Midterm Examination Questions - Spring 2016 1. The final examination grades of random samples of students from three different classes are shown below. Class A Class B Class C 92 91 85 85 85 93 96 90 82 95 86 84 At the = .05 level of significance, is there any difference in the mean grades of the three classes? 2. Individuals were randomly assigned to three different production processes. The hourly units of production for the three processes are shown below. Process 1 33 30 28 29 Production Process Process 2 33 35 30 38 Process 3 28 36 30 34 Use the analysis of variance procedure with = 0.05 to determine if there is a significant difference in the mean hourly units of production for the three types of production processes. Use both the critical and p-value approaches. 3. Random samples of employees from three different departments of MNM Corporation showed the following yearly incomes (in $1,000). Department A 40 37 43 41 35 38 Department B 46 41 43 33 41 42 Department C 46 40 41 48 39 44 At = .05, test to determine if there is a significant difference among the average incomes of the employees from the three departments. Use both the critical and p-value approaches. 4. The heating bills for a selected sample of houses using various forms of heating are given below (values are in dollars). Gas Heated Homes 83 80 82 83 82 Central Electric 90 88 87 82 83 Heat Pump 81 83 80 82 79 At = 0.05, test to see if there is a significant difference among the average bills of the homes. Use both the critical and p-value approaches. 5. Three universities in your state decided to administer the same comprehensive examination to the recipients of MBA degrees from the three institutions. From each institution, MBA recipients were randomly selected and were given the test. The following table shows the scores of the students from each university. Northern University 75 80 84 85 81 Central University 85 89 86 88 Southern University 80 81 84 79 83 85 At = 0.01, test to see if there is any significant difference in the average scores of the students from the three universities. (Note that the sample sizes are not equal.) Use both the critical and p-value approaches. 6. The three major automobile manufacturers have entered their cars in the Indianapolis 500 race. The speeds of the tested cars are given below. Manufacturer A 180 175 179 176 190 Manufacturer B 177 180 167 172 Manufacturer C 175 176 177 At = .05, test to see if there is a significant difference in the average speeds of the cars of the auto manufacturers. Use both the critical and p-value approaches. 7. Part of an ANOVA table is shown below. Source of Variation Between Treatments Within Treatments (Error) a. Sum of Squares 90 120 Degrees of Freedom Mean Square 3 20 _____? _____? F _____? Compute the missing values and fill in the blanks in the above table. Use = .01 to determine if there is any significant difference among the means. b. How many groups have there been in this problem? c. What has been the total number of observations? 8. MNM, Inc. has three stores located in three different areas. Random samples of the daily sales of the three stores (in $1,000) are shown below. Store 1 9 8 7 8 Store 2 10 11 10 13 Store 3 6 7 8 11 At 95% confidence, test to see if there is a significant difference in the average sales of the three stores. Use both the critical and p-value approaches. 9. Eight observations were selected from each of 3 populations (total of 24 observations), and an analysis of variance was performed on the data. The following are part of the results. Source of Variation Between Treatments Within Treatments (Error) Sum of Squares 216 Degrees of Freedom Mean Square F 252 Using = .05, test to see if there is a significant difference among the means of the three populations. 10. Random samples of individuals from three different cities were asked how much time they spend per day watching television. The results (in minutes) for the three groups are shown below. City I 260 280 240 260 300 City II 178 190 220 240 City III 211 190 250 At = 0.05, test to see if there is a significant difference in the averages of the three groups. 11. Three different brands of tires were compared for wear characteristics. From each brand of tire, ten tires were randomly selected and subjected to standard wear-testing procedures. The average mileage obtained for each brand of tire and sample variances (both in 1,000 miles) are shown below. Brand A Average Mileage Sample Variance 37 3 Brand B 38 4 Brand C 33 2 At 95% confidence, test to see if there is a significant difference in the average mileage of the three brands. 12. Nancy, Inc. has three stores located in three different areas. Random samples of the sales of the three stores (In $1,000) are shown below. Store 1 46 47 45 42 45 Store 2 34 36 35 39 Store 3 33 31 35 a. Compute the overall mean . b. At 95% confidence, test to see if there is a significant difference in the average sales of the three stores. 13. In a completely randomized experimental design, 11 experimental units were used for each of the 4 treatments. Part of the ANOVA table is shown below. Source of Variation Between Treatments Within Treatments Total Sum of Squares 1500 Degrees of Freedom Mean Square F _____? _____? _____? _____? 5500 _____? _____? Fill in the blanks in the above ANOVA table. 14. Samples were selected from three populations. The data obtained are shown below. Sample 1 Sample 2 Sample 3 10 13 12 13 16 14 15 15 18 Sample Mean 12 15 16.5 Sample Variance 2.0 1.0 4.5 a. Compute the overall mean . b. Set up an ANOVA table for this problem. c. At 95% confidence, test to determine whether there is a significant difference in the means of the three populations. Use both the critical and p-value approaches. 15. In a completely randomized experimental design, 14 experimental units were used for each of the 5 levels of the factor (i.e., 5 treatments). Fill in the blanks in the following ANOVA table. Source of Variation Between Treatments Error (Within Treatments) Total Sum of Squares _____? Degrees of Freedom Mean Square _____? 800? _____? _____? _____ 10600? _____? F _____ 16. Random samples of several days' sales of handguns per day in three different states are shown below. We are interested in determining whether or not there is a significant difference in the average sales of guns in the three states Tennessee 12 13 17 10 18 Kentucky 15 19 20 Texas 16 18 a. Compute the overall mean . b. State the null and alternative hypotheses to be tested. c. Show the complete ANOVA table for this test including the test statistic. d. The null hypothesis is to be tested at 95% confidence. Determine the critical value for this test. What do you conclude? e. Determine the p-value and use it for the test. 17. Information regarding random samples of annual salaries (in thousands of dollars) of doctors in three different specialties is shown below. Sample size Average salary Sample variance Pediatrics 12 120 16 Radiology 10 186 18 Pathology 11 240 20 a. Compute the overall mean . b. State the null and alternative hypotheses to be tested. c. Show the complete ANOVA table for this test including the test statistic. d. The null hypothesis is to be tested at 95% confidence. Determine the critical value for this test. What do you conclude? e. Determine the p-value and use it for the test. 18. Information regarding the ACT scores of samples of students in four different majors is given below. Majors Management Marketing Finance Accounting 29 22 29 28 27 22 27 26 21 25 27 25 28 26 28 20 22 27 24 21 28 20 20 19 28 23 20 27 23 25 30 24 28 27 29 21 24 28 23 29 27 31 27 24 Sums Means Variances 318 26.50 10.09 245 24.50 6.94 234 26.00 14.50 312 24.00 9.00 a. Set up the ANOVA table for this problem. b. At 95% confidence test to determine whether there is a significant difference in the means of the three populations. 19. Information regarding the ACT scores of samples of students in three different majors is given below. Major Management Finance 28 22 26 23 25 24 27 22 21 24 19 26 27 27 17 29 17 28 23 Sums Means Variances a. 230 23 18 225 25 6.75 Accounting 29 27 26 28 25 26 28 20 20 24 28 28 29 338 26 9.33 Set up the ANOVA table for this problem. b. At 95% confidence test to determine whether there is a significant difference in the means of the three populations. 20. The manager of Ahmadi Corporation, wants to determine whether or not the type of work schedule for her employees has any effect on their productivity. She has selected 15 production employees at random and then randomly assigned 5 employees to each of the 3 proposed work schedules. The following table shows the units of production (per week) under each of the work schedules. Schedule 1 50 60 70 40 45 a. Work Schedule (Treatments) Schedule 2 60 65 65 58 57 Schedule 3 75 75 55 40 55 Compute the overall sample mean . b. At 95% confidence, determine if there is a significant difference in the mean weekly units of production for the three types of work schedules. 21. Six observations were selected from each of three populations. The data obtained is shown below. Sample 1 31 28 34 32 26 29 a. Sample 2 31 32 33 30 32 34 Sample 3 37 36 39 40 35 35 Compute the overall sample mean . b. Test at the = 0.05 level to determine if there is a significant difference in the means of the three populations. 22. The test scores for selected samples of statistics students who took the course from three different instructors are shown below. Instructor A 81 62 82 87 73 Instructor B Instructor C 90 85 55 90 84 90 91 95 85 80 At = 0.05, test to see if there is a significant difference among the averages of the three groups. Show the complete ANOVA table. 23. The following information regarding the yearly salaries (in $1,000) of CEO's in 2010 and 2009 are provided. Sample size Sample Mean Population Standard Deviation Year 2010 190 560 80 Year 2009 145 540 90 a. At 95% confidence, perform a test to determine if there has been a significant increase in the salaries of CEO's. Use the Critical Value Approach. b. Compute the p-value. 24. The management of Chattanooga Paper Corporation wants to determine whether there is a significant difference in the thickness of papers produced at their two existing plants. The following data has been accumulated for this test. The degrees of freedom (df) for this problem are given to be df = 80. Sample Mean Sample Standard Deviation Sample Size a. Plant 1 30.8 8.0 55 Plant 2 30.0 9.0 41 Compute a 95% confidence interval for the difference in the average thickness of papers produced at the two plants. b. Is there conclusive evidence that the average thickness in one plant is significantly more than the other? If yes, which plant? Explain, using the results of part (a). Do not perform any test. 25. (Randomized Block Design)In order to determine whether or not a special tutoring service improves the scores of students in a Business Statistics examination, a sample of 6 students were given the exam before and after using the tutorial service. The results are shown below. Let d = Score After - Score Before. Student A B C D E F Score Before 80 82 75 78 90 85 Score After 84 86 82 83 95 84 At = 0.10, test to see if the tutorial service actually increased scores on the examination. 26. The following information shows the yearly salaries (in $1,000) of samples of physicians for 2013 and 2012. Sample Size Sample Mean Population Standard Deviation () (1) Year 2013 280 790 100 (2) Year 2012 244 685 110 We want to perform a test to determine if there has been a significant increase in the salaries of physicians. In your computations, please use \"1\" to represent year 2013. a. State the null and alternative hypotheses to be tested. H0: Ha: b. Compute the test statistic. c. The null hypothesis is to be tested at 95% confidence. Determine the critical value from the table. d. What do you conclude? Fully explain and answer the question. e. Compute the p-value. 27. The management of Chattanooga Paper Corporation wants to determine whether there is a significant difference in the thickness of papers produced at their two existing plants. The following data has been accumulated for this test The degrees of freedom (df) for this problem are df = 51. Plant 1 23.6 25 Sample Variance (S2) Sample Size a. b. 4 42 Sample Mean Plant 2 23.1 53 Compute a 95% confidence interval for the difference in the average thickness of papers produced at the two plants. Please use the same order as given above. Is there conclusive evidence that the average thickness in one plant is significantly more than the other? If yes, which plant? Explain, using the results of part (a). Do not perform any tests. 28. Samples were taken from the morning (AM) and the afternoon (PM) shifts of a production process. The results are shown below. Sample Size Sample Mean Population Variance (1) AM Shift 900 810 144 (2) PM Shift 800 600 196 Determine the following for the above data. a. Standard error of the mean b. Point estimate of the difference between the two means. c. Develop a 97% confidence interval estimate for the difference between the two means. 29. The following data represent a company's yearly sales volume and its advertising expenditure over a period of 8 years. (Y) Sales in Millions of Dollars 15 16 18 17 16 19 19 24 (X) Advertising in ($10,000) 32 33 35 34 36 37 39 42 a. Develop a scatter diagram of sales versus advertising and explain what it shows regarding the relationship between sales and advertising. b. Use the method of least squares to compute an estimated regression line between sales and advertising. c. If the company's advertising expenditure is $400,000, what are the predicted sales? Give the answer in dollars. d. What does the slope of the estimated regression line indicate? e. Compute the coefficient of determination and fully interpret its meaning. f. Use the F test to determine whether or not the regression model is significant at = 0.05. g. Use the t test to determine whether the slope of the regression model is significant at = 0.05. h. Develop a 95% confidence interval for predicting the average sales for the years when $400,000 was spent on advertising. i. Compute the correlation coefficient. 30. Given below are five observations collected in a regression study on two variables x (independent variable) and y (dependent variable). x 10 20 30 40 50 y 7 5 4 2 1 a. Develop the least squares estimated regression equation b. At 95% confidence, perform a t test and determine whether or not the slope is significantly different from zero. c. Perform an F test to determine whether or not the model is significant. Let = 0.05. d. Compute the coefficient of determination. e. Compute the coefficient of correlation. 31. Below you are given a partial computer output based on a sample of 14 observations, relating an independent variable (x) and a dependent variable (y). Predictor Constant X Coefficient 6.428 0.470 Standard Error 1.202 0.035 Analysis of Variance SOURCE Regression Error (Residual) Total SS 958.584 1021.429 a. Develop the estimated regression line. b. At = 0.05, test for the significance of the slope. c. At = 0.05, perform an F test. d. Determine the coefficient of determination. e. Determine the coefficient of correlation. 32. Below you are given a partial computer output based on a sample of 21 observations, relating an independent variable (x) and a dependent variable (y). Predictor Constant X Coefficient 30.139 -0.252 Standard Error 1.181 0.022 Analysis of Variance SOURCE Regression Error a. Develop the estimated regression line. b. At = 0.05, test for the significance of the slope. c. At = 0.05, perform an F test. d. Determine the coefficient of determination. e. Determine the coefficient of correlation. SS 1,759.481 259.186 33. An automobile dealer wants to see if there is a relationship between monthly sales and the interest rate. A random sample of 4 months was taken. The results of the sample are presented below. The estimated least squares regression equation is = 75.061 - 6.254X Y Monthly Sales 22 20 10 45 X Interest Rate (In Percent) 9.2 7.6 10.4 5.3 a. Obtain a measure of how well the estimated regression line fits the data. b. You want to test to see if there is a significant relationship between the interest rate and monthly sales at the 1% level of significance. State the null and alternative hypotheses. c. At 99% confidence, test the hypotheses. d. Construct a 99% confidence interval for the average monthly sales for all months with a 10% interest rate. e. Construct a 99% confidence interval for the monthly sales of one month with a 10% interest rate. 34. Jason believes that the sales of coffee at his coffee shop depend upon the weather. He has taken a sample of 6 days. Below you are given the results of the sample. Cups of Coffee Sold 350 200 210 100 60 40 Temperature 50 60 70 80 90 100 a. Which variable is the dependent variable? b. Compute the least squares estimated line. c. Compute the correlation coefficient between temperature and the sales of coffee. d. Is there a significant relationship between the sales of coffee and temperature? Use a .05 level of significance. Be sure to state the null and alternative hypotheses. e. Predict sales of a 90 degree day. 35. Researchers have collected data on the hours of television watched in a day and the age of a person. You are given the data below. Hours of Television 1 3 4 3 6 Age 45 30 22 25 5 a. Determine which variable is the dependent variable. b. Compute the least squares estimated line. c. Is there a significant relationship between the two variables? Use a .05 level of significance. Be sure to state the null and alternative hypotheses. d. Compute the coefficient of determination. How would you interpret this value? 36. Given below are seven observations collected in a regression study on two variables, X (independent variable) and Y (dependent variable). X 2 3 6 7 8 7 9 Y 12 9 8 7 6 5 2 a. Develop the least squares estimated regression equation. b. At 95% confidence, perform a t test and determine whether or not the slope is significantly different from zero. c. Perform an F test to determine whether or not the model is significant. Let = 0.05. d. Compute the coefficient of determination. 37. The owner of a retail store randomly selected the following weekly data on profits and advertising cost. Week Advertising Cost ($) Profit ($) 1 0 200 2 50 270 3 250 420 4 150 300 5 125 325 a. Write down the appropriate linear relationship between advertising cost and profits. Which is the dependent variable? Which is the independent variable? b. Calculate the least squares estimated regression line. c. Predict the profits for a week when $200 is spent on advertising. d. At 95% confidence, test to determine if the relationship between advertising costs and profits is statistically significant. e. Calculate the coefficient of determination. 38. The owner of a bakery wants to analyze the relationship between the expenditure of a customer and the customer's income. A sample of 5 customers is taken and the following information was obtained. Expenditure .45 10.75 5.40 7.80 5.60 The least squares estimated line is X Income (In Thousands) 20 19 22 25 14 = 4.348 + 0.0826 X. a. Obtain a measure of how well the estimated regression line fits the data. b. You want to test to see if there is a significant relationship between expenditure and income at the 5% level of significance. Be sure to state the null and alternative hypotheses. c. Construct a 95% confidence interval estimate for the average expenditure for all customers with an income of $20,000. d. Construct a 95% confidence interval estimate for the expenditure of one customer whose income is $20,000. 39. Below you are given information on annual income and years of college education. Income (In Thousands) 28 40 36 28 48 Years of College 0 3 2 1 4 a. Develop the least squares regression equation. b. Estimate the yearly income of an individual with 6 years of college education. c. Compute the coefficient of determination. d. Use a t test to determine whether the slope is significantly different from zero. Let = 0.05. e. At 95% confidence, perform an F test and determine whether or not the model is significant. 40. Below you are given information on a woman's age and her annual expenditure on purchase of books. Age 18 22 21 28 Annual Expenditure ($) 210 180 220 280 a. Develop the least squares regression equation. b. Compute the coefficient of determination. c. Use a t test to determine whether the slope is significantly different from zero. Let = 0.05. d. At 95% confidence, perform an F test and determine whether or not the model is significant. 41. The following sample data contains the number of years of college and the current annual salary for a random sample of heavy equipment salespeople. Years of College Annual Income (In Thousands) 2 2 3 4 3 1 4 3 4 4 20 23 25 26 28 29 27 30 33 35 a. Which variable is the dependent variable? Which is the independent variable? b. Determine the least squares estimated regression line. c. Predict the annual income of a salesperson with one year of college. d. Test if the relationship between years of college and income is statistically significant at the .05 level of significance. e. Calculate the coefficient of determination. f. Calculate the sample correlation coefficient between income and years of college. Interpret the value you obtain. 42. The following data shows the yearly income (in $1,000) and age of a sample of seven individuals. Income (in $1,000) 20 24 24 25 26 27 34 Age 18 20 23 34 24 27 27 a. Develop the least squares regression equation. b. Estimate the yearly income of a 30-year-old individual. c. Compute the coefficient of determination. d. Use a t test to determine whether the slope is significantly different from zero. Let = 0.05. e. At 95% confidence, perform an F test and determine whether or not the model is significant. 43. The following data show the results of an aptitude test (Y) and the grade point average of 10 students. Aptitude Test Score (Y) 26 31 28 30 34 38 41 44 40 43 GPA (X) 1.8 2.3 2.6 2.4 2.8 3.0 3.4 3.2 3.6 3.8 a. Develop a least squares estimated regression line. b. Compute the coefficient of determination and comment on the strength of the regression relationship. c. Is the slope significant? Use a t test and let = 0.05. d. At 95% confidence, test to determine if the model is significant (i.e., perform an F test). 44. Shown below is a portion of the computer output for a regression analysis relating sales (Y in millions of dollars) and advertising expenditure (X in thousands of dollars). Predictor Constant X Coefficient 4.00 0.12 Standard Error 0.800 0.045 Analysis of Variance SOURCE Regression Error DF 1 18 SS 1,400 3,600 a. What has been the sample size for the above? b. Perform a t test and determine whether or not advertising and sales are related. Let = 0.05. c. Compute the coefficient of determination. d. Interpret the meaning of the value of the coefficient of determination that you found in Part c. Be very specific. e. Use the estimated regression equation and predict sales for an advertising expenditure of $4,000. Give your answer in dollars. 45. A company has recorded data on the daily demand for its product (Y in thousands of units) and the unit price (X in hundreds of dollars). A sample of 15 days demand and associated prices resulted in the following data. X = 75 Y = 135 (Y - )2 = 100 (Y - )(X - ) = -59 (X - )2 = 94 SSE = 62.9681 a. Using the above information, develop the least-squares estimated regression line and write the equation. b. Compute the coefficient of determination. c. Perform an F test and determine whether or not there is a significant relationship between demand and unit price. Let = 0.05. d. Would the demand ever reach zero? If yes, at what price would the demand be zero? 46. A regression and correlation analysis resulted in the following information regarding an independent variable (x) and a dependent variable (y). X = 42 Y = 63 n=7 (Y - )(X - ) = 37 (X - )2 = 84 (Y (Y - )2 = 11.7024 )2 = 28 a. Develop the least squares estimated regression equation. b. At 95% confidence, perform a t test and determine whether or not the slope is significantly different from zero. c. Perform an F test to determine whether or not the model is significant. Let = 0.05. d. Compute the coefficient of determination. 47. A regression analysis between a dependent variable (Y) and an independent variable (X) was performed and part of the Excel results is shown below. ANOVA Regression Residual (Error) Total df 1 10 11 SS 552.0 80.0 632.0 Intercept X Coefficients 4.3939 1.9650 Standard Error 1.7569 0.2387 MS 552.0 8.0 F 69.0 t Stat 2.5009 8.2315 P-value 0.0314 0.0000 Significance F 0.0000 Answer the following questions based on the above information and use a 95% confidence. a. Is the regression model significant at 95% confidence? Why or why not. Fully explain. b. Is X significant? Why or why not. Fully explain. c. Compute the value of R-square. d. Determine the multiple R. e. Compute the standard error. f. What has been the sample size for this