Chapter 10.2, Problem 30E

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.

30. SAT Scores Educational researchers desired to find out if a relationship exists between the average SAT verbal score and the average SAT mathematical score. Several states were randomly selected, and their SAT average scores are recorded below. Is there sufficient evidence to conclude a relationship between the two scores?

To determine

To construct: The scatterplot for the variables the average SAT verbal score and the average SAT mathematical score.

Answer to Problem 30E

Output using the MINITAB software is given below:

Explanation of Solution

Given info:

The data shows the average SAT verbal score (x) and the average SAT mathematical score (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 of Verbal.
• Under X variables, enter a column of Math.
• Click OK.

To determine

To compute: The value of the correlation coefficient.

Answer to Problem 30E

The value of the correlation coefficient is 0.990.

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 0.990.

To determine

To state: The hypothesis.

Answer to Problem 30E

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 between the average SAT verbal score and the average SAT mathematical score.

Alternative hypothesis:

$H_1:\rho \ne 0$

That is, there is a linear relation between the average SAT verbal score and the average SAT mathematical score.

To determine

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

Answer to Problem 30E

The conclusion is that, there is sufficient evidence to support the claim that linear relation between the average SAT verbal score and the average SAT mathematical score.

Explanation of Solution

Given info:

The level of significance is $\alpha =0.05$ and the sample size is 6.

Calculation:

The sample size is 6.

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

That is,

$\begin{array}{c}n-2=6-2\\ =4\end{array}$

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 0.811.

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 is 0.990 That is the absolute value of r is 0.990.

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

That is, $\left|r\right|\left(=0.990\right)>\text{critical value}\left(=0.811\right)$.

By the rejection rule, reject the null hypothesis.

There is sufficient evidence to support the claim that “there is a linear relation between the average SAT verbal score and the average SAT mathematical score”.

To determine

To find: The regression equation for the given data.

Answer to Problem 30E

The regression equation for the given data is $\widehat{y}=63.5+0.900x$.

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 of Math.
• In Predictors, enter the column of Verbal.
• Click OK.

Output using the MINITAB software is given below:

Thus, regression equation for the given data is $\widehat{y}=63.5+0.900x$.

To determine

To construct: The scatterplot for the variables the average SAT verbal score and the average SAT mathematical score with regression line.

Answer to Problem 30E

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 Verbal.
• Under X variables, enter a column of Math.
• Click OK.

To determine

To summarize: The results.

Answer to Problem 30E

Explanation of Solution

Justification:

Thus, there is sufficient evidence to support the claim that “there is a linear relation between the average SAT verbal score and the average SAT mathematical score”.