Stats Midterm 1 - Student Name Discussion Section Number or...

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Unformatted text preview: Student Name: Discussion Section Number or TA Name: Student ID: Midterm 1: Form B 6E:071 Spring 2007 Please write legibly. There are two forms for this exam. If an answer from the other form appears on this exam, we will consider that evidence that you copied from another exam and you will receive a grade of zero on the entire exam. You may use any calculator that the College Board allows and one cheat sheet, 8.5 x 11. Please be sure to sign the honor pledge after you complete the exam. On my honor, I pledge that during this examination I neither gave nor received any assistance, and I did not witness any violation(s) of the honor code. Signature: 1) Fast food restaurants are interested in delivering accurate orders to their customers both for cost savings and for customer service. In 2002, a study was conducted by a marketing firm working for QSR Magazine in which several fast food restaurants were visited by mystery customers. Each customer ordered a similar meal, and then recorded whether the meal they received was what they ordered (accurate) or not what they ordered (inaccurate). QSR Magazine was interested in whether the proportion of accurate orders was different for different fast food restaurants. a) The analyst for the firm decided to conduct a chi-squared analysis. Does that seem appropriate here? Why or why not? 4 points. This is an appropriate technique to use because the data is categorical and we are trying to decide whether there is a relationship between the two different categorical variables. b) The analyst used Minitab and produced the following data, although some data was garbled. Fill in the four missing values in the chi-square table: Chi-Square Test: Inaccurate Order, Accurate Order Inaccurate Accurate Order Order Jack in 32 238 the Box 46.44 223.56 4.492 0.933 Church’s Chicken KFC Long John Silver’s Total Total 270 38 35.78 0.138 170 172.22 0.029 208 126 112.50 1.621 528 541.50 0.337 654 40 41.28 0.040 200 198.72 0.008 240 236 1136 1372 One point per answer, for four points total. Chi-Sq = 7.597, DF = 3, P-Value = 0.055 c) What is the null hypothesis that is being tested (in symbols or plain English, but be specific)? 6 points. H0: The proportion of accurate and inaccurate orders does not vary across the restaurants. H0: pJack = pChurch = pKFC = pLong. d) Which restaurant had more orders than expected that were labeled “accurate”? 2 Points. Jack in the Box and Long John Silver’s. e) Errors i) If we test at the 10% significance level, should we reject or fail to reject the null hypothesis? In plain English, what does this mean? 6 Points. We should reject the null hypothesis. The accuracy of orders does vary across the restaurants. Page 2 of 6 ii) Explain what a Type I error is for this test. At a 10% significance level, what is the probability of committing a Type I error? 5 Points. A Type I error occurs when we reject the null hypothesis even though it is true. Here, we’d conclude that accuracy differs across the restaurants when it really doesn’t. The probability is equal to the significance level, so it’s 10%. iii) Explain what a Type II error is for this test. If we decrease the significance level of the test to 5%, what happens to the probability of a Type II error? 5 Points. A Type II error occurs when we fail to reject the null hypothesis even though it is false. Here, we’d conclude that accuracy does not differ across the restaurants when it really does. If we reduce the sig. level, we increase the probability of a type II error. iv) Which error do you believe to be more costly for this situation? Based on your answer, which significance level should you choose, 5% or 10%? 5 Points. The answer can go either way, but must be consistent. Pick 5% if Type I is more costly. Pick 10% if Type II is more costly. f) Which entry had the biggest influence on the test results? If you ran that particular fast food chain, what might you do with this information? 10 Points. Jack in the Box had many fewer inaccurate orders than expected, and this entry had the biggest influence (contribution = 4.492). I would use this information to launch a marketing campaign to say that our restaurants delivered what the customer ordered. 2) Americans consume an average of 13 pounds of ready-to-eat cereal per person per year; most grocery stores devote an entire aisle to breakfast cereals. Shelf placement is a critical issue for both manufacturers and grocery store managers. Do stores place cereals strategically? In general, shelf 2 is thought to be reserved for children's cereals, while the top shelf (3) is the most sought-after space. We can examine this by looking at the placement of cereals based on their nutritional value. In particular, we want to know whether mean vitamin content of cereals differs by shelf. We have a random sample of the placement and nutritional content of 77 cereals at one grocery store. Minitab produces the following partial results. Complete the three missing entries in the ANOVA table: Page 3 of 6 One-way ANOVA: Vitamins versus Shelf Source Shelf Error Total DF 2 74 76 S = 21.53 Level 1 2 3 N 20 21 36 SS 3624 34314 37938 MS F 1812 3.91 463.70 R-Sq = 9.55% Mean 20.00 23.81 35.42 StDev 10.26 5.46 30.10 P 0.024 3 Points. One point per answer R-Sq(adj) = 7.11% Individual 95% CIs For Mean Based on Pooled StDev +---------+---------+---------+--------(---------*---------) (---------*--------) (------*-------) +---------+---------+---------+--------10 20 30 40 a) How many populations are there in this study? Provide a description of the populations. 6 Points. There are three populations: Cereals put on Shelf 1, Cereals put on Shelf 2 and Cereals put on Shelf 3. b) What null hypothesis is being tested? 2 Points. H0: the population mean vitamin content is the same across the three shelves. H0: mu(1) = mu(2) = mu(3), where mu(i) is the population mean vitamin content of cereals on shelf i. c) What is the formula for the f-statistic? Explain why a large f-statistic provide evidence against the null hypothesis. 8 Points. F = MSG/MSE = (SSG/DFG)/(SSE/DFE). A large f-statistic means that most of the variance in the data can be attributed to variance across the group means rather than to variance within the groups. d) At a 5% significance level, should we reject or fail to reject the null hypothesis? In plain English, explain your conclusion. 6 Points. The p-value is less than 5%, so we should reject the null hypothesis. We can conclude that the mean vitamin content of cereals is different across the three shelves. Page 4 of 6 The following output was produced by Minitab using the Tukey procedure for multiple comparisons: Tukey 95% Simultaneous Confidence Intervals All Pairwise Comparisons among Levels of Shelf Individual confidence level = 98.06% Shelf = 1 subtracted from: Shelf 2 3 Lower -12.27 1.06 Center 3.81 15.42 Upper 19.89 29.77 -------+---------+---------+---------+-(----------*---------) (--------*---------) -------+---------+---------+---------+--15 0 15 30 Shelf = 2 subtracted from: Shelf 3 Lower -2.52 Center 11.61 Upper 25.74 -------+---------+---------+---------+-(---------*--------) -------+---------+---------+---------+--15 0 15 30 e) Suppose we wish to test the null hypothesis that mean vitamin content for cereals on shelf 1 is the same as mean vitamin content for cereals on shelf 2. i) State the null hypothesis in symbols, using μ1 for mean vitamin content for cereals on shelf 1 and μ2 for mean vitamin content for cereals on shelf 2. 4 Points. H0: μ2 = μ1 ii) What is the confidence interval that is relevant to this hypothesis? 2 Points. [-12.27,19.88] iii) What conclusion do you reach about the hypothesis (reject or fail to reject)? Explain your answer. 6 Points. Fail to reject the null hypothesis because the confidence interval contains zero. There is no evidence that there is a difference in mean vitamin content between shelves 1 and 2. Page 5 of 6 f) Given the confidence intervals above, which shelf has the highest vitamin content? How do you know? 4 Points. Based on the confidence intervals, we can conclude that shelf 3 cereals have higher mean vitamin content than shelf 1 cereals, but the other comparisons show that shelf 1 and 2 are the same and shelf 2 and 3 are the same in terms of mean vitamin content. So the most we can say is that 3 is higher than 1. g) Suppose we were interested in testing whether the vitamin content of cereals on shelf three is different from the average vitamin content of cereals on shelves 1 and 2. Write a contrast that is relevant for testing this hypothesis and calculate the sample value of the contrast and its standard error. 10 Points. The contrast is ψ = μ3 – ½(μ1 + μ2). The sample value is 13.515 and the standard error is 4.918. Page 6 of 6 ...
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