Statistical Hypothesis Testing: Homework Solutions for Stat 204 (Exam 3) - Prof. Mark G. D, Study notes of Business Statistics

Download Statistical Hypothesis Testing: Homework Solutions for Stat 204 (Exam 3) - Prof. Mark G. D and more Study notes Business Statistics in PDF only on Docsity! Homework, Stat 204 (Exam 3) Solutions to selected problems are located at the end of the document. The problems assigned for each lab are due at the end of that lab meeting. HW 7 1. Consider the hypothesis test: H0 : µ ≥ 60 Ha : µ < 60 Assume we have a sample of size n = 85 and the population standard deviation is 11. Find the p-value and state your conclusion for each of the following sample means. Use α = 0.01. (a) x̄ = 59 (b) x̄ = 58 (c) x̄ = 57 (d) x̄ = 62 (think) 2. At the start of 2006, Fortune magazine reported that Wall Street securities firms had recently paid the average employee $125,500 in end-of-year bonuses (money.cnn.com). We want to know if the mean year-end bonus for an employee at Company X is different from the reported population mean of $125,500. (a) Write down the appropriate null and alternative hypotheses. (b) A sample of 35 employees at Company X has a mean year-end bonus of $121,000. Find the p-value for this test if the population standard deviation is $19,000. (c) State your conclusion at the α = 0.05 level of significance. 3. Consider the hypothesis test: H0 : µ = 47 Ha : µ 6= 47 A sample of size 18 has a mean of 45 and a standard deviation of 3.5. (a) Find the test statistic. (b) Find a range for the p-value using the t distribution table. (c) At α = 0.05, state your conclusion. 4. For the month of June in 2010, the U.S. Department of Labor reported a national mean unemployment insurance benefit of $306 per week (dol.gov). A researcher in Colorado suspected that sample data would provide evidence that the mean weekly benefit in her state was above the national level. (a) Write down the appropriate null and alternative hypotheses to test her claim. (b) A sample of 27 individuals from Colorado had a mean weekly unemployment insurance benefit of $349 and a standard deviation of $119. Find the p-value for this test. (c) State your conclusion at the α = 0.05 level of significance. 1 5. The Bureau of Labor Statistics reported that 12.3% of U.S. workers belonged to unions in 2009 (bls.gov). To test the claim that union membership was declining, a researcher sampled 479 U.S. workers in 2010 and found that 57 belonged to unions. (a) Write down the appropriate null and alternative hypotheses to test his claim that union membership declined in 2010. (b) Find the p-value for this test. (c) State your conclusion at the α = 0.05 level of significance. 6. A researcher predicted that during the second week of June in 2011, less than 5.1% of U.S. households watching television would watch the show Hawaii Five-0. A sample consistent with the results found by the Nielsen Company showed that of 25,000 households watching television that week, 1225 watched Hawaii Five-0 (nielsen.com). (a) For U.S. households watching television that week, find the point estimate of the pro- portion that watched Hawaii Five-0. (b) Test the claim at α = 0.05. Carefully show all five steps. *************** [ END OF HW 7 PROBLEMS ] *************** 2 HW 9 1. Complete the following ANOVA table. For this design we have samples of size seven from each of five different populations. Response: y Df Sum Sq Mean Sq F value Treatments 59.3 Residuals ----------------------------------------- Total 205.9 What is the p-value? 2. For the previous problem, (a) Write down the implied null and alternative hypotheses. (b) At α = 0.05, state your conclusion. 3. In March 2009, Motor Trend conducted a road test comparison of the 2010 Chevrolet Camaro SS, the 2009 Dodge Challenger R/T, and the 2010 Ford Mustang GT (motortrend.com). One of the criteria measured was the time required to travel one-quarter mile from a stand still. For purposes of this problem, assume six cars of each brand were tested by the same driver and the standing quarter mile time (in seconds) was recorded for each car. Results consistent with Motor Trend’s analysis are provided. Chevy 12.9 13.1 13.1 13.0 13.0 12.9 Dodge 13.8 13.4 13.6 13.5 13.6 13.7 Ford 13.3 13.5 13.5 13.6 13.7 13.4 This is the ANOVA table for the data: Response: Acceleration Df Sum Sq Mean Sq F value Pr(>F) Treatments 2 1.240 0.620 38.75 1.187e-06 Residuals 15 0.240 0.016 (a) Find the Total df. (b) Find the Total SS. (c) Find the point estimate for the mean standing quarter mile time for each of the three cars. (d) At α = 0.05, carefully show all five steps of the hypothesis test for significant difference in the population means of standing quarter mile times. (e) Is it likely that the three cars have the same mean standing quarter mile times? Explain. *NOTE: Motor Trend selected the Chevy Camaro as the “Prime Pony of the 21st Century.” (This has no bearing on the calculations for the problem.) 5 4. The following five observations were taken for two variables. xi 2 3 5 7 11 yi 4 6 10 8 12 (a) Make a scatter diagram for the data. (b) Does the scatter diagram indicate a possible relationship between x and y? Explain. (c) Find the sample covariance. Use the alternate formula: sxy = ∑ xiyi− ∑ xi ∑ yi n n−1 . (d) Find the sample correlation coefficient. Interpret. 5. Nielsen Media Research provides two values for each television show. The rating measures the percentage of viewers with televisions watching the show, and the share measures the percentage of viewers watching the show among those viewers whose televisions are on during that time slot. The following data show the rating and share for a sample of six of the 45 top-rated programs of all-time (wikipedia.com). Rating (x) 43 44 46 48 51 60 Share (y) 62 63 71 67 71 77 (a) Make a scatter diagram for the data. (b) Explain the relationship between rating and share. (c) Find the sample covariance. Use the alternate formula given in the previous problem. (d) Find the sample correlation coefficient. Interpret. *************** [ END OF HW 9 PROBLEMS ] *************** 6 Additional problems These problems will not be collected for a grade, but the material they cover will be on the exam. 1. Consider the five observations for two variables, x and y. xi 8 9 5 6 7 yi 12 9 25 14 18 (a) Make a scatter diagram for the data. (b) Does the scatter diagram indicate a possible relationship between x and y? Explain. (c) Find the estimated regression equation by computing b0 and b1. (d) Use the estimated regression equation to find the predicted value of y when x = 6. 2. A glider-car is pushed along a nearly frictionless track and is released at the 10 cm mark on the track. At the time of release, a stopwatch is started. At two-second intervals the location of the car is recorded. The data appear in the following table. Time (x), seconds 2 4 6 8 10 Location (y), cm 12.8 16.9 22.2 25.3 29.2 (a) Make a scatter diagram for Location versus Time. (b) Does the scatter diagram indicate a possible relationship between Time and Location? Explain. (c) Find the estimated regression equation by computing b0 and b1. (d) In this problem, β0 represents the location of the car when the stopwatch was started. How does b0 compare to the true value? (e) The slope, β1, represents the constant velocity of the car in cm/sec. What is the esti- mated velocity of the car, based on our data? (f) Use the estimated regression equation to find the predicted Location after five seconds. 7