Consider a regression analysis with n = 34 and four potential independent variables. Suppose that one of the independent variables has a correlation of 0.23 with the dependent variable. Does this imply that this independent variable will have a very small Student’s t statistic in the regression analysis with all four predictor variables?

Let f satisfy f(t+T)=f(t)for all t≥0 and for some fixed positive number T; f is said to be periodic with period Ton 0≤t<∞. Show that L{f(t)}=∫T0e−stf(t)dt1−e−sT

In a survey of 27 undergraduates at the University of Illinois the accompanying results were obtained with grade point averages $(y)$, the number of hours per week spent studying $\left(x_1\right)$, the average number of hours spent preparing for tests $\left(x_2\right)$, the number of hours per week spent in bars $\left(x_3\right)$, whether students take notes or mark highlights when reading texts $\left(x_4=1\right.$ if yes, 0 if no $)$, and the average number of credit hours taken per semester $\left(x_5\right)$. Estimate the regression of grade point average on the five independent variables, and write a report on your findings. The data are in the data file Student Performance.

A regression model was estimated to compare performance of students taking a business statistics course— either as a standard 14-week course or as an intensive 3-week course. The following model was estimated from observations of 350 students (Van Scyoc and Gleason 1993):

$$\hat{y}=-.7052+\underset{(0.4568)}{1.4170 x_1}+\underset{(0.3287)}{2.1624 x_2}+\underset{(.4393)}{.8680 x_3}+\underset{(0.3766)}{1.0845 x_4}$$

$$+\underset{(0.0628)}{0.4694 x_5}+\underset{(0.0094)}{0.0038 x_6}+\underset{(0.0776)}{0.0484 x_7} \quad R^2=0.344$$

where

$\hat{y}$ = score on a standardized test of understanding of statistics after taking the course

$x_1$ = dummy variable taking the value 1 if the 3-week course was taken and 0 if the 14-week course was taken

$x_2$ = student's grade point average

$x_3$ = dummy variable taking the value 0 or 1, depending on which of two teachers had taught the course

$x_4$ = dummy variable taking the value 1 if the student is male and 0 if female

$x_5$ = score on a standardized test of understanding of mathematics before taking the course

$x_6$ = number of semester credit hours the student had completed

$x_7$ = age of student

The numbers in parentheses under the coefficients are the estimated coefficient standard errors. Write a report discussing what can be learned from this fitted regression.

In a study of the influence of financial institutions on bond interest rates in Germany, quarterly data over a period of 12 years were analyzed. The postulated model was

$$y_i=\beta_0+\beta_1 x_{1 i}+\beta_2 x_{2 i}+\varepsilon_i$$

where

$y_i=$ change over the quarter in the bond interest rates

$x_{1 i}=$ change over the quarter in bond purchases by financial institutions

$x_{2 i}=$ change over the quarter in bond sales by financial institutions

The estimated regression coefficients were as follows:

$$b_1=0.057 \quad b_2=-0.065$$

Interpret these estimates.

Random samples of 900 people in the United States and in Great Britain indicated that 60% of the people in the United States were positive about the future economy, whereas 66% of the people in Great Britain were positive about the future economy. Does this provide strong evidence that the people in Great Britain are more optimistic about the economy?

A consulting group offers courses in financial management for executives. At the end of these courses participants are asked to provide overall ratings of the value of the course. For a sample of 25 courses, the following regression was estimated by least squares:

$$\hat{y}=42.97+\underset{(0.29)}{0.38 x_1}+\underset{(0.21)}{0.52 x_2}-\underset{(0.11)}{0.08 x_3}+\underset{(0.359)}{6.21 x_4} \quad R^2=0.569$$

where

$\hat{y}$ = average rating by participants of the course

$x_1$ = percentage of course time spent in group discussion sessions

$x_2$ = money, in dollars, per course member spent on preparing course material

$x_3$ = money, in dollars, per course member spent on food and drinks

$x_4$ = dummy variable taking the value 1 if a visiting guest lecturer is brought in and 0 otherwise

The numbers in parentheses under the coefficients are the estimated coefficient standard errors.

a. Interpret the estimated coefficient on $x_4$.

b. Test, against the alternative that it is positive, the null hypothesis that the true coefficient on $x_4$ is 0.

c. Interpret the coefficient of determination, and use it to test the null hypothesis that, taken as a group, the four independent variables do not linearly influence the dependent variable.

d. Find and interpret a $95 \%$ confidence interval for $\beta_2$.

The following model was fitted to data on 50 states:

$$\hat{y}=13,472+\underset{(124.3)}{547 x_1}+\underset{(1.858)}{5.48 x_2}+\underset{(208.9)}{493 x_3}+\underset{(234)}{32.7 x_4}$$

$$\begin{aligned} &+5,793 x_5-3,100 x_6 \quad R^2=.54 \\ &\begin{array}{ll} (2,897) & (1,761) \end{array} \\ & \end{aligned}$$

where

$\hat{y}$ = annual salary of the attorney general of the state

$x_1$ = average annual salary of lawyers, in thousands of dollars

$x_2$ = number of bills enacted in previous legislative session

$x_3$ = number of due process reviews by state courts that resulted in overturn of legislation in previous 40 years

$x_4$ = length of term of the attorney general of the state

$x_5$ = dummy variable taking value 1 if justices of the state supreme court can be removed from office by the governor, judicial review board, or majority vote of the supreme court and 0 otherwise

$x_6$ = dummy variable taking value 1 if supreme court justices are elected on partisan ballots and 0 otherwise

The numbers in parentheses under the coefficients are the estimated coefficient standard errors.

a. Interpret the estimated coefficient on the dummy variable $x_5$.

b. Interpret the estimated coefficient on the dummy variable $x_6$.

c. Test, at the $5 \%$ level, the null hypothesis that the true coefficient on the dummy variable $x_5$ is 0 against the alternative that it is positive.

d. Test, at the $5 \%$ level, the null hypothesis that the true coefficient on the dummy variable $x_6$ is 0 against the alternative that it is negative.

e. Find and interpret a $95 \%$ confidence level for the parameter $\beta_1$.

Tell whether the sampling strategy will result in a random sample from the population under consideration. If it is not random, explain why it is not. To get a sample of 50 students for a student services survey, an assistant principal uses the calculator command "randInt(1, 72, 5)" to pick five numbers between 1 and 72. He goes to those five pages in the 72-page student directory and chooses the ten students pictured on each of those five pages.

A business school dean wanted to assess the importance of factors that might help in predicting success in law school. For a random sample of 50 students, data were obtained when students graduated from law school, and the following model was fitted:

$$y=\alpha+\beta_1 x_1+\beta_2 x_2+\beta_3 x_3+\varepsilon$$

where

$y=$ score reflecting overall performance while in law school

$x_1=$ undergraduate grade point average

$x_2=$ score on GMAT

$x_3=$ dummy variable taking the value 1 if a student's letters of recommendation are unusually strong and 0 otherwise

Use the portion of the computer output from the estimated regression shown here to write a report summarizing the findings of this study.

$$\begin{aligned} &\begin{array}{lrrrcc} \hline \text { Source } & \text { DF } & \begin{array}{c} \text { Sum of } \\ \text { Squares } \end{array} & \begin{array}{c} \text { Mean } \\ \text { Square } \end{array} & \begin{array}{c} F \\ \text { Value } \end{array} & \begin{array}{c} R- \\ \text { Square } \end{array} \\ \hline \text { Model } & 3 & 641.04 & 212.68 & 8.48 & 0.356 \\ \text { Error } & 46 & 1,159.66 & 25.21 & & \end{array}\\ &\begin{array}{lll} \text { Total } & 49 & 1,800.70 \\ \hline \end{array} \end{aligned}$$

$$\begin{array}{cccc} \hline \text { Parameter } & \text { Estimate } & \begin{array}{c} t \text { for } H_0: \\ \beta_j=0 \end{array} & \begin{array}{c} \text { Std. Error of } \\ \text { Estimate } \end{array} \\ \hline \text { Intercept } & 6.512 & & \\ \text { X1 } & 3.502 & 1.45 & 2.419 \\ \text { X2 } & 0.491 & 4.59 & 0.107 \\ \text { X3 } & 10.327 & 2.45 & 4.213 \\ \hline \end{array}$$

The following model was fitted to data on 32 insurance companies:

$$\hat{y}=7.62-\underset{(0.008)}{0.16 x_1}+\underset{(0.496)}{1.23 x_2} \quad R^2=0.37$$

where

$\hat{y}=$ price-earnings ratio

$x_1=$ size of insurance company assets, in billions of dollars

$x_2=$ dummy variable taking the value 1 for regional companies and 0 for national companies

The numbers in parentheses under the coefficients are the estimated coefficient standard errors.

a. Interpret the estimated coefficient on the dummy variable.

b. Test against a two-sided alternative. the null hypothesis that the true coefficient on the dummy variable is 0.

c. Test, at the 5% level, the null hypothesis $\beta_1=\beta_2=0$, and interpret your result.

The following model was fitted to explain the selling prices of condominiums in a sample of 815 sales:

$$\hat{y}=\underset{(0.91)}{-1264}+\underset{(515)}{48.18 x_1}+\underset{(488)}{3382 x_2}-\underset{(947)}{1859 x_3}+\underset{(768)}{3219 x_4}$$

$$+\underset{(768)}{2005 x_5} \quad \bar{R}^2=0.86$$

where

$\hat{y}=$ selling price of condo, in dollars

$x_1=$ square footage of living area

$x_2=$ size of garage, in number of cars

$x_3=$ age of condo, in years

$x_4=$ dummy variable taking the value 1 if the condo has a fireplace and 0 otherwise

$x_5=$ dummy variable taking the value 1 if the condo has hardwood floors and 0 if it has vinyl floors

a. Interpret the estimated coefficient of $x_4$.

b. Interpret the estimated coefficient of $x_5$.

c. Find a $95 \%$ confidence interval for the impact of a fireplace on selling price, all other things being equal.

d. Test the null hypothesis that type of flooring has no impact on selling price against the alternative that, all other things equal, condos with hardwood floors have a higher selling price than those with vinyl flooring.

The following model was fitted to observations from 1972 to 1979 in an attempt to explain oil-pricing behavior:

$$\hat{y}=\underset{(0.029)}{37 x_1}+\underset{(0.50)}{5.22 x_2}$$

where

$\hat{y}=$ difference between price in the current year and price in the previous year, in dollars per barrel

$x_1=$ difference between spot price in the current year and spot price in the previous year

$x_2=$ dummy variable taking the value 1 in 1974 and 0 otherwise to represent the specific effect of the oil embargo of that year

The numbers in parentheses under the coefficients are the estimated coefficient standard errors.

Interpret verbally and graphically the estimated coefficient on the dummy variable.

What are the model constant and the slope coefficient of $x_1$ when the dummy variable equals 1 in the following equations, where $x_1$ is a continuous variable and $x_2$ is a dummy variable with a value of 0 or 1?

a. $\hat{y}=4+9 x_1+1.78 x_2+3.09 x_1 x_2$

b. $\hat{y}=-3+7 x_1+4.15 x_2+2.51 x_1 x_2$

c. $\hat{y}=10+5 x_1+3.67 x_2+3.98 x_1 x_2$