Question 1: It is believed that, the average numbers of hours spent studying per day (HOURS) during undergraduate education should have a positive linear relationship with the starting salary (SALARY, measured in thousands of dollars per month) after graduation. Given below is the Excel output from regressing starting salary on number of hours spent studying per day for a sample of 51 students.
Note: Some of the numbers in the output are purposely erased.
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Regression Statistics
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Multiple R
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0.8857
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R Square
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0.7845
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Adjusted R Square
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0.7801
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Standard Error
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1.3704
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Observations
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51
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ANOVA
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df
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SS
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MS
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F
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Significance F
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Regression
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1
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335.0472
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335.0473
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178.3859
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Residual
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1.8782
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Total
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50
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427.0798
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Coefficients
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Standard Error
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t Stat
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P-value
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Lower 95%
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Upper 95%
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Intercept
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-1.8940
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0.4018
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-4.7134
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2.051E-05
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-2.7015
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-1.0865
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Hours
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0.9795
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0.0733
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13.3561
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5.944E-18
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0.8321
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1.1269
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f) The 90% confidence interval for the average change in SALARY (in thousands of dollars) as a result of spending an extra hour per day studying is
A) wider than [-2.70159, -1.08654].
B) narrower than [-2.70159, -1.08654].
C) wider than [0.8321927, 1.12697].
D) narrower than [0.8321927, 1.12697].
Explain your reasoning.
g) To test the claim that average SALARY depends positively on HOURS against the null hypothesis that average SALARY does not depend linearly on HOURS, what is the p-value of the test statistic? What are the results of the test? Explain your answer.
Question 2: The management of a chain electronic store would like to develop a model for predicting the weekly sales (in thousands of dollars) for individual stores based on the number of customers who made purchases. A random sample of 12 stores yields the following results:
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Customers
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Sales (Thousands of Dollars)
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907
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11.20
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926
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11.05
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713
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8.21
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741
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9.21
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780
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9.42
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898
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10.08
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510
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6.73
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529
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7.02
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460
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6.12
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872
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9.52
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650
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7.53
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603
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7.25
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a) Estimate a linear regression. What are the values of the estimated intercept and slope? Show how you obtain your answer.
b) What is the value of the coefficient of determination?
c) What is the value of the coefficient of correlation?
d) What is the value of the standard error of the estimate?
e) Which of the following is the correct null hypothesis for testing whether the number of customers who make purchases affects weekly sales?
A) H0 : β0 = 0
B) H0 : β1 = 0
C) H0 : μ = 0
D) H0 : ρ = 0
f) What is the value of the t test statistic when testing whether the number of customers who make purchases affects weekly sales?
g) What are the degrees of freedom of the t test statistic when testing whether the number of customers who make purchases affects weekly sales?
h) Construct a 95% confidence interval for the change in average weekly sales when the number of customers who make purchases increases by one. Show how you obtain your answer.
i) Construct a 95% confidence interval for the average weekly sales when the number of customers who make purchases is 600. Show how you obtain your answer.
j) Construct a 95% prediction interval for the weekly sales of a store that has 600 purchasing customers. Show how you obtain your answer.