Chapter 23, Problem 23.38E

(a)

To determine

To explain:

The scatterplot of white matter change and the explanatory of the given data.

Explanation of Solution

Given:

The table of the exposure and change.

Concept used:

Formula

The binomial distribution

  $p\left[X=x\right]=\left(\begin{array}{l}n\\ x\end{array}\right)p^x{q}^{n-x}$

Calculation:

Scatter plot is given as the table

The weak is negative relationship exists.

Because the value of $r=-0.30$

Where $r$ is negative

Let us perform the regression

Regressing sub concussion exposure on white matter change with below mentioned steps in excel

  $X$ And $Y$ are two variables.

SUMMARY OUTPUT

Regression Statistics

Multiple R $=0.368868063$

R Square $=0.136063648$

Adjusted R Square $=0.09286683$

Standard Error $= 3.467466463$

Observations $=22$

Draw the table

	Degree	SS	MS	F	Significant
Regression	$1$	$\text{37}\text{.87170833}$	$\text{37}\text{.8717}$	$\text{3}\text{.149854}$	$\text{0}\text{.091157025}$
Residual	$\text{20}$	$\text{240}\text{.4664735}$	$\text{12}\text{.02332}$
Total	$\text{21}$	$\text{278}\text{.3381818}$

Draw the second table

	Coefficient	Standard Error	t Stat	P-value	Lower $95\%$	Upper $95\%$	Lower $95.0\%$	Upper $95.0\%$
Intercept	$\text{2}\text{.272015257}$	$\text{1}\text{.201555964}$	$\text{1}\text{.890894}$	$\text{0}\text{.073215}$	$\text{-0}\text{.234386557}$	$\text{4}\text{.778417071}$	$\text{-0}\text{.234386557}$	$\text{4}\text{.778417071}$
$X$	$\text{-6}\text{.912959936}$	$\text{3}\text{.895102237}$	$\text{-1}\text{.77478}$	$\text{0}\text{.091157}$	$\text{-15}\text{.0380008}$	$\text{1}\text{.212080931}$	$\text{-15}\text{.0380008}$	$\text{1}\text{.212080931}$

(b)

To determine

To find:

Theregression standard error of the given data.

Explanation of Solution

Given:

The table of the exposure and change.

Concept used:

Formula

The binomial distribution

  $p\left[X=x\right]=\left(\begin{array}{l}n\\ x\end{array}\right)p^x{q}^{n-x}$

Calculation:

Scatter plot is given as the table

The weak is negative relationship exists.

Because the value of $r=-0.30$

Where $r$ is negative

Let us perform the regression

Regressing sub concussion exposure on white matter change with below mentioned steps in excel

  $X$ And $Y$ are two variables.

SUMMARY OUTPUT

Regression Statistics

Multiple R $=0.368868063$

R Square $=0.136063648$

Adjusted R Square $=0.09286683$

Standard Error $= 3.467466463$

Observations $=22$

Draw the table

	Degree	SS	MS	F	Significant
Regression	$1$	$\text{37}\text{.87170833}$	$\text{37}\text{.8717}$	$\text{3}\text{.149854}$	$\text{0}\text{.091157025}$
Residual	$\text{20}$	$\text{240}\text{.4664735}$	$\text{12}\text{.02332}$
Total	$\text{21}$	$\text{278}\text{.3381818}$

Draw the second table

	Coefficient	Standard Error	t Stat	P-value	Lower $95\%$	Upper $95\%$	Lower $95.0\%$	Upper $95.0\%$
Intercept	$\text{2}\text{.272015257}$	$\text{1}\text{.201555964}$	$\text{1}\text{.890894}$	$\text{0}\text{.073215}$	$\text{-0}\text{.234386557}$	$\text{4}\text{.778417071}$	$\text{-0}\text{.234386557}$	$\text{4}\text{.778417071}$
$X$	$\text{-6}\text{.912959936}$	$\text{3}\text{.895102237}$	$\text{-1}\text{.77478}$	$\text{0}\text{.091157}$	$\text{-15}\text{.0380008}$	$\text{1}\text{.212080931}$	$\text{-15}\text{.0380008}$	$\text{1}\text{.212080931}$

Here $X=$ exposure

  $Y=$ Change as variable

So least squares regression line

Change ( $Y$ ) $=-6.913\ast$ Exposure $\left(X\right)+2.272$

Regression standard error $=3.467$

The low p-value of $0.0912$ of the Exposure coefficient, indicates that a confidence level of $90\%$ (where p-value of $0.1$ ) that the Exposure is a significant predictor of white matter Change.

Hence, the null hypothesis of no relationship between exposure and white matter change can be rejected in favor of the alternative hypothesis at a confidence level of $10\%.$

(c)

To determine

To explain:

Theequation for the least-squares regression line without data point and test null hypothesis of the given data.

Explanation of Solution

Given:

The table of the exposure and change.

Concept used:

Formula

The binomial distribution

  $p\left[X=x\right]=\left(\begin{array}{l}n\\ x\end{array}\right)p^x{q}^{n-x}$

Calculation:

Scatter plot is given as the table

The weak is negative relationship exists.

Because the value of $r=-0.30$

Where $r$ is negative

Let us perform the regression

Regressing sub concussion exposure on white matter change with below mentioned steps in excel

  $X$ And $Y$ are two variables.

SUMMARY OUTPUT

Regression Statistics

Multiple R $=0.368868063$

R Square $=0.136063648$

Adjusted R Square $=0.09286683$

Standard Error $= 3.467466463$

Observations $=22$

Draw the table

	Degree	SS	MS	F	Significant
Regression	$1$	$\text{37}\text{.87170833}$	$\text{37}\text{.8717}$	$\text{3}\text{.149854}$	$\text{0}\text{.091157025}$
Residual	$\text{20}$	$\text{240}\text{.4664735}$	$\text{12}\text{.02332}$
Total	$\text{21}$	$\text{278}\text{.3381818}$

Draw the second table

	Coefficient	Standard Error	t Stat	P-value	Lower $95\%$	Upper $95\%$	Lower $95.0\%$	Upper $95.0\%$
Intercept	$\text{2}\text{.272015257}$	$\text{1}\text{.201555964}$	$\text{1}\text{.890894}$	$\text{0}\text{.073215}$	$\text{-0}\text{.234386557}$	$\text{4}\text{.778417071}$	$\text{-0}\text{.234386557}$	$\text{4}\text{.778417071}$
$X$	$\text{-6}\text{.912959936}$	$\text{3}\text{.895102237}$	$\text{-1}\text{.77478}$	$\text{0}\text{.091157}$	$\text{-15}\text{.0380008}$	$\text{1}\text{.212080931}$	$\text{-15}\text{.0380008}$	$\text{1}\text{.212080931}$

Here $X=$ exposure

  $Y=$ Change as variable

So least squares regression line

Change ( $Y$ ) $=-6.913\ast$ Exposure $\left(X\right)+2.272$

Regression standard error $=3.467$

The low p-value of $0.0912$ of the Exposure coefficient, indicates that a confidence level of $90\%$ (where p-value of $0.1$ ) that the Exposure is a significant predictor of white matter Change.

Hence, the null hypothesis of no relationship between exposure and white matter change can be rejected in favor of the alternative hypothesis at a confidence level of $10\%.$

SUMMARY OUTPUT

Regression Statistics

Multiple R $=\text{0}\text{.104318418}$

R Square $=\text{0}\text{.010882332}$

Adjusted R Square $=\text{-0}\text{.041176492}$

Standard Error $= \text{3}\text{.471079833}$

Observations $=21$

Draw the table

	Degree	SS	MS	F	Significant
Regression	$1$	$\text{2}\text{.518586244}$	$\text{2}\text{.518586}$	$\text{0}\text{.209039}$	$\text{0}\text{.652706126}$
Residual	$19$	$\text{228}\text{.919509}$	$\text{12}\text{.0484}$
Total	$\text{20}$	$\text{231}\text{.4380952}$

Draw the second table

	Coefficient	Standard Error	t Stat	P-value	Lower $95\%$	Upper $95\%$	Lower $95.0\%$	Upper $95.0\%$
Intercept	$\text{1}\text{.47586}$	$\text{1}\text{.45193}$	$\text{1}\text{.01648}$	$\text{0}\text{.32216}$	$\text{-1}\text{.56306}$	$\text{4}\text{.51480}$	$\text{-1}\text{.56306}$	$\text{4}\text{.51480}$
$X$	$\text{-2}\text{.6666}$	$\text{5}\text{.83246}$	$\text{-0}\text{.45721}$	$\text{0}\text{.65270}$	$\text{-14}\text{.8741}$	$\text{9}\text{.5408}$	$\text{-14}\text{.8741}$	$\text{9}\text{.54083}$

Standard error is same

But the p value is $0.65$

Exposure is not good predictor of Change in white matter.

Also, the Exposure coefficient has come down from $-6.91\text{ to }-2.67$ ,

(d)

To determine

To explain:

Theimpact of the usual observation on the regression analysis of the given data.

Explanation of Solution

Given:

The table of the exposure and change.

Concept used:

Formula

The binomial distribution

  $p\left[X=x\right]=\left(\begin{array}{l}n\\ x\end{array}\right)p^x{q}^{n-x}$

Calculation:

Scatter plot is given as the table

The weak is negative relationship exists.

Because the value of $r=-0.30$

Where $r$ is negative

Let us perform the regression

Regressing sub concussion exposure on white matter change with below mentioned steps in excel

  $X$ And $Y$ are two variables.

SUMMARY OUTPUT

Regression Statistics

Multiple R $=0.368868063$

R Square $=0.136063648$

Adjusted R Square $=0.09286683$

Standard Error $= 3.467466463$

Observations $=22$

Draw the table

	Degree	SS	MS	F	Significant
Regression	$1$	$\text{37}\text{.87170833}$	$\text{37}\text{.8717}$	$\text{3}\text{.149854}$	$\text{0}\text{.091157025}$
Residual	$\text{20}$	$\text{240}\text{.4664735}$	$\text{12}\text{.02332}$
Total	$\text{21}$	$\text{278}\text{.3381818}$

Draw the second table

	Coefficient	Standard Error	t Stat	P-value	Lower $95\%$	Upper $95\%$	Lower $95.0\%$	Upper $95.0\%$
Intercept	$\text{2}\text{.272015257}$	$\text{1}\text{.201555964}$	$\text{1}\text{.890894}$	$\text{0}\text{.073215}$	$\text{-0}\text{.234386557}$	$\text{4}\text{.778417071}$	$\text{-0}\text{.234386557}$	$\text{4}\text{.778417071}$
$X$	$\text{-6}\text{.912959936}$	$\text{3}\text{.895102237}$	$\text{-1}\text{.77478}$	$\text{0}\text{.091157}$	$\text{-15}\text{.0380008}$	$\text{1}\text{.212080931}$	$\text{-15}\text{.0380008}$	$\text{1}\text{.212080931}$

Here $X=$ exposure

  $Y=$ Change as variable

So least squares regression line

Change ( $Y$ ) $=-6.913\ast$ Exposure $\left(X\right)+2.272$

Regression standard error $=3.467$

The low p-value of $0.0912$ of the Exposure coefficient, indicates that a confidence level of $90\%$ (where p-value of $0.1$ ) that the Exposure is a significant predictor of white matter Change.

Hence, the null hypothesis of no relationship between exposure and white matter change can be rejected in favor of the alternative hypothesis at a confidence level of $10\%.$

SUMMARY OUTPUT

Regression Statistics

Multiple R $=\text{0}\text{.104318418}$

R Square $=\text{0}\text{.010882332}$

Adjusted R Square $=\text{-0}\text{.041176492}$

Standard Error $= \text{3}\text{.471079833}$

Observations $=21$

Draw the table

	Degree	SS	MS	F	Significant
Regression	$1$	$\text{2}\text{.518586244}$	$\text{2}\text{.518586}$	$\text{0}\text{.209039}$	$\text{0}\text{.652706126}$
Residual	$19$	$\text{228}\text{.919509}$	$\text{12}\text{.0484}$
Total	$\text{20}$	$\text{231}\text{.4380952}$

Draw the second table

	Coefficient	Standard Error	t Stat	P-value	Lower $95\%$	Upper $95\%$	Lower $95.0\%$	Upper $95.0\%$
Intercept	$\text{1}\text{.47586}$	$\text{1}\text{.45193}$	$\text{1}\text{.01648}$	$\text{0}\text{.32216}$	$\text{-1}\text{.56306}$	$\text{4}\text{.51480}$	$\text{-1}\text{.56306}$	$\text{4}\text{.51480}$
$X$	$\text{-2}\text{.6666}$	$\text{5}\text{.83246}$	$\text{-0}\text{.45721}$	$\text{0}\text{.65270}$	$\text{-14}\text{.8741}$	$\text{9}\text{.5408}$	$\text{-14}\text{.8741}$	$\text{9}\text{.54083}$

Standard error is same

But the p value is $0.65$

Exposure is not good predictor of Change in white matter.

Also, the Exposure coefficient has come down from $-6.91\text{ to }-2.67$ ,

The Change in white matter will not actually change as fast

So,eliminate of the outlier data points.

The Outlier data points tend to increase the correlation between the two variables.