Question

Suppose an environmental agency would like to investigate the relationship between the engine size of​ sedans, x, and the miles per gallon​ (MPG), y, they get. The accompanying table shows the engine size in cubic liters and rated miles per gallon for a selection of sedans. The regression line for the data is y hat=36.7920−4.1547x.

Use this information to complete the parts below.

Engine Size   MPG
2.4 27
2.1 31
2.3 26
3.4 22
3.5 24
2.2 28
2.2 24
2.1 29
3.9 20

a) Calculate the coefficient of determination. R²=? (Round to three decimal places as​ needed.)

b) Using α=0.05​, test the significance of the population coefficient of determination.

Determine the null and alternative hypotheses.

c) The​ F-test statistic is? ​(Round to two decimal places as​needed.)

d) the p-value is? ​(Round to three decimal places as​needed.)

e) Construct a​ 95% confidence interval for the average MPG of a 2.5​-cubic liter engine.

UCL= ? ​(Round to two decimal places as​ needed.)

LCL= ​? (Round to two decimal places as​ needed.)

f) Construct a​ 95% prediction interval for the MPG of a 2.5​-cubic liter engine.

UPL= ? ​(Round to two decimal places as​ needed.)

LPL= ? ​(Round to two decimal places as​ needed.)

Answer 1

using excel>addin>phstat>Regression

we have

Simple Linear Regression Analysis
Regression Statistics
Multiple R	0.8431
R Square	0.7108
Adjusted R Square	0.6695
Standard Error	2.0121
Observations	9
ANOVA
	df	SS	MS	F	Significance F
Regression	1	69.6611	69.6611	17.2070	0.0043
Residual	7	28.3389	4.0484
Total	8	98.0000
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	36.7921	2.7646	13.3082	0.0000	30.2548	43.3294
Engine	-4.1547	1.0016	-4.1481	0.0043	-6.5231	-1.7863

Confidence Interval Estimate
Data
X Value	2.5
Confidence Level	95%
Intermediate Calculations
Sample Size	9
Degrees of Freedom	7
t Value	2.364624
XBar, Sample Mean of X	2.677778
Sum of Squared Differences from XBar	4.035556
Standard Error of the Estimate	2.012068
h Statistic	0.118943
Predicted Y (YHat)	26.40529
For Average Y
Interval Half Width	1.6409
Confidence Interval Lower Limit	24.7644
Confidence Interval Upper Limit	28.04615
For Individual Response Y
Interval Half Width	5.0328
Prediction Interval Lower Limit	21.3725
Prediction Interval Upper Limit	31.43808

a) the coefficient of determination. R2= 0.711

b) the null and alternative hypotheses is

Ho:

Ha:

Correlation: Engine, MPG

Pearson correlation of Engine and MPG = -0.843
P-Value = 0.004
p value is less than 0.05 so the population coefficient of determination is significant.

) The​ F-test statistic is 17.21

d) the p-value is 0.004

e) 95% confidence interval for the average MPG of a 2.5​-cubic liter engine.

UCL= 24.764

LCL= 28.046

f) a​ 95% prediction interval for the MPG of a 2.5​-cubic liter engine.

UPL= 21.373

LPL= 31.438