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Southerly Showtime Movie Theaters, Inc. owns and operates a chain of cinemas in several markets in the southern United States. The owners would like to estimate weekly gross revenue as a function of advertising expenditures. Data for a sample of eight markets for a recent week follow. $($Let x$1$ represent Television Advertising $($$$100$s$),$ x$2$ represent Newspaper Advertising $($$$100$s$),$ and y represent Weekly Gross Revenue $($$$100$s$).)$

Market Weekly Gross Revenue $($$$100$s$)$ Television Advertising $($$$100$s$)$ Newspaper Advertising $($$$100$s$)$

Market $1101\mathrm{.}34\mathrm{.}91\mathrm{.}4$

Market $251\mathrm{.}93\mathrm{.}13\mathrm{.}1$

Market $374\mathrm{.}84\mathrm{.}11\mathrm{.}6$

Market $4126\mathrm{.}24\mathrm{.}44\mathrm{.}4$

Market $5137\mathrm{.}83\mathrm{.}54\mathrm{.}1$

Market $6101\mathrm{.}43\mathrm{.}42\mathrm{.}2$

Market $7237\mathrm{.}84\mathrm{.}98\mathrm{.}3$

Market $8219\mathrm{.}66\mathrm{.}85\mathrm{.}7$

$($a$)$

Develop an estimated linear regression equation with the amount of television advertising as the independent variable. $($Round your numerical values to four decimal places.$)$

$=$

Test for a significant relationship between the amount spent on television advertising and weekly gross revenue at the $0\mathrm{.}05$ level of significance. $($Use the t test.$)$

Find the p$-$value. $($Round your answer to four decimal places.$)$

p$-$value $=$

State your conclusion. $($Make your conclusion regardless of any validity concerns.$)$

We fail to reject H$0.$ We can conclude that there is a relationship between the amount spent on television advertising and weekly gross revenue.

We reject H$0.$ We cannot conclude that there is a relationship between the amount spent on television advertising and weekly gross revenue.

We fail to reject H$0.$ We cannot conclude that there is a relationship between the amount spent on television advertising and weekly gross revenue.

We reject H$0.$ We can conclude that there is a relationship between the amount spent on television advertising and weekly gross revenue.

What is the interpretation of this relationship?

Our best estimate is that a $$100$ increase in television advertising corresponds to a decrease of b$1$ hundred dollars in weekly gross revenue.

Our best estimate is that a $$100$ increase in weekly gross revenue corresponds to an increase of b$1$ hundred dollars in television advertising.

Our best estimate is that a $$100$ increase in weekly gross revenue corresponds to a decrease of b$1$ hundred dollars in television advertising.

Our best estimate is that a $$100$ increase in television advertising corresponds to an increase of b$1$ hundred dollars in weekly gross revenue.

$($b$)$

How much of the variation in the sample values of weekly gross revenue $($in $\%)$ does the model in part $($a$)$ explain? $($Round your answer to two decimal places.$)$

$\%$

$($c$)$

Develop an estimated linear regression equation with both television advertising and newspaper advertising as the independent variables. $($Round your numerical values to four decimal places.$)$

$=$

Test whether the parameter $0$ is equal to zero at a $0\mathrm{.}05$ level of significance.

Find the p$-$value. $($Round your answer to four decimal places.$)$

p$-$value $=$

State your conclusion. $($Make your conclusion regardless of any validity concerns.$)$

We reject H$0.$ We can conclude that the y$-$intercept is not equal to zero.

We fail to reject H$0.$ We cannot conclude that the y$-$intercept is not equal to zero.

We reject H$0.$ We cannot conclude that the y$-$intercept is not equal to zero.

We fail to reject H$0.$ We can conclude that the y$-$intercept is not equal to zero.

Test whether the parameter $1$ is equal to zero at a $0\mathrm{.}05$ level of significance.

Find the p$-$value. $($Round your answer to four decimal places.$)$

p$-$value $=$

State your conclusion. $($Make your conclusion regardless of any validity concerns.$)$

We fail to reject H$0.$ We can conclude that there is a relationship between the amount spent on television advertising and weekly gross revenue.

We reject H$0.$ We can conclude that there is a relationship between the amount spent on television advertising and weekly gross revenue.

We reject H$0.$ We cannot conclude that there is a relationship between the amount spent on television advertising and weekly gross revenue.

We fail to reject H$0.$ We cannot conclude that there is a relationship between the amount spent on television advertising and weekly gross revenue.

Test whether the parameter $2$ is equal to zero at a $0\mathrm{.}05$ level of significance.

Find the p$-$value. $($Round your answer to four decimal places.$)$

p$-$value $=$

State your conclusion. $($Make your conclusion regardless of any validity concerns.$)$

We fail to reject H$0.$ We can conclude that there is a relationship between the amount spent on newspaper advertising and weekly gross revenue.

We reject H$0.$ We cannot conclude that there is a relationship between the amount spent on newspaper advertising and weekly gross revenue.

We reject H$0.$ We can conclude that there is a relationship between the amount spent on newspaper advertising and weekly gross revenue.

We fail to reject H$0.$ We cannot conclude that there is a relationship between the amount spent on newspaper advertising and weekly gross revenue.

Interpret the estimated parameter b$0$ and determine if this is reasonable.

The intercept occurs when both independent variables are one. Thus, b$0$ is the estimate of the weekly gross revenue when there is $$100$ spent on both television and newspaper advertising. This estimated parameter was based on extrapolation, so it is reasonable.

The intercept occurs when both independent variables are zero. Thus, b$0$ is the estimate of the weekly gross revenue when there is no money spent on television or newspaper advertising. This estimated parameter was based on extrapolation, so it is reasonable.

The intercept occurs when both independent variables are one. Thus, b$0$ is the estimate of the weekly gross revenue when there is $$100$ spent on both television and newspaper advertising. This estimated parameter was based on extrapolation, so it is not reasonable.

The intercept occurs when both independent variables are zero. Thus, b$0$ is the estimate of the weekly gross revenue when there is no money spent on television or newspaper advertising. This estimated parameter was based on extrapolation, so it is not reasonable.

Interpret the estimated parameter b$1$ and determine if this is reasonable.

b$1$ describes the change in y when there is a one$-$unit increase of x$2$ and x$1$ is held constant. Thus, b$1$ is the estimated change in the weekly gross revenue when television advertising is held constant and there is a $$100$ increase in newspaper advertising. This estimated parameter is not reasonable.

b$1$ describes the change in y when there is a one$-$unit increase of x$1$ and x$2$ is held constant. Thus, b$1$ is the estimated change in the weekly gross revenue when newspaper advertising is held constant and there is a $$100$ increase in television advertising. This estimated parameter is reasonable.

b$1$ describes the change in y when there is a one$-$unit increase of x$1$ and x$2$ is held constant. Thus, b$1$ is the estimated change in the weekly gross revenue when newspaper advertising is held constant and there is a $$100$ increase in television advertising. This estimated parameter is not reasonable.

b$1$ describes the change in y when there is a one$-$unit increase of x$2$ and x$1$ is held constant. Thus, b$1$ is the estimated change in the weekly gross revenue when television advertising is held constant and there is a $$100$ increase in newspaper advertising. This estimated parameter is reasonable.

Interpret the estimated parameter b$2$ and determine if this is reasonable.

b$2$ describes the change in y when there is a one$-$unit increase of x$1$ and x$2$ is held constant. Thus, b$2$ is the estimated change in the weekly gross revenue when newspaper advertising is held constant and there is a $$100$ increase in television advertising. This estimated parameter is reasonable.

b$2$ describes the change in y when there is a one$-$unit increase of x$2$ and x$1$ is held constant. Thus, b$2$ is the estimated change in the weekly gross revenue when television advertising is held constant and there is a $$100$ increase in newspaper advertising. This estimated parameter is reasonable.

b$2$ describes the change in y when there is a one$-$unit increase of x$1$ and x$2$ is held constant. Thus, b$2$ is the estimated change in the weekly gross revenue when newspaper advertising is held constant and there is a $$100$ increase in television advertising. This estimated parameter is not reasonable.

b$2$ describes the change in y when there is a one$-$unit increase of x$2$ and x$1$ is held constant. Thus, b$2$ is the estimated change in the weekly gross revenue when television advertising is held constant and there is a $$100$ increase in newspaper advertising. This estimated parameter is not reasonable.

$($d$)$

How much of the variation in the sample values of weekly gross revenue $($in $\%)$ does the model in part $($c$)$ explain? $($Round your answer to two decimal places.$)$

$\%$

$($e$)$

Given the results in parts $($a$)$ and $($c$),$ what should your next step be$?$ Explain.