\">boxplot<span class=\"hljs-punctuation\">(<\/span>FulfillmentTimeDays <span class=\"hljs-operator\">~<\/span> size_group<span class=\"hljs-punctuation\">,<\/span> data <span class=\"hljs-operator\">=<\/span> dfB<span class=\"hljs-punctuation\">)<\/span><\/span><br \/>\n<span style=\"font-size: 12pt\"><br \/>\nby<span class=\"hljs-punctuation\">(<\/span>dfB<span class=\"hljs-operator\">$<\/span>FulfillmentTimeDays<span class=\"hljs-punctuation\">,<\/span> dfB<span class=\"hljs-operator\">$<\/span>size_group<span class=\"hljs-punctuation\">,<\/span> sd<span class=\"hljs-punctuation\">)<\/span>  <span class=\"hljs-comment\"># group spreads<\/span><\/span><br \/>\n<span class=\"hljs-comment\" style=\"font-size: 12pt\"><br \/>\n# Regression diagnostics<\/span><br \/>\n<span style=\"font-size: 12pt\"><br \/>\nplot<span class=\"hljs-punctuation\">(<\/span>m<span class=\"hljs-punctuation\">,<\/span> which <span class=\"hljs-operator\">=<\/span> <span class=\"hljs-number\">1<\/span><span class=\"hljs-punctuation\">)<\/span>  <span class=\"hljs-comment\"># Residuals vs Fitted<\/span><\/span><br \/>\n<span style=\"font-size: 12pt\"><br \/>\nplot<span class=\"hljs-punctuation\">(<\/span>m<span class=\"hljs-punctuation\">,<\/span> which <span class=\"hljs-operator\">=<\/span> <span class=\"hljs-number\">2<\/span><span class=\"hljs-punctuation\">)<\/span>  <span class=\"hljs-comment\"># Normal Q-Q<\/span><\/span><br \/>\n<\/span><\/code><\/div>\n<\/div>\n<p data-start=\"3625\" data-end=\"3635\">Regression<\/p>\n<ul data-start=\"3636\" data-end=\"3923\">\n<li data-start=\"3636\" data-end=\"3706\">\n<p data-start=\"3638\" data-end=\"3706\">Linearity: residuals vs. fitted should look like noise, not a curve.<\/p>\n<\/li>\n<li data-start=\"3707\" data-end=\"3814\">\n<p data-start=\"3709\" data-end=\"3814\">Constant variance: avoid funnel shapes; consider a variance-stabilizing approach or segmenting if needed.<\/p>\n<\/li>\n<li data-start=\"3815\" data-end=\"3923\">\n<p data-start=\"3817\" data-end=\"3923\">Influence: in small samples, a single unusual order can sway the fit; inspect points with large residuals.<\/p>\n<\/li>\n<\/ul>\n<p data-start=\"3925\" data-end=\"4021\">Transition: with checks in place, synthesize findings in a brief memo that a manager can act on.<\/p>\n<h2 data-start=\"4028\" data-end=\"4068\"><strong data-start=\"4031\" data-end=\"4068\">9.6 Applied assignment: Pemi memo<\/strong><\/h2>\n<p data-start=\"4070\" data-end=\"4181\">Goal: use the chapter workflow to answer a specific operations question and communicate a clear recommendation.<\/p>\n<p data-start=\"4183\" data-end=\"4191\">Datasets<\/p>\n<ul data-start=\"4192\" data-end=\"4285\">\n<li data-start=\"4192\" data-end=\"4235\">\n<p data-start=\"4194\" data-end=\"4235\">Chapter9_Dataset_A_FulfillmentTimes.csv<\/p>\n<\/li>\n<li data-start=\"4236\" data-end=\"4285\">\n<p data-start=\"4238\" data-end=\"4285\">Chapter9_Dataset_B_OrderSize_vs_Fulfillment.csv<\/p>\n<\/li>\n<\/ul>\n<p data-start=\"4287\" data-end=\"4292\">Tasks<\/p>\n<ol data-start=\"4293\" data-end=\"4970\">\n<li data-start=\"4293\" data-end=\"4537\">\n<p data-start=\"4296\" data-end=\"4303\">Excel<\/p>\n<ul data-start=\"4307\" data-end=\"4537\">\n<li data-start=\"4307\" data-end=\"4387\">\n<p data-start=\"4309\" data-end=\"4387\">Dataset A: mean, median, standard deviation; histogram of fulfillment times.<\/p>\n<\/li>\n<li data-start=\"4391\" data-end=\"4472\">\n<p data-start=\"4393\" data-end=\"4472\">Dataset B: correlation (items vs. time); two-sample t-test (Small vs. Large).<\/p>\n<\/li>\n<li data-start=\"4476\" data-end=\"4537\">\n<p data-start=\"4478\" data-end=\"4537\">Optional: scatterplot with trendline, show equation and R\u00b2.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"4539\" data-end=\"4707\">\n<p data-start=\"4542\" data-end=\"4545\">R<\/p>\n<ul data-start=\"4549\" data-end=\"4707\">\n<li data-start=\"4549\" data-end=\"4654\">\n<p data-start=\"4551\" data-end=\"4654\">Replicate Excel results with <span style=\"font-size: 12pt\"><code data-start=\"4580\" data-end=\"4588\">mean()<\/code>, <code data-start=\"4590\" data-end=\"4600\">median()<\/code>, <code data-start=\"4602\" data-end=\"4608\">sd()<\/code>, <code data-start=\"4610\" data-end=\"4618\">hist()<\/code>, <code data-start=\"4620\" data-end=\"4627\">cor()<\/code>, <code data-start=\"4629\" data-end=\"4639\">t.test()<\/code>, and <code data-start=\"4645\" data-end=\"4651\">lm()<\/code>.<\/span><\/p>\n<\/li>\n<li data-start=\"4658\" data-end=\"4707\">\n<p data-start=\"4660\" data-end=\"4707\">Confirm that Excel and R agree within rounding.<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<li data-start=\"4709\" data-end=\"4970\">\n<p data-start=\"4712\" data-end=\"4746\">Managerial memo (1\u20132 paragraphs)<\/p>\n<ul data-start=\"4750\" data-end=\"4970\">\n<li data-start=\"4750\" data-end=\"4810\">\n<p data-start=\"4752\" data-end=\"4810\">Is the difference in mean time by size group meaningful?<\/p>\n<\/li>\n<li data-start=\"4814\" data-end=\"4868\">\n<p data-start=\"4816\" data-end=\"4868\">About how much time does each additional item add?<\/p>\n<\/li>\n<li data-start=\"4872\" data-end=\"4970\">\n<p data-start=\"4874\" data-end=\"4970\">One concrete action you recommend (e.g., staffing during peaks, batching rules, target setting).<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<p data-start=\"4972\" data-end=\"4984\">Deliverables<\/p>\n<ul data-start=\"4985\" data-end=\"5167\">\n<li data-start=\"4985\" data-end=\"5067\">\n<p data-start=\"4987\" data-end=\"5067\">One Excel file with clearly labeled sheets (A_Summary, B_ttest, B_Regression).<\/p>\n<\/li>\n<li data-start=\"5068\" data-end=\"5145\">\n<p data-start=\"5070\" data-end=\"5145\">One R script (.R) or R Markdown (.Rmd) with commands and printed outputs.<\/p>\n<\/li>\n<li data-start=\"5146\" data-end=\"5167\">\n<p data-start=\"5148\" data-end=\"5167\">A PDF or DOCX memo.<\/p>\n<\/li>\n<\/ul>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Key Takeaways<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ul>\n<li data-start=\"230\" data-end=\"319\">\n<p data-start=\"232\" data-end=\"319\">Mean and median summarize central tendency; the median is less sensitive to outliers.<\/p>\n<\/li>\n<li data-start=\"320\" data-end=\"431\">\n<p data-start=\"322\" data-end=\"431\">Variance and standard deviation measure variability around the mean, indicating consistency in performance.<\/p>\n<\/li>\n<li data-start=\"432\" data-end=\"509\">\n<p data-start=\"434\" data-end=\"509\">Correlation quantifies the strength of association between two variables.<\/p>\n<\/li>\n<li data-start=\"510\" data-end=\"618\">\n<p data-start=\"512\" data-end=\"618\">A two-sample t-test compares group means and can show whether differences are statistically significant.<\/p>\n<\/li>\n<li data-start=\"619\" data-end=\"744\">\n<p data-start=\"621\" data-end=\"744\">Linear regression estimates the relationship between predictors (e.g., order size) and outcomes (e.g., fulfillment time).<\/p>\n<\/li>\n<li data-start=\"745\" data-end=\"856\">\n<p data-start=\"747\" data-end=\"856\">Diagnostic checks (shape, variance, residuals) ensure statistical results are reliable for decision-making.<\/p>\n<\/li>\n<li data-start=\"857\" data-end=\"978\">\n<p data-start=\"859\" data-end=\"978\">Excel and R can be used together to calculate, visualize, and cross-verify these statistics in an operations context.<\/p>\n<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<h2>Chapter Nine References<\/h2>\n<p data-start=\"207\" data-end=\"317\">Microsoft. (n.d.). <em data-start=\"226\" data-end=\"262\">Statistical functions (reference).<\/em> Microsoft.<a href=\"https:\/\/support.microsoft.com\/en-us\/excel\"> https:\/\/support.microsoft.com\/en-us\/excel<\/a><\/p>\n<p data-start=\"320\" data-end=\"467\">R Core Team. (2023). <em data-start=\"341\" data-end=\"399\">R: A language and environment for statistical computing.<\/em> R Foundation for Statistical Computing. <a href=\"https:\/\/www.r-project.org\">https:\/\/www.r-project.org<\/a><span aria-hidden=\"true\" class=\"ms-0.5 inline-block align-middle leading-none\"><\/span><\/p>\n<p data-start=\"470\" data-end=\"621\">Wickham, H., &amp; Grolemund, G. (2017). <em data-start=\"507\" data-end=\"580\">R for data science: Import, tidy, transform, visualize, and model data.<\/em> O\u2019Reilly Media. <a data-start=\"597\" data-end=\"619\" rel=\"noopener\" target=\"_new\" class=\"decorated-link cursor-pointer\" href=\"https:\/\/r4ds.had.co.nz\">https:\/\/r4ds.had.co.nz<\/a><\/p>\n","protected":false},"author":1,"menu_order":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-53","chapter","type-chapter","status-publish","hentry"],"part":51,"_links":{"self":[{"href":"https:\/\/openbooks.spmvv.ac.in\/businessopsanalytics\/wp-json\/pressbooks\/v2\/chapters\/53","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/openbooks.spmvv.ac.in\/businessopsanalytics\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/openbooks.spmvv.ac.in\/businessopsanalytics\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/openbooks.spmvv.ac.in\/businessopsanalytics\/wp-json\/wp\/v2\/users\/1"}],"version-history":[{"count":1,"href":"https:\/\/openbooks.spmvv.ac.in\/businessopsanalytics\/wp-json\/pressbooks\/v2\/chapters\/53\/revisions"}],"predecessor-version":[{"id":54,"href":"https:\/\/openbooks.spmvv.ac.in\/businessopsanalytics\/wp-json\/pressbooks\/v2\/chapters\/53\/revisions\/54"}],"part":[{"href":"https:\/\/openbooks.spmvv.ac.in\/businessopsanalytics\/wp-json\/pressbooks\/v2\/parts\/51"}],"metadata":[{"href":"https:\/\/openbooks.spmvv.ac.in\/businessopsanalytics\/wp-json\/pressbooks\/v2\/chapters\/53\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/openbooks.spmvv.ac.in\/businessopsanalytics\/wp-json\/wp\/v2\/media?parent=53"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/openbooks.spmvv.ac.in\/businessopsanalytics\/wp-json\/pressbooks\/v2\/chapter-type?post=53"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/openbooks.spmvv.ac.in\/businessopsanalytics\/wp-json\/wp\/v2\/contributor?post=53"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/openbooks.spmvv.ac.in\/businessopsanalytics\/wp-json\/wp\/v2\/license?post=53"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}