Chapter 28, Problem 28.14AYK

(a)

To determine

To make a scatterplot with taste on the y axis and find the correlation coefficient and explain which relationships are linear and which have the strongest correlation with taste.

Answer to Problem 28.14AYK

The strongest correlation with taste is hydrogen sulfide.

Explanation of Solution

In the question, it is given that experimenters assessed the concentration of lactic acid, acetic acid and hydrogen sulfide in thirty randomly chosen pieces of cheddar cheese. The table is given which shows the data. The scatterplot with taste on the y axis is as follows:

As we can see in the scatterplot that all the lines are almost parallel and also that the R -square of the hydrogen sulfide is largest with taste so the correlation is largest for the hydrogen sulfide. The correlation is given in the scatterplot above by finding square root, the calculation is as:

Acetic	=SQRT(0.302)
Lactic	=SQRT(0.3055)
H2S	=SQRT(0.5712)

And the result is as:

Acetic	0.549545
Lactic	0.552721
H2S	0.755778

(b)

To determine

To use a software to obtain the regression equation and run inference for a regression model that includes all three explanatory variables and interpret the software output, including the meaning of the value taken by $R^2$ .

Answer to Problem 28.14AYK

The equation is $\widehat{y}=113.935-3.483x_1-94.631x_2+5.241x_1{x}_2$ .

Explanation of Solution

In the question, it is given that experimenters assessed the concentration of lactic acid, acetic acid and hydrogen sulfide in thirty randomly chosen pieces of cheddar cheese. The table is given which shows the data. Now, run inference for a regression model that includes all three explanatory variables and interpret the software output by using the Excel, the result will be as:

Regression Statistics
Multiple R	0.800438
R Square	0.640701
Adjusted R Square	0.599243
Standard Error	10.29053
Observations	30

ANOVA
	df	SS	MS	F	Significance F
Regression	3	4909.619	1636.54	15.45438	5.68E-06
Residual	26	2753.268	105.8949
Total	29	7662.887

	Coefficients	Standard Error	t Stat	P-value
Intercept	-32.8566	20.2335	-1.62387	0.116466
Acetic	2.000654	4.346475	0.460294	0.649132
H2S	4.566348	1.176917	3.879925	0.000639
Lactic	13.67117	6.643259	2.057902	0.049755

And the equation is as:

  $\begin{array}{l}\widehat{y}=b_0+b_1{x}_1+b_2{x}_2+b_3{x}_3\\ \Rightarrow \widehat{y}=-32.86+2.001x_1+4.567x_2+13.671x_3\end{array}$

And $R^2=64.07\%$ explains the variations in the model by the explanatory variables.

(c)

To determine

To explain which explanatory variable does it describe and create a new regression model that excludes this explanatory variable and interpret the software output and compare it with your findings in (b).

Answer to Problem 28.14AYK

That explanatory variable is Acetic.

Explanation of Solution

In the question, it is given that experimenters assessed the concentration of lactic acid, acetic acid and hydrogen sulfide in thirty randomly chosen pieces of cheddar cheese. The table is given which shows the data. In the above result in part (b), we can see that the explanatory variable Acetic has a P-value greater than the level of significance so it is not significant. Thus, we will remove this variable and run this test with the other two variables using Excel and the result will be as:

Regression Statistics
Multiple R	0.798607
R Square	0.637773
Adjusted R Square	0.610941
Standard Error	10.13922
Observations	30

ANOVA
	df	SS	MS	F	Significance F
Regression	2	4887.183	2443.592	23.76946	1.11E-06
Residual	27	2775.704	102.8038
Total	29	7662.887

	Coefficients	Standard Error	t Stat	P-value
Intercept	-24.4609	8.629104	-2.8347	0.008581
H2S	4.858662	0.976305	4.976581	3.24E-05
Lactic	14.28672	6.411593	2.228263	0.034385

In this all the explanatory variables are statistically significant but in the above model in (b) all are not statistically significant but the variations explained are approximately equal.

(d)

To determine

To explain which explanatory variable of the two has the less significant or larger value and create a new regression model that excludes this explanatory variable and keeps only significant one and explain how does this last model compare with the model in (c).

Answer to Problem 28.14AYK

The explanatory variable of the two has the less significant or larger value is lactic.

Explanation of Solution

In the question, it is given that experimenters assessed the concentration of lactic acid, acetic acid and hydrogen sulfide in thirty randomly chosen pieces of cheddar cheese. The table is given which shows the data. In the above result in part (c), we can see that the P-value for the Lactic is larger than the hydrogen sulfide thus, we will remove the Lactic variable and then run the regression analysis using the Excel as:

Regression Statistics
Multiple R	0.755752
R Square	0.571162
Adjusted R Square	0.555846
Standard Error	10.83338
Observations	30

ANOVA
	df	SS	MS	F	Significance F
Regression	1	4376.746	4376.746	37.29265	1.37E-06
Residual	28	3286.141	117.3622
Total	29	7662.887

	Coefficients	Standard Error	t Stat	P-value
Intercept	-9.78684	5.95791	-1.64266	0.111638
H2S	5.776089	0.94585	6.10677	1.37E-06

In this as we compare it with the model in part (c), we can see that the coefficient of determination or the variations explained are less in this model then in part (c) and all the slopes are statistically significant.

(e)

To determine

To explain which model best explains cheddar taste and check the conditions for inference for this model and conclude.

Answer to Problem 28.14AYK

Model (b) best explains cheddar taste and conditions are met.

Explanation of Solution

In the question, it is given that experimenters assessed the concentration of lactic acid, acetic acid and hydrogen sulfide in thirty randomly chosen pieces of cheddar cheese. The table is given which shows the data. By looking at the model (b), (c) and (d), we can say that the variations explained is more in part (b) than in (c) and (d). Thus, the model in (b) best explains cheddar taste. The conditions for inferences are as: as we can see in the scatterplot, it shows the linearity and as we look at the data it shows the normality and constant variance by looking at the model regression analysis using Excel’s residual plot and the data is randomly selected so it shows independence. Thus, the conditions are met.