STAT 400 Lecture AL1 1 Practice Problems 7 8 Part 2 Spring 2015 Dalpiaz Taxes R Us Block is a national chain that prepares income tax returns for individuals The president of the company wants an average number of 5 customers per clerk per day On April 15 a random sample of 64 clerks shows the average number of customers per clerk at 4 75 Assume that the overall standard deviation of 1 15 a Test whether the average number of customers per clerk on April 15 was 5 or it was different Use 0 05 b Suppose the actual overall average number of customers per clerk on April 15 was 4 8 Was the decision you made in part a correct If not what type of error has been made c What is the p value of the test in part a d Using the p value obtained in part c would you accept or reject the null hypothesis from part a at 0 07 2 In its advertising Taxes R Us Block claims that the overall average bill for its services does not exceed 75 A consumer agency obtains a random sample of 20 tax returns prepared by the company and finds the average preparation bill for those 20 to be 79 82 with a sample standard deviation of 10 30 Assume the amounts of the service fees are normally distributed a We would like to test the claim made in the ad Perform the appropriate test at a 10 significance level b What is the p value of the test in part a c Was the decision you made in part a correct If not what type of error has been made d Construct a 95 confidence interval for the average bill for services at this firm 3 Taxes R Us Block is currently under an IRS investigation for possible underreporting of income by its customers on federal income tax returns a A random sample of 225 Taxes R Us Block s customers is obtained of which 72 underreported their income Construct a 90 confidence interval for the proportion of all Taxes R Us Block s customers who underreport their income b IRS claims that at least 35 of all Taxes R Us Block s customers underreported their income Perform the appropriate test using a 5 level of significance c What is the p value of the test in part b d Find the minimum sample size required for estimating the proportion of all Taxes R Us Block s customers who underreport their income to within 3 with 90 confidence if it is known that this proportion is between 0 25 and 0 35 4 In its advertising Taxes R Us Block claims that the average tax return for a family with two children is at least 700 A random sample of 81 tax returns shows a sample mean of 674 and a sample standard deviation of 117 a Perform the appropriate test at a 1 level of significance b What is the p value of the test in part a c Using the p value obtained in part b would you accept or reject the null hypothesis from part a at 0 04 5 Suppose the time it takes to prepare a tax return at Taxes R Us Block is normally distributed with mean 27 minutes and standard deviation 5 minutes a What proportion of tax returns take more than 30 minutes to prepare b What is the probability that the average preparation time for a random sample of 16 customers will be greater than 29 minutes Answers 1 15 1 X 4 75 a Claims 5 average number of customers per clerk was 5 n 64 and 5 it was different H0 5 vs H 1 5 Z is known Test Statistic Z z Rejection Region 0 05 Two tail 2 X 0 n or Z z 4 75 5 1 74 1 15 64 2 z 0 025 1 960 2 0 025 The value of the test statistic does not fall into the Rejection Region Accept H 0 Do not reject H 0 at 0 05 OR The confidence interval X z is known 0 05 z 1 960 95 conf level 4 75 1 960 1 15 4 75 0 28 64 Since 5 is covered by the confidence interval 2 4 47 5 03 Accept H 0 Do not reject H 0 at 0 05 b Given 4 8 4 8 makes H 0 false Yet in part a H 0 was not rejected H 0 true Do not Reject H 0 Reject H 0 H 0 false Type II Error Type I Error Therefore Type II Error was made 2 n c p value 2 P Z 1 74 2 1 74 2 0 0409 0 0818 d p value Accept H 0 p value Reject H 0 Since 0 0818 0 07 Accept H 0 Do not reject H 0 at 0 07 s 10 30 2 X 79 82 a Claim 75 overall average bill does not exceed 75 H 0 75 vs Test Statistic Rejection Region n 20 H 1 75 is unknown Right tail T X 0 s n 79 82 75 2 093 10 30 20 T t number of degrees of freedom n 1 20 1 19 0 10 t 0 100 1 328 The value of the test statistic does fall into the Rejection Region Reject H 0 at 0 10 b p value area to the right of T 2 093 19 degrees of freedom 0 025 c Since we do not know the value of the overall average preparation bill we do not know whether the decision in part a is correct or not H 0 true Do not Reject H 0 Reject H 0 H 0 false Type II Error Type I Error If H 0 is FALSE then the decision is correct If H 0 is TRUE then the Type I Error is made d is unknown X t The confidence interval 2 s n number of degrees of freedom n 1 20 1 19 95 conf level t 0 025 2 093 79 82 2 093 10 30 79 82 4 82 20 p 3 X 72 n 225 a The confidence interval b X n p z 2 72 0 32 225 p 1 p n z 0 05 1 645 90 conf level 0 32 1 645 75 00 84 64 0 32 0 68 225 0 32 0 051 0 269 0 371 Claim p 0 35 at least 35 of all Taxes R Us Block s customers H 0 p 0 35 Z vs H 1 p 0 35 Left tail 0 05 p p0 0 32 0 35 0 94 0 35 0 65 p0 1 p0 225 n Rejection Region z 0 05 1 645 Z z The value of the test statistic does not fall into the Rejection Region Accept H 0 Do not reject H 0 at 0 10 c p value P Z 0 94 0 94 0 1736 d It is known that this proportion is between 0 …
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