Based on a sample of 25 observations, the population regression model

$$y_i=\beta_0+\beta_1 x_1+\varepsilon_i$$

was estimated. The least squares estimates obtained were as follows:

$$b_0=15.6 \text { and } b_1=1.3$$

The total and error sums of squares were as follows:

$$S S T=268 \text { and } \quad S S E=204$$

a. Find and interpret the coefficient of determination.

b. Test, against a two-sided alternative at the $5 \%$ significance level, the null hypothesis that the slope of the population regression line is 0 .

c. Find a $95 \%$ confidence interval for $\beta_1$.

An attempt was made to evaluate the inflation rate as a predictor of the spot rate in the German treasury bill market. For a sample of 79 quarterly observations, the estimated linear regression

$$\hat{y}=0.0027+0.7916 x$$

was obtained, where

y = actual change in the spot rate

x = change in the spot rate predicted by the inflation rate

The coefficient of determination was 0.097, and the estimated standard deviation of the estimator of the slope of the population regression line was 0.2759.

a. Interpret the slope of the estimated regression line.

b. Interpret the coefficient of determination.

c. Test the null hypothesis that the slope of the population regression line is 0 against the alternative that the true slope is positive, and interpret your result.

d. Test, against a two-sided alternative, the null hypothesis that the slope of the population regression line is 1, and interpret your result.

Based on a sample on $n$ observations, $\left(x_1, y_1\right)$, $\left(x_2, y_2\right), \ldots,\left(x_n, y_n\right)$, the sample regression of $y$ on $x$ is calculated. Show that the sample regression line passes through the point $(x=\bar{x}, y=\bar{y})$, where $\bar{x}$ and $\bar{y}$ are the sample means.

For a sample of 66 months, the correlation between the returns on Canadian and Singapore 10-year bonds was found to be 0.293. Test the null hypothesis that the population correlation is 0 against the alternative that it is positive.

For a random sample of 526 firms, the sample correlation between the proportion of a firm’s officers who are directors and a risk-adjusted measure of return on the firm’s stock was found to be 0.1398. Test, against a two-sided alternative, the null hypothesis that the population correlation is 0.

For a random sample of 53 building supply stores in a chain, the correlation between annual sales per square meter of floor space and annual rent per square meter of floor space was found to be 0.37. Test the null hypothesis that these two quantities are uncorrelated in the population against the alternative that the population correlation is positive.

Given the regression equation

$$Y = 100 + 10X$$

a. What is the change in Y when X changes by +3?

b. What is the change in Y when X changes by -4?

c. What is the predicted value of Y when X = 12?

d. What is the predicted value of Y when X = 23?

e. Does this equation prove that a change in X causes a change in Y?

Charlie Ching has asked you to analyze the possibility of including Seneca Foods and Safeco in his portfolio. Data for this task are contained in the data file Return on Stock Price 60 Months. Compute the beta coefficients for the stock price growth for each stock. Then construct a portfolio that includes equal dollar value for both stocks. Compute the beta coefficient for that portfolio. Compare the mean and variance for the portfolio with the S & P 500. What is your recommendation regarding the inclusion of these two stocks in Charlie’s portfolio?

Allied Financial is considering the possibility of adding one or more computer industry stocks to its portfolio. You are asked to consider the possibility of Seagate, Microsoft, and Tata Information systems. Data for this task are contained in the data file Return on Stock Price 60 Months. Compare the return on these three stocks by computing the beta coefficients and the mean and variance of the returns. What is your recommendation regarding these three stocks?

An investor is considering the possibility of including TCF Financial in her portfolio. Data for this task are contained in the data file Return on Stock Price 60 Months. Compare the mean and variance of the monthly return with the S & P 500 mean and variance. Then, estimate the beta coefficient. Based on this analysis, what would you recommend to the investor?

In this exercise you are asked to determine the beta coefficient for Senior Housing Properties Trust. Data for this task are contained in the data file Return on Stock Price 60 Months. Interpret this coefficient.

In an advertising study the researchers wanted to determine if there was a relationship between the per capita cost and the per capita revenue. The following variables were measured for a random sample of advertising programs:

$x_i=$ Cost of Advertisement $\div$ Number of Inquiries Received

$y_i=$ Revenue from Inquiries $\div$ Number of Inquiries Received

The sample data results are shown in the data file Advertising Revenue. Find the sample correlation and test, against a two-sided alternative, the null hypothesis that the population correlation is 0 .

A college administers a student evaluation questionnaire for all its courses. For a random sample of 12 courses, the accompanying table and the data file Student Evaluation show both the average student ratings of the instructor (on a scale of 1 to 5), and the average expected grades of the students (on a scale of A = 4 to F = 0).

$$\begin{array}{lllllllllllll} \hline \begin{array}{l} \text { Instructor } \\ \text { rating } \end{array} & 2.8 & 3.7 & 4.4 & 3.6 & 4.7 & 3.5 & 4.1 & 3.2 & 4.9 & 4.2 & 3.8 & 3.3 \\ \hline \begin{array}{l} \text { Expected } \\ \text { grade } \end{array} & 2.6 & 2.9 & 3.3 & 3.2 & 3.1 & 2.8 & 2.7 & 2.4 & 3.5 & 3.0 & 3.4 & 2.5 \\ \hline \end{array}$$

a. Find the sample correlation between instructor ratings and expected grades.

b. Test, at the 10% significance level, the hypothesis that the population correlation coefficient is zero against the alternative that it is positive.

The accompanying table and the data file Dow Jones show percentage changes $\left(x_i\right)$ in the Dow Jones index over the first five trading days of each of 13 years and also the corresponding percentage changes $\left(y_i\right)$ in the index over the whole year.

a. Calculate the sample correlation.

b. Test, at the 10% significance level against a two-sided alternative, the null hypothesis that the population correlation is 0.

$$\begin{array}{rrrr} \hline x & y & x & y \\ \hline 1.5 & 14.9 & 5.6 & 2.3 \\ 0.2 & -9.2 & -1.4 & 11.9 \\ -0.1 & 19.6 & 1.4 & 27.0 \\ 2.8 & 20.3 & 1.5 & -4.3 \\ 2.2 & -3.7 & 4.7 & 20.3 \\ -1.6 & 27.7 & 1.1 & 4.2 \\ -1.3 & 22.6 & & \\ \hline \end{array}$$