Question 1: The managing partner of an advertising agency believes that his company's sales are related to the industry sales. He uses Microsoft Excel's Data Analysis tool to analyze the last 4 years of quarterly data (i.e., n = 16) with the following results:

Regression Statistics

Multiple R 0.802

R Square 0.643

Adjusted R Square 0.618

Standard Error SYX 0.9224

Observations 16

ANOVA

df SS MS F Sig.F

Regression 1 21.497 21.497 25.27 0.000

Error 14 11.912 0.851

Total 15 33.409

Predictor Coef StdError t Stat P-value

Intercept 3.962 1.440 2.75 0.016

Industry 0.040451 0.008048 5.03 0.000

Durbin-Watson Statistic 1.59

a) What is the value of the quantity that the least squares regression line minimizes? Explain your answer.

b) What is the prediction of Y for a quarter in which X = 120? Show how you obtain your answer.

c) What is the value for the coefficient of determination?

d) What is the value of the correlation coefficient?

Question 2: An investment specialist claims that if one holds a portfolio that moves in the opposite direction to the market index like the S&P 500, then it is possible to reduce the variability of the portfolio's return. In other words, one can create a portfolio with positive returns but less exposure to risk.

A sample of 26 years of S&P 500 index and a portfolio consisting of stocks of private prisons, which are believed to be negatively related to the S&P 500 index, is collected. A regression analysis was performed by regressing the returns of the prison stocks portfolio (Y) on the returns of S&P 500 index (X) to prove that the prison stocks portfolio is negatively related to the S&P 500 index at a 5% level of significance. The results are given in the following EXCEL output.

	Coefficients	Standard Error	T Stat	P-value
Intercept	4.8660	0.3574	13.6136	8.7932E-13
S&P	-0.5025	0.0716	-7.0186	2.94942E-07

a) To test whether the prison stocks portfolio is negatively related to the S&P 500 index, the appropriate null and alternative hypotheses are, respectively,

A) H0: 0 vs. H1: < 0

B) H0: 0 vs. H1: > 0

C) H0: r 0 vs. H1: r < 0

D) H0: r 0 vs. H1: r > 0

b) To test whether the prison stocks portfolio is negatively related to the S&P 500 index, what is the measured value of the test statistic?

c) To test whether the prison stocks portfolio is negatively related to the S&P 500 index, what is the p-value of the associated test statistic?

d) Which of the following will be a correct conclusion? Explain your answer.

A) We cannot reject the null hypothesis and, therefore, conclude that there is sufficient evidence to show that the prisons stock portfolio and S&P 500 index are negatively related.

B) We can reject the null hypothesis and, therefore, conclude that there is sufficient evidence to show that the prisons stock portfolio and S&P 500 index are negatively related.

C) We cannot reject the null hypothesis and, therefore, conclude that there is not sufficient evidence to show that the prisons stock portfolio and S&P 500 index are negatively related.

D) We can reject the null hypothesis and conclude that there is not sufficient evidence to show that the prisons stock portfolio and S&P 500 index are negatively related.

Question 3: It is believed that GPA (grade point average, based on a four point scale) should have a positive linear relationship with ACT scores. Given below is the Excel output from regressing GPA on ACT scores using a data set of 8 randomly chosen students from a Big-Ten university.

Regressing GPA on ACT
Regression Statistics
Multiple R	0.7598
R Square	0.5774
Adjusted R Square	0.5069
Standard Error	0.2691
Observations	8
ANOVA
	df	SS	MS	F	Significance F
Regression	1	0.5940	0.5940	8.1986	0.0286
Residual	6	0.4347	0.0724
Total	7	1.0287
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	0.5681	0.9284	0.6119	0.5630	-1.7036	2.8398
ACT	0.1021	0.0356	2.8633	0.0286	0.0148	0.1895

a) The interpretation of the coefficient of determination in this regression is

A) 57.74% of the total variation of ACT scores can be explained by GPA.

B) ACT scores account for 57.74% of the total fluctuation in GPA.

C) GPA accounts for 57.74% of the variability of ACT scores.

D) None of the above.

b) What is the value of the measured test statistic to test whether there is any linear relationship between GPA and ACT?

c) What is the predicted average value of GPA when ACT = 20? Show how you obtain your answer.

d) What are the decision and conclusion on testing whether there is any linear relationship at 1% level of significance between GPA and ACT scores? Explain your answer.