90% Confidence Intervals and Hypothesis Testing Explained

1. We are interested in the mean scores on an exam. A random sample of 36 scores is taken and gives a sample mean 76. Suppose we know that population standard deviation is 3 points. What is the 90% confidence interval estimate for the population mean exam score? a. [75.2, 76.8] b. [75.0, 77.0] c. [77.4, 78.6] d. None of the other choices. Lower limit=76-1.645*3/sqrt(36)=75.2; Upper limit=76+1.645*3/sqrt(36)=76.8 2. A financial officer for a company wants to estimate the percent of accounts receivable that are more than 30 days overdue. He surveys 500 accounts and finds that 300 are more than 30 days overdue. What is the 90% confidence interval for the true percent of accounts receivable that are more than 30 days overdue? a. [56.4%, 63.6%] b. [57.2%, 62.8%] c. [55.7%, 64.3%] d. None of the other choices. Lower limit=0.6-1.645*sqrt(0.6*0.4/500)=0.564; Upper limit=0.6+1.645*sqrt(0.6*0.4/500)=0.636; 3. A cost accountant is developing a regression model to predict the total cost of producing a batch of printed circuit boards as a linear function of batch size (the number of boards produced in one lot or batch), production plant (Kitchener and Hamilton), and production shift (day and evening). In this model, "shift" is ___. a) a response variable b) an independent variable c) a quantitative variable d) a dependent variable

4. Suppose the earnings per share (EPS) for a population of industrial stocks are Normally distributed. A random sample of 16 stocks is taken. The sample mean is 1.90 and the sample standard deviation is 0.40. What is the 95% confidence interval for the average EPS of this population of industrial stocks? a. [1.50, 2.30] b. [1.72, 2.08] c. [1.68, 2.12] d. None of the other choices. 5. A certain experimental drug claims a cure rate of 75% for males with prostate cancer. If the null hypothesis is p = 75%, what is the Type II error? a. α b. β c. Rejecting the claim when the cure rate actually is 75%. d. Accepting the claim when the cure rate is actually not 75%. 6. A manufacturer of salad dressings uses machines to dispense liquid ingredients into bottles that move along a filling line. The machine that dispenses salad dressings is working properly when 8.0 ounces are dispensed. So Ho is µ=8.0. Suppose that the average amount dispensed in a particular sample of 24 bottles is 7.95 ounces with a sample variance of 0.03 ounces squared. What should be the test statistic to use? a. Z test statistic b. t test statistic with 23 as the degrees of freedom c. t test statistic with 24 as the degrees of freedom d. None of the other choices 7. The mortgage department of a large bank is interested in the nature of loans of first-time borrowers. This information will be used to tailor their marketing strategy. They believe that more than 50% of

first-time borrowers take out smaller loans than other borrowers. To support their belief, they sample 100 first-time borrowers and find 53 of these loans are smaller than the other borrowers. For the hypothesis test, they choose a 5% level of significance. What should be the Ho? a. p ≤ 0.50 b. p ≤ 50 c. µ ≥ 50 d. p > 0.50 Using the following information, answer questions 8 and 9: An interesting research question is the effect, if any, that different types of teaching formats have on the grade outcomes of students. Under a current study, we consider two teaching formats: a hybrid format (Population 1) and a standard lecture format (Population 2). We want to prove that the hybrid format leads to a higher average student grade. For hypothesis testing, a sample of 35 students from the hybrid format is taken with mean of 76. A sample of 40 students from the standard lecture format is taken with mean of 70. Assume σ 1 is 16 and σ 2 is 9 and α = 0.01. 8. What is the test statistic based on the samples? a. Not enough information to tell because the population distributions are unknown. b. t = 1.31 c. z = 1.96 e. z = 1.31 9. What is the p-value? a. Not enough information to tell because the population distributions are unknown. b. p-value = 0.025 c. p-value = 0.095 e. p-value = 0.190 Use the following information to answer questions 10 and 11: A researcher is conducting a matched-pairs study to see if the quality level will change after some quality management program. She gathers data on each pair in the study, which are shown in the table below. The

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sample statistic will be the before-treatment quality minus the after-treatment quality. Assume that the data are normally distributed in the populations. Pair Before-treatment quality After-treatment quality Difference 1 9 12 -3 2 8 10 -2 3 11 12 -1 4 8 10 -2 5 9 12 -3 10. The sample mean is ___. a) 9.0 b) 11.2 c) 2.20 d ) -2.20 11. Assume that the data are normally distributed in the population. The sample standard deviation of the difference is 0.837. A 95% confidence interval of the population difference would be ___. a) –3.02 to –0.18 b) –1.6 to –1.09 c) –3.24 to -1.16 d) –2.11 to –1.09 Confidence interval: Lower limit=-2.20 – critica_t*0.837/sqrt(5)=-2.20 – 2.776 *0.837/sqrt(5)=- 3.24. Upper limit=-1.16. critica_t=2.776 because upper tail probability is 0.025 and DF=4.

12. Maureen Jackson, CEO of a mail order business for fashion shoes, is reviewing the order filling operations at her warehouses. Her goal is 100% of orders shipped within 24 hours. In previous years, the East Coast Warehouse performed about equally well with the West Coast Warehouse. This year, her staff randomly selected 200 orders from the West Coast Warehouse (population 1) and 400 orders from the East Coast Warehouse (population 2), and reports that 190 of the West Coast Orders were shipped within 24 hours, and the East Coast Warehouse shipped 350 orders within 24 hours. Maureen would like to show the West Coast outperforms the East Coast. Assuming α = 0.05, the observed z value is ___. a) –3.15 b) 2.42 c) 1.53 d) 2.89 West: p1bar=190/200=0.95; East: p2bar=350/400=0.875; pbar=(190+350)/600=0.9 Z=(0.95-0.875)/sqrt(0.9*0.1*(1/200+1/400))=2.89 13. Supermart Inc. operates 151 stores. Cathy Ng, Marketing Manager, feels that the level of lighting may affect store sales. She chooses three levels of lighting (soft, medium, and bright) and randomly assigns 10 stores to each lighting level. In Cathy’s experimental design "store sales" is ___. a) the dependent variable b) a concomitant variable c) a treatment variable d) a blocking variable Supermart Inc. operates 151 stores. Cathy Ng, Marketing Manager, feels that the level of lighting may affect store sales. She chooses three levels of lighting (soft, medium, and bright)

and randomly assigns ten stores to each lighting level. Analysis of Cathy’s data yielded the following ANOVA table. Use this ANOVA table to answer questions 14 and 15. Source of Variation SS df MS F Treatmen t 48411.1 1 2 Cell_II: 48411.11/2 =24205.555 Cell_IV: 24205.555/1278.86 =18.93 Error 34529.1 7 Cell_I : 30- 3=27 Cell_III: 34529.17/27 =1278.86 Total 84940.2 8 14. What is the value of Cell_III? a. 4841.11 b. 2420.56 c. 3429.17 d . 1278.86 C=3, n_T=30. Cell_III= 34529.17 /(30-3)= 1278.86 15. What is the value of Cell_IV? a. 18.93: 24205.555/1278.86=18.93 b. 11.43 c. 1.39 d. None of the other choices.

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16. In multiple regression analysis involving 10 independent variables and 100 observations, the critical value of t for testing individual coefficients in the model will have a. 99 degrees of freedom b. 90 degrees of freedom c. 89 degrees of freedom: DF=n-k-1=100-10-1=89 d. 98 degrees of freedom 17. A test of independence is to be performed. The contingency table has 4 rows and 5 columns. Using α = 0.05, the critical chi-square value is ___. a) 20 b) 9 c) 26.30 d) 21.03: DF=(4-1)*(5-1)=12; Critical value (α = 0.05, DF=12)=21.03 18. The numerical value of the coefficient of correlation must be ___. a) between –1 and +1 b) between –1 and 0 c) between 0 and 1 d) between 0 and –1 19. A simple regression model developed for 12 pairs of data resulted in a sum of squares of error, SSE = 400. The standard error of the estimate is ___. a) 4.47 b) 6.32: Se=sqrt(400/(12-2))=6.32 c) 5.30 d) 4.74

Use the following information to answer questions 20-22: Louis Tang, a cost accountant at Prime Plastics, Inc. (PPI), is analyzing the manufacturing costs of a moulded plastic telephone handset produced by PPI. Louis's independent variable is production lot size (in 1,000's of units), and his dependent variable is the total cost of the lot (in $100's). Regression analysis of the data yielded the following tables. Note that the numbers in some cells are to be determined. Coefficients Standard Error t statistic Intercept 3.996 1.161268 3.441065 X 0.358 0.102397 3.496205 Source Df SS MS F Regressio n 1 9.858769 9.858769 Residual 11 0.806545 Total 12 18.73077 20. Using α = 0.01, Louis should ____. a) increase the sample size b) not reject H 0 : β 1 = 0 c) reject H 0 : β 1 = 0 d) do not reject H 0 : β 0 = 0 In the final exam, we only have the F-table for α = 0.05. So we can’t use the F test. We can only use the t-test. Observed t-statistic=0.358/0.102397=3.496 Critical t (0.01/2, DF=n-k-1=n-2=11)=3.106 So we reject H 0 : β 1 = 0. 21. To test the validity of the overall linear model using the F test, what is the observed F test statistic? a) 8.87 b) 1.11 c) 12.22: observed F=9.858769/0.806545=12.22 d) Not enough information to tell

22. To test the validity of the overall linear model using the F test with α = 0.05, which of the following statements is FALSE? a) The F-test is a two-tailed test. b) The F-test is an upper-tailed test. c) The null hypothesis is β 1 = 0 d) The null hypothesis states there is no linear relationship between the independent variable and the dependent variable. 23. A researcher has developed a simple regression model from 13 pairs of data points. He wants to test to determine if the slope is significantly larger than zero. He uses α = 0.01. The critical value of the test statistic t is ___. a) 2.650 b) 3.012 c) 2.681 d) 2.718 critical t (0.01, DF=n-2=11)=2.718 Note that Ha is: β 1 > 0. So this is a upper-tailed test. 24. Which of the following statements is true about using simple linear regression to develop a forecasting trend line? a) There can be multiple dependent variables. b) The independent variable must be discrete and its possible values must be integers starting at 1. c) The trend line indicates a causation between x and y. d) Unlike regular simple regression, extrapolating outside the original time frame in the sample data is required.

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25. The Least Square Method in multiple regression analysis is designed to ___. a) minimize the total sum of squares (SST) b) minimize the sum of squares of errors (SSE) c) maximize the standard error of the estimate d) maximize the p value for the calculated F value 26. The following ANOVA table is from a multiple regression analysis with n = 35 and four independent variables: Source Df SS MS F P Regressio n K=4 600 Error n-k- 1=35-4- 1=30 1000- 600=400 MSE=400/3 0 =13.33 Total 1000 The MSE value is ___. a) 8.57 b) 13.33 c) 10.00 d) 75.00

27. The following ANOVA table is from a multiple regression analysis: Source Df SS MS F P Regressio n 4 2000 Error 19 Total n-1=23 2500 The sample size of the analysis is ___. a) 30 b) 25 c) 15 d) 24

28. If a qualitative variable has "c" categories, how many dummy variables must be created and used in the regression analysis? a) c – 1 b) c c) c + 1 d) c 2 29. A multiple regression analysis produced the following tables: Predictor Coefficients Standard Error t statistic p value Intercept 752.0833 336.3158 2.236241 X 1 11.87375 5.32047 2.231711 X 2 1.908183 0.662742 2.879226 Source Df SS MS F p value Regressio n 2 203693.3 101846.7 6.745406 Residual 12 181184.1 15098.67 Total 14 384877.4 The adjusted R 2 is ___. a) 0.4508 b) 0.5511 c) 0.4809 d) 0.5232 Adjusted R 2 ¿ 1 − SSE /( n − k − 1 ) SS yy /( n − 1 ) = 1-(181184.1/12)/ ( 384877.4/14)=0.4508

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30. Alan Ho, a market analyst for Clear Sound Inc., is analyzing sales of heavy metal CD’s. Alan’s dependent variable is annual heavy metal CD sales (in $100,000's), and his independent variables are teenage population (in 1,000's) and type of sales district (0 = urban, 1 = rural). Regression analysis of the data yielded the following tables: Coefficients Standard Error t statistic p value Intercept 1.7 0.384212 4.424638 0.00166 X 1 (teenagers) 0.04 0.014029 2.851146 0.019054 X 2 (district) -1.5666667 0.20518 -7.63558 3.21E-05 For two sales districts with the same number of teenagers one urban and one rural, Alan's model predicts ___. a) $1,7000,000 lower average sales in the rural district b) the same average sales in both districts c) $156,666 lower average sales in the rural district d) $156,666 higher average sales in the rural district 31. Use the following set of observed frequencies to test the independence of the two variables. Variable one has the two categories of 'A' and 'B'; variable two has the categories of 'C', 'D', and 'E'. C D E A 12 10 8 30 B 20 24 26 70 32 34 34 100 What is the expected frequency of the intersection between Category “B” and Category “E”? a) 34.0 b) 70.0 c) 23.8: 34*70/100=23.8 d) None of the other choices.

BU255 Mock Final Exam Solution

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