STAT 212 Business Statistics II 1

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1 STAT 1 Business Statistics II 1 KING FAHD UNIVERSITY OF PETROLEUM & MINERALS DEPARTMENT OF MATHEMATICAL SCIENCES DHAHRAN, SAUDI ARABIA STAT 1: BUSINESS STATISTICS II Semester 091 Final Exam Thursday Feb 4, 010 7:00 10:00 pm Please circle your Instructor s name & section number Raid Anabosi 5 (11am) Abdul Kadir Husein 1(8am) 3(9am) Mohammad H. Omar (10am) 4 (1.50pm) Name: Student ID#: Serial #: Directions: 1) You must show all work to obtain full credit for questions on this exam. ) DO NOT round your answers at each step. Round answers only if necessary at your final step to 4 decimal places. 3) Use the default α = 0.05 unless specified otherwise. Question No Full Marks Marks Obtained Total 10

2 STAT 1 Business Statistics II 1. (3 marks = ) Can demographic information be helpful in predicting sales at sporting goods stores? The monthly sales totals are recorded for a sample of 38 stores in a large chain of nationwide sporting goods stores. For each of the 38 stores, eight variables are recorded, namely; Sales (Y): Latest one-month sales total ($). Age (X1): Median age of customers (years). HS (X): Percentage of customers with high school diploma. College (X3): Number of customers with college diploma. Growth (X4): Annual population growth rate of customers over the past ten years (%). Income (X5): Median family income of customers ($). Location of store: X6 = 1 if located in Down Town & 0 otherwise. X7 = 1 if located in Urban Area & 0 otherwise. Default location = Rural Area. For the following output, answer the questions that follow: Regression Analysis: Y versus X1; X; X3; X4; X5; X6; X7 The regression equation is Y = X X X X X X X7 Predictor Coef SE Coef T P VIF Constant I X X X II X X X6 III X S = R-Sq = 83.7% R-Sq(adj) = 79.9% Analysis of Variance Source DF SS MS F P Regression E E Residual Error E E+11 Total E+13 Predicted Values for New Observations New Obs Fit SE Fit 99% CI 99% PI ( IV ; ) ( V ; 55690) Values of Predictors for New Observations New Obs X1 X X3 X4 X5 X6 X a. Write down below the quantities that should be in the boxes labeled I-V in the MINITAB output above. I II III IV V b. What is the slope of the Number of customers with college diploma? Interpret it. Slope = - Interpret:

3 STAT 1 Business Statistics II 3 c. Is there any potential multicolinearity problem with this regression analysis? Explain your answer with appropriate details from the MINITAB output. A potential multicollinearity exists Yes / No (circle one) Explanation: d. Is the overall model statistically significant at the 0.05 level? H 0 : The test hypotheses are: H A : The test statistic value = The critical value(s) = Reject H 0 if The decision rule is: The decision is: Conclusion is: We have (enough/not enough) evidence that e. Is the Median family income significantly linearly related to Sales? Explain. H 0 : The test hypotheses are: H A : The test statistic value = Reject H 0 if The decision rule is: The decision is: f. Write the regression model for stores located in the Down Town area. Model: - g. With Age=30, HS =1, College =9000, Growth = 31, Income =75, and a store located at an urban area, what is the fitted Sales value? Fitted value = h. Find 99% C.I. for the regression coefficient of Growth, and interpret this C.I. A 99% C.I. for the regression coefficient of Growth is Interpretation

4 STAT 1 Business Statistics II 4. (7 marks) For the same variables considered in question, the following MINITAB output provides best subset regression models for predicting sales. Use the output to answer the accompanying questions. Best Subsets Regression: Sales versus Age; Growth;... Response is Sales C G I o r n l U o c l T r A w o e o b Mallows g t m H g w a Vars R-Sq R-Sq(adj) Cp S e h e S e n n X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X What is the best model to be selected? Explain your choice. The group of predictors to be used is: R adj = is Explanation: C p = is S ε = is

5 STAT 1 Business Statistics II 5 3. (13 marks = ) Use the MINITAB output below to answer the accompanying questions. Regression Analysis: Profit versus Sales; Cost; x3; x4; Sales*x4 The regression equation is Profit = Sales Cost x x Sales*x4 Predictor Coef SE Coef T P VIF Constant Profit Cost x x Sales*x S = R-Sq = 99.5% Analysis of Variance Source DF SS MS F P Regression Residual Error 59. Total a. Fill in the shaded areas in the MINITAB output above. b. Compute the adjusted R of the model and interpret it Adjusted R = - Interpret: c. The variables x3 (Newspaper) and x4 (Internet) are dummy variables for a categorical variable, Marketing Strategy. How many categories does the variable Marketing strategy has? If the two categories are Newspaper and Internet, what does the reference (or default) category represent? No categories = Reference (default) Category = d. Is the Sales*x4 variable significantly linearly related to Profit? H 0 : The test hypotheses are: H A : The test statistic value = The decision rule is: The decision is: Reject H 0 if e. Provide the correct regression models for each Marketing Strategy categories below i) Model for Internet: ii) Model for Newspaper: iii) Model for Reference (default) category:

6 STAT 1 Business Statistics II 6 4. (10 marks) The data in the table below represent the mean price for three types of energy products in the US on each of the one- thirds period of the year (every four-month period) from 003 to 006. Included are prices of Electricity ($ per 500 kwh), natural gas ($ per 40 therms), and fuel oil ($ per gallon) and their respective consumption Amounts. Energy Prices Consumption Amounts Year Period Electric price Natural Gas price Fuel Oil price Electric (kwh) Natural Gas (therms) Fuel Oil (gallons) First (F) Middle(M) Last(L) F M L F M L F M L

7 STAT 1 Business Statistics II 7 a. Based on the period F of 005, compute the simple price index number of electricity for the period F of 003, and interpret it. Interpretation b. Based on the first period of 003, compute the Laspeyres index of the Energy prices in the first period of 004 and interpret it. Interpretation c. Based on the first period of 003, compute the Paasche index of the Energy prices in the first period of 004 and interpret it. Interpretation d. Based on period L of 003, compute the unweighted aggregate price index number of Energy prices in the last period L of 005 and interpret it. Interpretation e. The single-exponential smoothing forecast for period F of 003, using α =0.4 f. With the regression model Y= X, forecast the price of electricity in the period M of 003 using a doubleexponential smoothing method with a trend-smoothing constant of 0.3 and a constant-process smoothing constant of 0.4 Year CPI g. Assuming the same CPI index can be used within the same year, find the deflated price of electricity in period F on 005 and period M of 006.

8 STAT 1 Business Statistics II 8 5. (14 marks = +++8) Hotel Occupancy Percentages (HOP) in Madinah over the quarter-periods of 003 to 007 were observed as follows. Year Quarter t Hotel Occupancy Rate Use the MINITAB output below to answer the accompanying questions. Regression Analysis: Hotel Occupancy Percentage versus t The regression equation is Hotel Occupancy Percentage = t Predictor Coef SE Coef T P Constant t S = R-Sq = 9.3% R-Sq(adj) = 4.3% Analysis of Variance Source DF SS MS F P Regression Residual Error Total Durbin-Watson statistic = 1.99 a. Find the forecast for Hotel Occupancy Rate in the 3 rd quarter of 003. b. Find the forecast error in the 3 rd quarter of 003 c. Find the mean squared error of the forecast model d. Test that there is a significant positive serial correlation in the forecast model, and comment on your findings. H 0 : The test hypotheses are: H A : The test statistic value = The critical value(s) = Reject H 0 if The decision rule is: The decision is: Conclusion is: We have (enough/not enough) evidence that

9 STAT 1 Business Statistics II 9 6. (1 marks ) Quarterly hotel occupancy percentage (HOP) in Madinah for the period 005 up to 007 are given in the table below. Fill in the blank shaded areas below. Year Quarter HOP Moving Average (MA) Centered MA Ratio to MA Normalized Seasonal Index Deseasoned HOP

10 STAT 1 Business Statistics II 10 Part 7. MULTIPLE CHOICE QUESTIONS (30 = points each). Record (by circling) your answers in the following table. 1 A B C D E A B C D E 3 A B C D E 4 A B C D E 5 A B C D E 6 A B C D E 7 A B C D E 8 A B C D E 9 A B C D E 10 A B C D E 11 A B C D E 1 A B C D E 13 A B C D E 14 A B C D E 15 A B C D E

11 STAT 1 Business Statistics II When performing a chi-square hypothesis test, what happens if the assumption is violated? a. The value of the test statistic will be underestimated b. The null hypothesis will be more likely to be rejected than it should be c. The degrees of freedom are reduced d. a & b e. None of the above. The expected frequency in a chi-square contingency table cell can be calculated from the expected proportion for that cell by multiplying by which of the following? a. that column s total b. that row s total c. the total sample size d. 1, simply e. None of the above 3. If you were given n 1 = 15, n = 1, and a = 0.1, to test H 0 : σ1 = σ against H0 : σ1 > σ, then the test statistic will be compared with which of the following critical value? a. F 0.05;14,11 b. F 0.1;14,11 c. F 0.05;11,14 d. F 0.1;11,14 e. None of the above 4. If α is increased then: a. The probability of rejecting H o when it is true will be decreased b. The probability of accepting H A when it is false will be decreased c. The observed significance level will be increased d. The assumed significance level will be decreased e. None of the above 5. In a simple linear regression problem, the following sum of squares are produced: ( y i y) = 00, ( y ˆ i y i ) = 50, and ( yˆ i y) = 150. What is the percentage of Y variation that is explained by x variation? a. 5% b. 33% c. 50% d. 75% e. 150%

12 STAT 1 Business Statistics II 1 6. A contingency table test with 4 rows and 30 degrees of freedom implies that the table must have how many columns? a. 8 b. 9 c. 10 d. 11 e When the necessary conditions are met, a two-tailed test is being conducted to test the difference between two population means, but MINITAB provides only a one-tail area of.036. The correct p-value for this test will be which of the following? a b c d e ANSWER: d 8. To test a claim against the alternative that the actual proportion of doctors who recommend aspirin is less than 0.90, a random sample of 100 doctors results in 83 who indicate that they recommend aspirin. The test statistic in this problem is approximately equal to which value below? a b..33 c d e..33 ANSWER: b 9. Given the regression model ŷ = x, where x = sales and y = profit, which of the following gives the correct interpretation of the regression slope? a. A 1 unit increase in profit results in a 0.34 unit increase in sales b. A 1 unit increase in sales results in a unit increase in profits c. When sales increases units, profit increases by 1.1 units d. When sales decreases 1 unit, profit increases by units e. A 1 unit increase in profit results in a unit increase in sales 10. You are given the regression model, profit = cost, with SSE=103.5 obtained through a least square method. For the other models below, which of the following may be correct? a. profit = cost can have SSE < b. profit = cost can have SSE = c. profit = 1cost can have SSE = d. profit = cost can have SSE = e. All other models must have SSE 103.5

13 STAT 1 Business Statistics II In testing H : µ 1 µ = 5 vs. H a : µ 1 µ > 5, the test statistic value z is found to be What is the p value of the test? a b c d e For a linear regression model profit = cost where all data points are on this regression line, which of the following is true? a. r xy = - 1 b. r xy = c. b 1 = 0.95 d. r xy = 1 e. b 0 = In testing H o : σ = 5 vs. H a : σ > 5, which of the following is NOT true? a. The test statistics is chi-square with n- degrees of freedom b. The underlying variable must be normal c. The true variance is not known d. The sample variance is known e. The true variance must be greater than or equal to In contingency table analyses, which of the following is true? a. The variables must be continuous b. The degree of freedom is rc r c + 1 c. The variables must be normally distributed d. The attributes in the row and column are independent e. Attributes in the rows are not mutually exclusive 15. Which of the following may be a correct example of a simple linear regression model? a. profit = 0.55(1+1.3Cost) b. profit = Cost + error c. profit = Revenue Cost + 0.5Revenue*Cost d. profit = Revenue Revenue + error e. profit = Revenue Cost + error

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