Chapter 7, Problem 7.56SE

a.

To determine

To find: The equation of the least-squares line for predicting a neuron's call response from its pure tone response.

To construct: The scatter plot with tone as x and call as y with regression line.

To construct: The scatter plot with the point A which is the largest residual (either positive or negative) and also the point B that is an outlier in the x direction.

Answer to Problem 7.56SE

The equation of the least-squares line for predicting a neuron's call response from its pure tone response is $\widehat{y}=93.9+0.778x$.

Output using the MINITAB software is given below:

Output with marked point A and B using the MINITAB software is given below:

Explanation of Solution

The given data shows the pure-tone response and monkey call response values.

Calculation:

Software procedure:

Step by step procedure to find the equation of the least-squares line by using the MINITAB software:

• Choose Stat > Regression > Regression.
• In Responses, enter the column of Call.
• In Predictors, enter the column of Tone.
• Click OK.

Output using the MINITAB software is given below:

From the MINITAB output, it is clear that the regression equation is $\widehat{y}=93.9+0.778x$.

Scatterplot:

Software procedure:

Step by step procedure to construct the scatter plots using the MINITAB software:

• Choose Graph > Scatter plot.
• Choose with regression and groups, and then click OK.
• Under Y variables, enter a column of Call.
• Under X variables, enter a column of Tone.
• Click OK.

From the scatterplot, it is observed that the plot shows no distinct pattern. Also, the x (pure-tone response) values increase then the corresponding y (monkey call response) values increase. Also, the points are moderately scattered. Thus, there is moderately positive linear correlation between x and y.

Also, from the scatterplot, it is observed that the largest residual is marked as point A(241,485) and also the outlier in the x direction point is marked as B(474,500).

b.

To determine

To explain: The influences of the points A and B for the correlation value r.

Answer to Problem 7.56SE

The outlier and residual is removed then the correlation decreases.

Explanation of Solution

Calculation:

Correlation: With points A and B

Step by step procedure to obtain the correlation 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 output, the correlation with points A and B is 0.639.

Correlation: Without points A and B.

Step by step procedure to obtain the correlation 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 output, the correlation without points A and B is 0.359.

The correlation with points A and B is 0.639 and the correlation without points A and B is 0.359. Also, from the scatterplot in part (a), it is observed that the outlier and residual deviate from linear pattern of the other points.

Thus, it can be concluded that if the outlier and residual are removed then the correlation decreases. Thus, the outlier and residual influence the correlation.

c.

To determine

To explain: The influences of the points A and B for the least-squares line.

Answer to Problem 7.56SE

The effect of the outlier and residual influences on the line is small.

Explanation of Solution

Calculation:

Regression: With Fidelity Gold Fund.

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

• Choose Stat > Regression > Regression.
• In Responses, enter the column of y.
• In Predictors, enter the column of x.
• Click OK.

Output using the MINITAB software is given below:

From the output, the least-squares line for predicting y from x with points A and B is $\widehat{y}=93.9+0.778\left(x\right)$.

Regression: Without points A and B.

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

• Choose Stat > Regression > Regression.
• In Responses, enter the column of y.
• In Predictors, enter the column of x.
• Click OK.

Output using the MINITAB software is given below:

From the output, the least-squares line for predicting y from x without points A and B is $\widehat{y}=116+0.466\left(x\right)$.

Scatter plot:

Step by step procedure to construct the scatter plots with two regression line using the MINITAB software:

• Choose Graph > Scatter plot.
• Choose with regression and groups, and then click OK.
• Under Y variables, enter a column of Y.
• Under X variables, enter a column of X.
• Click OK.

Output using the MINITAB software is given below:

Observation:

From the scatterplot with two regression lines, it is observed that the horizontal axis represents the pure-tone response values and vertical axis represents the monkey call response values. Also, the regression line with all points is marked with solid line and the regression line without points A and B is marked with dashed line.

Also, it is observed that the effect of the points A and B on the line is small. Hence, the residual and outlier was not particularly extreme in the pure-tone response.

Thus, it can be concluded that the outlier and residual do not influence the least-square regression line.