Chapter 11.2, Problem 35E

a.

To determine

Calculate the correlation between math score and verbal SAT score.

Answer to Problem 35E

The correlation between math score and verbal SAT score is 0.75.

Explanation of Solution

Calculation:

The verbal SAT scores (x) and the math SAT scores (y) of six fishermen are given.

Correlation:

The correlation coefficient, r, between ordered pairs of variables, (x, y) having sample means $\overline{x}$ and $\overline{y}$, sample standard deviations $s_x$ and $s_y$ for a sample of size n is given as $r=\frac{1}{n-1}{\displaystyle \sum \left(\frac{x-\overline{x}}{s_x}\right)\left(\frac{y-\overline{y}}{s_y}\right)}$. The correlation coefficient is useful for measuring the strength of the linear relationship between the two variables.

Software procedure:

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

• Choose Stat > Basic Statistics > Correlation.
• In Variables, enter the columns of x and y.
• Click OK.

Output using the MINITAB software is given below:

Hence, the correlation between math score and verbal SAT score is 0.75.

b.

To determine

Calculate the mean and standard deviation, $\overline{x}$ and $s_x$ respectively, of the verbal scores.

Answer to Problem 35E

The mean, $\overline{x}$, of the verbal scores is 424.

The standard deviation, $s_x$, of the verbal scores is 123.3.

Explanation of Solution

Calculation:

Denote $\overline{x}$ as the mean of x and $s_x$ as the standard deviation of x.

Descriptive statistics:

Software procedure:

Step-by-step procedure to obtain the descriptive statistics using the MINITAB software:

• Choose Stat > Basic Statistics > Display Descriptive Statistics, click OK.
• In Variables, enter the columns of x.
• Choose Statistics, select Mean, Standard deviation and click OK.
• Click OK.

Output using the MINITAB software is given below:

From the above output, it is evident that the mean, $\overline{x}$, of the verbal scores is 424 and the standard deviation, $s_x$, of the verbal scores is 123.3.

c.

To determine

Calculate the mean and standard deviation, $\overline{y}$ and $s_y$ respectively, of the math scores.

Answer to Problem 35E

The mean, $\overline{y}$, of the math scores is 499.3.

The standard deviation, $s_y$, of the math scores is 143.9.

Explanation of Solution

Calculation:

Denote $\overline{y}$ as the mean of y and $s_y$ as the standard deviation of y.

Descriptive statistics:

Software procedure:

Step-by-step procedure to obtain the descriptive statistics using the MINITAB software:

• Choose Stat > Basic Statistics > Display Descriptive Statistics, click OK.
• In Variables, enter the columns of y.
• Choose Statistics, select Mean, Standard deviation and click OK.
• Click OK.

Output using the MINITAB software is given below:

From the above output, it is evident that the mean, $\overline{y}$, of the math scores is 499.3 and the standard deviation, $s_y$, of the math scores is 143.9.

d.

To determine

Find the least-squares regression line to predict the math score from the verbal score.

Answer to Problem 35E

The least-squares regression line to predict the math score from the verbal score is $\underset{\_}{\widehat{y}=128+0.875x}$.

Explanation of Solution

Calculation:

Least-squares regression:

For an ordered pairs of values of variables, (x, y) with respective means $\overline{x}$ and $\overline{y}$, respective standard deviations $s_x$ and $s_y$, correlation coefficient r, the least-squares regression line for predicting the response variable, y, by using the predictor variable, x, is given as:

$\widehat{y}=b_0+b_{1}x$, where the slope is $b_1=r\frac{s_y}{s_x}$  and the y-intercept is $b_0=\overline{y}-b_1\overline{x}$.

Regression:

Software procedure:

Step by step procedure to obtain regression using Minitab software is given as,

• Choose Stat > Regression > Regression > Fit Regression Model.
• In Responses, enter the numeric column containing the response data y.
• In Continuous Predictors, enter the numeric column containing the predictor variable x.
• Choose Results, select Regression equation and click OK.
• Click OK.

Output using MINITAB software is given below:

From the output, the least-squares regression line to predict the math score from the verbal score is $\underset{\_}{\widehat{y}=128+0.875x}$.

e.

To determine

Find the z-score for each value of x.

Answer to Problem 35E

The z-scores for the values of x are:

$z_x$

0.0324

­–0.3082

1.8573

–0.8759

0.1135

–0.8191

Explanation of Solution

Calculation:

The z-score for an x-value is denoted as $z_x=\frac{x-\overline{x}}{s_x}$.

Here, $\overline{x}=424$, $s_x=123.3$. Thus, the expression for the z-scores would be:

$z_x=\frac{x-424}{123.3}$.

The calculation for the z-scores for the values of x is shown in the following table:

x	$x-\overline{x}=x-424$	$\begin{array}{c}{z}_x=\frac{x-\overline{x}}{s_x}\\ =\frac{x-424}{123.3}\end{array}$
428	4	0.0324
386	–38	­–0.3082
653	229	1.8573
316	–108	–0.8759
438	14	0.1135
323	–101	–0.8191

f.

To determine

Find the z-score for each value of y.

Answer to Problem 35E

The z-score for the values of y are:

$z_y$

–0.8777

0.4983

1.2974

–1.2530

0.7484

–0.4121

Explanation of Solution

Calculation:

The z-score for a y-value is denoted as $z_y=\frac{y-\overline{y}}{s_y}$.

Here, $\overline{y}=499.3$, $s_y=143.9$. Thus, the expression for the z-scores would be:

$z_y=\frac{y-499.3}{143.9}$.

The calculation for the z-scores for the values of y is shown in the following table:

y	$y-\overline{y}=y-499.3$	$\begin{array}{c}{z}_y=\frac{y-\overline{y}}{s_y}\\ =\frac{y-499.3}{143.9}\end{array}$
373	–126.3	–0.8777
571	71.7	0.4983
686	186.7	1.2974
319	–180.3	–1.2530
607	107.7	0.7484
440	–59.3	–0.4121

g.

To determine

Find the correlation coefficient, r, between $z_x$ and $z_y$.

Explain whether this correlation is the same as the correlation between math and verbal SAT scores.

Answer to Problem 35E

The correlation, r, between $z_x$ and $z_y$ is 0.75.

The correlation between $z_x$ and $z_y$ is the same as the correlation between math and verbal SAT scores.

Explanation of Solution

Calculation:

Correlation:

Software procedure:

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

• Choose Stat > Basic Statistics > Correlation.
• In Variables, enter the columns of zx and zy.
• Click OK.

Output using the MINITAB software is given below:

Hence, the correlation, r, between $z_x$ and $z_y$ is 0.75.

Observe that the correlation between math and verbal SAT scores obtained in part a is 0.75.

Hence, the correlation between $z_x$ and $z_y$ is the same as the correlation between math and verbal SAT scores.

h.

To determine

Find the least-squares regression line to predict $z_y$ from $z_x$.

Explain the reason the equation of the line is ${\widehat{z}}_y=r{z}_x$.

Answer to Problem 35E

The least-squares regression line to predict $z_y$ from $z_x$ is $\underset{\_}{{\widehat{z}}_y=0.750z_x}$.

Explanation of Solution

Calculation:

Regression:

Software procedure:

Step by step procedure to obtain regression using Minitab software is given as,

• Choose Stat > Regression > Regression > Fit Regression Model.
• In Responses, enter the numeric column containing the response data zy.
• In Continuous Predictors, enter the numeric column containing the predictor variable zx.
• Choose Results, select Regression equation and click OK.
• Click OK.

Output using MINITAB software is given below:

From the output, the least-squares regression line obtained is:

$\begin{array}{c}{\widehat{z}}_y=0.000+0.750z_x\\ {\widehat{z}}_y=0.750z_x.\end{array}$

Hence, the least-squares regression line to predict $z_y$ from $z_x$ is $\underset{\_}{{\widehat{z}}_y=0.750z_x}$.

Consider the least-squares regression equation:

$\widehat{y}=b_0+b_{1}x$, where the slope is $b_1=r\frac{s_y}{s_x}$  and the y-intercept is $b_0=\overline{y}-b_1\overline{x}$.

Now,

$z_x=\frac{x-\overline{x}}{s_x}$ and $z_y=\frac{y-\overline{y}}{s_y}$.

Here, $z_x$ and $z_y$ are standardized variables. As a result,

${\overline{z}}_x=0$,

${\overline{z}}_y=0$,

$s_{z_x}=1$,

$s_{z_y}=1$.

Now, in terms of $z_x$ and $z_y$, it is evident that:

$\begin{array}{c}{b}_1=r\frac{s_y}{s_x}\\ =r\frac{1}{1}\\ =r.\end{array}$

$\begin{array}{c}{b}_0={\overline{z}}_y-b_1{\overline{z}}_x\\ =0-0\\ =0.\end{array}$

Hence, the equation of the line in terms of $z_x$ and $z_y$ is ${\widehat{z}}_y=r{z}_x$.