Chapter 10, Problem 10.1.6RE

For Exercises 1 through 7, do a complete regression analysis by performing the following steps.

a. Draw the scatter plot.

b. Compute the value of the correlation coefficient.

c. Test the significance of the correlation coefficient at α = 0.01, using Table I.

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

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

f. Predict y′ for a specific value of x, if appropriate.

6. Protein and Diastolic Blood Pressure A study was conducted with vegetarians to see whether the number of grams of protein each ate per day was related to diastolic blood pressure. The data are given here. If there is a significant relationship, predict the diastolic pressure of a vegetarian who consumes 8 grams of protein per day. (This information will be used for Exercises 10 and 12.)

To determine

To construct: The scatterplot for the variables the number of grams of protein and the diastolic blood pressure.

Answer to Problem 10.1.6RE

Output using the MINITAB software is given below:

Explanation of Solution

Given info:

The data shows the number of grams of protein (x) and the diastolic blood pressure (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 Pressure.
• Under X variables, enter a column of Grams.
• Click OK.

To determine

To compute: The value of the correlation coefficient.

Answer to Problem 10.1.6RE

The value of the correlation coefficient is 0.916.

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 0.916.

To determine

To test: The significance of the correlation coefficient at $\alpha =0.01$ using Table I.

Answer to Problem 10.1.6RE

The conclusion is that, there is a linear relation between the number of grams of protein and the diastolic blood pressure.

Explanation of Solution

Given info:

The level of significance is $\alpha =0.01$ and the sample size is 9.

Calculation:

The hypotheses are given below:

Null hypothesis:

$H_0:\rho =0$

That is, there is no linear relation between the number of grams of protein and the diastolic blood pressure.

Alternative hypothesis:

$H_1:\rho \ne 0$

That is, there is a linear relation between the number of grams of protein and the diastolic blood pressure.

The sample size is 9.

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

That is,

$\begin{array}{c}n-2=9-2\\ =7\end{array}$

From the “TABLE –I: Critical Values for the PPMC”, the critical value for 7 degrees of freedom and $\alpha =0.01$ level of significance is 0.798.

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 0.916 that is the absolute value of r is 0.916.

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

That is, $\left|r\right|\left(=0.916\right)<\text{critical value}\left(=0.798\right)$.

By the rejection rule, reject the null hypothesis.

There is a sufficient evidence to support the claim that “there is a linear relation between the number of grams of protein and the diastolic blood pressure.

To determine

To find: The regression equation for the given data.

Answer to Problem 10.1.6RE

The regression equation for the given data is $\widehat{y}=64.936+2.662x$.

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 Pressure.
• In Predictors, enter the column of Grams.
• Click OK.

Output using the MINITAB software is given below:

Thus, regression equation for the given data is $\widehat{y}=64.936+2.662x$.

To determine

To construct: The scatterplot for the variables the speed and time with regression line.

Answer to Problem 10.1.6RE

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 Pressure.
• Under X variables, enter a column of Grams.
• Click OK.

To determine

To obtain: The predicted value of the diastolic pressure of a vegetarian who consumes 8 grams of protein per day.

Answer to Problem 10.1.6RE

The predicted value of the diastolic pressure of a vegetarian is 86.232.

Explanation of Solution

Calculation:

Thus, regression equation for the given data is $\widehat{y}=64.936+2.662x$.

Substitute x as 8 in the regression equation

$\begin{array}{c}\widehat{y}=64.936+2.662x\\ =64.936+\left(2.662\times 8\right)\\ =64.936+21.296\\ =86.232\end{array}$

Thus, the predicted value of the diastolic pressure of a vegetarian is 86.232.