STAT work
STAT 3011 Homework 7 (Due: Tuesday April, 2, 2024) Spring 2024 Chapter covered: Lecture Notes Chapters 9/10 Show your work for full credit! Problem 1. Name the tests For each part below answer (i) State the name of relevant hypothesis test: • One sample z-test for proportion p • One sample t-test for mean µ • Matched pairs t-test for the mean of difference µD • Independent two-sample t-test for difference of means µ1 − µ2 (ii) Define parameter(s) of interest and state the null and alternative hypotheses. No need to conduct 5 step hypothesis test. Example: To test the claim that Startbucks Coffee’s daily sale in Twin Cities is higher than that in Des Moines, IA, you randomly select 30 stores from each location and calculate the sample daily sales mean for each state. Answer : (i) Independent two-sample t-test for difference of means. µ1 : mean daily sale of Starbucks coffee’s in Twin Cities µ2 : mean daily sale of Starbucks coffee’s in Des Moines, IA H0 : µ1 = µ2 vs H1 : µ1 > µ2 (a) The Drug Lipitor is meant to reduce total cholesterol and LDL cholesterol. In clinical trials, 19 out of 863 patients taking 10 mg of Lipitor daily complained of flulike symptoms. Is there evidence to conclude that less than 3.8% of Lipitor users experience flulike symptoms as a side effect at the α = 0.01 level of significance? STAT 3011 Homework 7 (Due: Tuesday April, 2, 2024) Spring 2024 (b) Researchers at the University of Mississippi wanted to determine whether the reaction time (in seconds) of males is less than that of females for a go/no-go stimulus. The researchers independently randomly selected 20 females and 15 males to participate in the study. The go/no stimulus required the student to respond to a particular stimulus and not to respond to other stimuli. (c) To test the belief that sons are not of the same height as their fathers, a student randomly selects 13 fathers who have adult male children and records the height of both the father and the son. Use the α = 0.01 level of significance. STAT 3011 Homework 7 (Due: Tuesday April, 2, 2024) Spring 2024 (d) Carl Reinhold August Wunderlich said that the mean temperature of humans is 98.6◦ F. Some researchers thought that the mean temperature is greater than 98.6 ◦ F. They measured the temperature of 700 healthy adults, obtaining 700 measurements. The sample data resulted in a sample mean of 98.2 ◦ F and a sample standard deviation of 0.7 ◦ F. Problem 2. Suppose that in past years, the average price per square meter for warehouses has been $ 347.46, A national real estate investor wants to determine whether that figure has decreased now. The investor hires a researcher who randomly samples 49 warehouses that are for sale across the country and finds the mean price per square meter is $ 340.89, with a standard deviation of $ 13.89. Assume that prices of warehouse areas are normally distributed in the population. In this problem, use α = 0.05 (a) What should be the conclusion about the claim that the mean price per square meter for warehouses has decreased since?α = 0.05? Explain. (b) Construct the 95% confidence interval about the mean average price per square meter for warehouses. What does this interval imply? STAT 3011 Homework 7 (Due: Tuesday April, 2, 2024) Spring 2024 (c) Give an example of the value of the sample mean that can be used to fail to reject the null hypothesis without computing the p-value. Explain (d) How many randomly selected warehouses should be included in the study to be 97 % confident that their estimated price per square meter is within $ 3.00 of the true mean price per square meter for warehouses? (Use the fact that for a previous study conducted, the sample standard deviation was $10.00) Problem 3. : Type I vs Type II error A study by Hewit Associates showed that 79% of companies offer employees flexible scheduling. Suppose a researcher believes that in accounting firms this figure is larger. (a) State the null and the alternative Hypotheses. STAT 3011 Homework 7 (Due: Tuesday April, 2, 2024) Spring 2024 (b) Provide the statements explaining what it would mean to make • a Type I error • a Type II error (c) If the researcher wanted to find out if this figure is lower in accounting firms. What should be the conclusion/ decision of the study if a type II error is made? Can we make type I and type II errors when using the same information in this case? Explain. STAT 3011 Homework 7 (Due: Tuesday April, 2, 2024) Spring 2024 STAT 3011 Homework 7 (Due: Tuesday April, 2, 2024) Spring 2024 Problem 4. A large production facility uses two machines to produce a key part of its main product. Inspectors have expressed concerns about the quality of the finished product. A quality-control investigation has revealed that the key part made by the two machines is defective at times. The inspectors independently randomly sampled 35 units of the key part from machine A and 70 units of the key part from machine B. Of those produced by machine A, five were defective. twenty-two of the 70 randomly sampled parts from machine B were defective. (a) Do these data show enough evidence to conclude that the proportion of parts that are defective for machine B is greater than the proportion of parts that are defective for machine A at α = 0.02 level of significance? • Assumption: • Hypotheses: • Test statistic: What is the distribution of test statistics? Based on the sample information provided in this problem, what is the value of the test statistic? STAT 3011 Homework 7 (Due: Tuesday April, 2, 2024) Spring 2024 • P-value • Conclusion and interpretation in context. • Refer to the conclusion from this Problem. Based on the conclusion, what type of error (either Type 1 or Type 2) could you have made? Explain. STAT 3011 Homework 7 (Due: Tuesday April, 2, 2024) Spring 2024 (b) Repeat this analysis in R. Hint: Use the prop.test() or t.test() command. In addition, use the output from this command to compute the observed test statistic. Do not compute the observed test statistic in the same way you did earlier, use another method (c) Would you have the same conclusion and type of error if α = 0.05? Explain. (d) Construct by hand and interpret the 90% confidence interval for the difference, pB − pA between the two population proportions and interpret. Let us assume for this question that of the 35 randomly sampled units of the key part from machine A, 11 were defective, and forty-seven of the 100 randomly sampled parts from machine B were defective. Hint:Don’t forget to check if the assumptions are met and note that pA = p2 is the population STAT 3011 Homework 7 (Due: Tuesday April, 2, 2024) Spring 2024 proportion of units of the key part from machine A that are defective and pB = p1 is the population proportion of units of the key part from machine A that are defective. (e) Find the 90% CI for pB − pA by using the prop.test() command in R STAT 3011 Homework 7 (Due: Tuesday April, 2, 2024) Spring 2024 (f) Use the previous questions to find the 90 % confidence interval for pA −pB . Even though no calculation is necessary, use the method of your choice. Problem 5. The vice president of marketing brought to the attention of sales managers the fact that most of the company’s manufacturer representatives contacted clients and maintained client relationships in a disorganized, haphazard way. The sales managers brought the reps in for a three-day seminar and training session on how to use an organizer to schedule visits and recall pertinent information about each client more effectively. Sales reps were taught how to schedule visits most efficiently to maximize their efforts. Sales managers were given data on the number of site visits by sales reps on a randomly selected day both before and after the seminar. We will use the data below to test whether significantly more visits were made after the seminar (α = 0.1) and to find confidence intervals. Assume that the differences in the number of site visits are normally distributed with no outliers. Rep Before,Xi After, Yi 1 2 3 4 5 6 7 8 2 4 3 3 4 2 2 3 4 5 3 3 3 5 6 4 Use the following R command to input the data. Bef
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