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Homework 12 Due Tuesday April 19 11 59pm STAT 400 Spring 2022 D Unger Each Exercise or lettered part of an Exercise is worth 5 points Homework assignments are worth 50 points Exercise 1 Solution Solution A hamburger claiming to be a Quarter Pounder should have a beef patty that weighs a quarter of a pound i e 4 ounces If it weighs more than that the restaurant is giving away more beef than they need to If it weighs less than that the customer is getting ripped off In a random sample of 11 Quarter Pounder with Cheese hamburgers at McDowell s the average weight of the beef patty was 3 90 ounces with a sample standard deviation of 0 12 ounces Assume the weights are normally distributed a Define the population parameter of interest State an appropriate null and alternative hypothesis for this situation the average weight of the beef in all Quarter Pound burgers at McDowell s H0 cid 3404 4 vs H1 cid 3405 4 b Calculate the test statistic for testing the null hypothesis you defined in part a With the population standard deviation unknown the test statistic is cid 3404 cid 3051 cid 2879 cid 3091 cid 3046 cid 3041 cid 4666 cid 3398 1 cid 4667 cid 3404 cid 4666 10 cid 4667 cid 3404 cid 3398 cid 3404 3 90 cid 3398 4 0 12 11 cid 3404 cid 3398 0 1 0 0362 cid 3404 cid 3398 2 764 c Calculate the p value for the test statistic Solution p value P observing a value as or more extreme than 2 764 2 P T10 2 764 2 0 01 0 02 d Report your decision regarding the null hypothesis with a level of significance of 0 05 Provide a concluding statement in the context of this situation Since p value 0 02 0 05 we reject H0 There is significant evidence to suggest that the average weight of the beef in all Quarter Pound burgers at McDowell s is not 4 ounces e In Homework 9 you constructed a 95 two sided confidence interval for the overall average weight of beef in a Quarter Pound burger at McDowell s Using that as evidence for the hypothesis test in part a do you come to the same conclusion that you did in part d Explain Solution Solution From the earlier assignment we found that we are 95 confident that the overall average weight of beef in a Quarter Pound burger at McDowell s is between 3 82 and 3 98 ounces Since the null hypothesis value of 4 ounces is not in the 95 confidence interval then it is not a plausible value for and we would similarly reject H0 at the complementary 0 05 significance level Exercise 2 A nationwide study showed that undergraduate students in the United States spend about 7 1 hours on average studying on weekends The study also revealed the population standard deviation to be 2 3 hours A random sample of 60 Illinois undergraduate students are selected to answer the following question How many hours do you spend on studying during the weekend The sample average of weekend studying time is 6 8 hours Assume the number of hours each student spends studying on the weekends is independent from others Conduct a hypothesis test to determine whether the weekend study times at Illinois are at least as many as their fellow students nationwide or not a State the null and alternative hypotheses calculate the test statistic and provide the p value Solution Step 1 the average hours spent studying on weekends by Illinois undergraduate students H0 7 1 vs H1 7 1 Step 2 cid 3364 the sample mean of study hours cid 3364 Normal 2 n with known We will use cid 3404 cid 3025 cid 3364 cid 2879 cid 3091 cid 3097 cid 3041 Step 3 cid 3404 cid 3398 cid 3404 6 8 cid 3398 7 1 2 3 60 cid 3404 cid 3398 0 3 0 297 cid 3404 cid 3398 1 01 p value P Z 1 01 0 1562 b Using a level of significance of 0 10 make a decision about the null hypothesis and provide a concluding statement in the context of this situation Step 4 Since p value 0 1562 0 10 we fail to reject H0 Step 5 There is not significant evidence to suggest that the average time spent studying on weekends by Illinois undergraduate students is less than 7 1 hours c State the rejection region for this test using a level of significance of 0 10 Based on your test statistic would you make the same decision as you did with the p value in part b Explain Solution Solution We would reject H0 for any test statistic z such that z z0 10 cid 3404 1 28 Since our test statistic is z 1 01 we would we fail to reject H0 This is the same decision in part b Exercise 3 A little girl named Sami is normally not allowed to have soda But for her 7th birthday she is allowed to have as much as she wants The restaurant serves drinks in pint glasses 16oz but Sami won t always consume a full pint of soda due to ice and refills On her birthday in one sitting she consumed the following quantities of a pint 16 oz of soda 15 4 oz 15 0 oz 15 1 oz 15 3 oz 15 2 oz Suppose that the amount of soda in a pint glass follows a Normal 2 distribution and that Sami s five drinks represent a random sample a Construct a 95 one sided upper bound confidence interval for the true mean volume in ounces of a pint of soda Interpret the confidence interval within the context of the exercise A 1 100 one sided upper bound confidence interval for is Solution cid 4672 cid 3398 cid 3397 cid 3080 cid 4666 cid 3398 1 cid 4667 cid 3046 cid 3041 cid 4673 The 95 upper bound is cid 3397 cid 3080 cid 4666 cid 3398 1 cid 4667 cid 3404 15 2 cid 3397 cid 2868 cid 2868 cid 2873 cid 4666 4 cid 4667 cid 3404 15 2 cid 3397 2 132 0 1581 5 0 1581 5 cid 3404 15 2 cid 3397 0 151 cid 3404 15 351 We are 95 confident that the true mean volume of a pint of soda is no more than 15 351 ounces b A one sided upper bound confidence interval can be used as evidence in hypothesis tests when the alternative hypothesis suggests that the population parameter is less than a claimed value For example suppose the restaurant claims that they put at least 15 5 oz of soda in each pint glass Thus we wish to test H0 15 5 versus H1 15 5 Using the confidence interval you constructed in part a make a decision about this test at the 0 05 level of significance Provide a concluding statement in the context of this situation Solution Since the null hypothesis value of 15 5 ounces …

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