Chapter 10.2, Problem 32E

For Exercises 28 through 33, do a complete regression analysis by performing these steps.

a. Draw a scatter plot.

b. Compute the correlation coefficient.

c. State the hypotheses.

d. Test the hypotheses at α = 0.05. Use Table I.

e. Determine the regression line equation if r is significant.

f. Plot the regression line on the scatter plot, if appropriate.

g. Summarize the results.

32. Television Viewers A television executive selects 10 television shows and compares the average number of viewers the show had last year with the average number of viewers this year. The data (in millions) are shown. Describe the relationship.

To determine

To construct: The scatterplot for the variablesthe average number of viewers the show hadlast year and the average number of viewers this year.

Answer to Problem 32E

Output using the MINITAB software is given below:

Explanation of Solution

Given info:

The data shows the average number of viewers the show hadlast year (x) and the average number of viewers this year(y) values.

Calculation:

Step by step procedure to obtain scatterplot using the MINITAB software:

• Choose Graph > Scatterplot.
• Choose Simple and then click OK.
• Under Y variables, enter a column ofViewers last year.
• Under X variables, enter a column of Viewers this year.
• Click OK.

To determine

To compute: The value of the correlation coefficient.

Answer to Problem 32E

The value of the correlation coefficientis 0.839.

Explanation of Solution

Calculation:

Correlation coefficient r:

Software Procedure:

Step-by-step procedure to obtain the ‘correlation coefficient’ using the MINITAB software:

• Select Stat >Basic Statistics > Correlation.
• In Variables, select x and y from the box on the left.
• Click OK.

Output using the MINITAB software is given below:

From the MINITAB output, the value of the correlation is 0.839.

To determine

To state: The hypothesis.

Answer to Problem 32E

The null hypothesis is $H_0:\rho =0$.

The alternative hypothesis is $H_1:\rho \ne 0$.

Explanation of Solution

Calculation:

The hypotheses are given below:

Null hypothesis:

$H_0:\rho =0$

That is, there is no linear relation betweenthe average number of viewers the show hadlast year and the average number of viewers this year.

Alternative hypothesis:

$H_1:\rho \ne 0$

That is, there is a linear relationbetween the average number of viewers the show hadlast year and the average number of viewers this year.

To determine

To test: The significance of the correlation coefficient at $\alpha =0.05$ using Table I.

Answer to Problem 32E

The conclusion is that, there is a sufficient evidence to support the claim that linear relation betweenthe average number of viewers the show had last year and the average number of viewers this year.

Explanation of Solution

Given info:

The level of significance is $\alpha =0.05$ and the sample size is 8.

Calculation:

The sample size is 10.

The formula to find the degrees of the freedom is $n-2$.

That is,

$\begin{array}{c}n-2=10-2\\ =8\end{array}$

From the “TABLE –I: Critical Values for the PPMC”, the critical value for 4 degrees of freedom and $\alpha =0.05$ level of significance is 0.632.

Rejection Rule:

If the absolute value of r is greater than the critical value then reject the null hypothesis.

Conclusion:

From part (b), the value of r is0.839 that is the absolute value of r is 0.839.

Here, the absolute value of r is greater than the critical value $\left|r\right|>\text{critical value}$.

That is, $\left|r\right|\left(=0.839\right)>\text{critical value}\left(=0.632\right)$.

By the rejection rule,reject the null hypothesis.

There is sufficient evidence to support the claim that “there is alinear relation betweenthe average number of viewers the show had last year and the average number of viewers this year”.

To determine

To find: The regression equation for the given data.

Answer to Problem 32E

The regression equation for the given datais $\widehat{y}=-3.67+1.28x$.

Explanation of Solution

Calculation:

Regression:

Software procedure:

Step by step procedure to obtain the regression equation using the MINITAB software:

• Choose Stat > Regression > Regression.
• In Responses, enter the column ofViewers this year.
• In Predictors, enter the column ofViewers last year.
• Click OK.

Output using the MINITAB software is given below:

Thus, regression equation for the given datais $\widehat{y}=-3.67+1.28x$.

To determine

To construct: The scatterplot for the variablesthe average number of viewers the show hadlast year and the average number of viewers this year.

Answer to Problem 32E

Output using the MINITAB software is given below:

Explanation of Solution

Calculation:

Step by step procedure to obtain scatterplot using the MINITAB software:

• Choose Graph > Scatterplot.
• Choose with line and then click OK.
• Under Y variables, enter a column of Viewers last year.
• Under X variables, enter a column of Viewers this year.
• Click OK.

To determine

To summarize: The results.

Answer to Problem 32E

Explanation of Solution

Justification:

Thus, there is a sufficient evidence to support the claim that linear relation betweenthe average number of viewers the show had last year and the average number of viewers this year.

To determine

To explain: The type of relation.

Answer to Problem 32E

The type of relation is the positivelinear relation.

Explanation of Solution

Justification:

From part (a), it is observed that there is a positive linear relation between the variables.

Thus, it can be conclude that there is the type of the relation is “linear relation”.