Question

Cutting Speed (meters per minute) Useful Life Brand A (Hours) Useful Life Brand B (Hours)

30 5.2 6.3

30 4.4 6.4

30 5.2 5.2

40 4.5 6.0

40 3.7 4.6

40 2.5 5.0

50 4.4 4.5

50 2.8 4.0

50 1.0 3.7

60 4.0 3.8

60 2.0 3.0

60 1.1 2.4

70 1.1 1.5

70 0.5 2.0

70 3.0 1.0

.

Use a 95​% confidence interval to estimate the mean useful life of a brand A cutting tool when the cutting speed is 45 meters per minute. Repeat for brand B. Compare the widths of the two intervals and comment on the reasons for any difference.

The mean useful life of a brand A cutting tool when the cutting speed is ___ to ____ hours. (Round to one decimal place as​ needed.)

The mean useful life of a brand B cutting tool when the cutting speed is ___ to ____ hours. (Round to one decimal place as​ needed.)

Compare the widths of the two intervals and comment on the reasons for any difference. Choose the correct answer below.

A. Brand A is wider than brand B. The estimated standard error of y^ is different for the two intervals.

B. Brand B is wider than brand A. The value of t α/2 is different for the two intervals.

C. Brand A is wider than brand B. The calue of y^ is different for the two intervals.

D. There is no difference in the widths of the two intervals.

b. Use a 95% prediction interval to predict the useful life of a brand A cutting tool when the cutting speed is 45 meters per minute. Repeat for brand B. Compare the widths of the two intervals to each other and to the two intervals you calculated in part a. Comment on the reasons for any difference.

The predicted useful life of a brand A cutting tool when the speed is 45 meters per minute is ___ to ____ hours. (Round to one decimal place as​ needed.)

The predicted useful life of a brand B cutting tool when the speed is 45 meters per minute ___ to ____ hours. (Round to one decimal place as​ needed.)

Compare the widths of the two intervals to each other. Choose the correct answer below.

A.The prediction interval for brand A is larger than the prediction interval for brand B because the estimated standard error of y^ is different for the two intervals.

B.The prediction interval for brand B is larger than the prediction interval for brand A because the value of  y^ is different for the two intervals.

C.The prediction intervals are the same size.

Compare the widths of the two prediction intervals to the two confidence intervals you calculated in part a.

Choose the correct answer below.

A.The prediction intervals are both larger than the corresponding confidence intervals.

B.The prediction intervals are both smaller than the corresponding confidence intervals.

C.The two prediction intervals are the same size as the corresponding confidence intervals.

D.There is no difference in the widths of the four intervals.

Comment on the reasons for any difference. Choose the correct answer below.

A. The value of t α/2 for the estimated mean value of y is smaller than the value of t α/2 for the predicted value of y.

B. The standard error for the estimated mean value of y is smaller than the standard error for the predicted value of y.

C. The standard error for the estimated mean value of y is larger than the standard error for the predicted value of y.

D. The value of t α/2 for the estimated mean value of y is larger than the value of t α/2 for the predicted value of y.

c. Suppose you were asked to predict the useful life of a brand A cutting tool for a cutting speed of x=100 meters per minute. Because the given value of x is outside the range of the sample​ x-values, the prediction is an example of extrapolation. Predict the useful life of a brand A cutting tool that is operated at 100 meters per minute and construct a 95​% prediction interval for the actual useful life of the tool. What additional assumption do you have to make in order to ensure the validity of an​ extrapolation?

The predicted useful life of a brand A cutting tool that is operated at 100 meters per minute is _____ hours. ​(Round to two decimal places as​ needed.)

The actual predicted useful life of a brand A cutting tool when the speed is 100 meters per minute is ______ to ____ hours. (Round to one decimal place as​ needed.)

What additional assumption do you have to make in order to ensure the validity of an​ extrapolation?

A.The linear regression is an accurate model when x=100.

B.The value of t α/2 can be found for x=100.

C.There is no additional assumption required.

Answer 1

Use a 95​% confidence interval to estimate the mean useful life of a brand A cutting tool when the cutting speed is 45 meters per minute. Repeat for brand B. Compare the widths of the two intervals and comment on the reasons for any difference.

The mean useful life of a brand A cutting tool when the cutting speed is 2.8 to 4.1 hours. (Round to one decimal place as​ needed.)

The mean useful life of a brand B cutting tool when the cutting speed is 4.2 to 4.9 hours. (Round to one decimal place as​ needed.)

Compare the widths of the two intervals and comment on the reasons for any difference. Choose the correct answer below.

Answer: A. Brand A is wider than brand B. The estimated standard error of y^ is different for the two intervals.

B. Brand B is wider than brand A. The value of t α/2 is different for the two intervals.

C. Brand A is wider than brand B. The calue of y^ is different for the two intervals.

D. There is no difference in the widths of the two intervals.

b. Use a 95% prediction interval to predict the useful life of a brand A cutting tool when the cutting speed is 45 meters per minute. Repeat for brand B. Compare the widths of the two intervals to each other and to the two intervals you calculated in part a. Comment on the reasons for any difference.

The predicted useful life of a brand A cutting tool when the speed is 45 meters per minute is 0.9 to 6.0 hours. (Round to one decimal place as​ needed.)

The predicted useful life of a brand B cutting tool when the speed is 45 meters per minute 3.2 to 5.8 hours. (Round to one decimal place as​ needed.)

Compare the widths of the two intervals to each other. Choose the correct answer below.

Answer: A.The prediction interval for brand A is larger than the prediction interval for brand B because the estimated standard error of y^ is different for the two intervals.

B.The prediction interval for brand B is larger than the prediction interval for brand A because the value of  y^ is different for the two intervals.

C.The prediction intervals are the same size.

Compare the widths of the two prediction intervals to the two confidence intervals you calculated in part a.

Choose the correct answer below.

Answer: A.The prediction intervals are both larger than the corresponding confidence intervals.

B.The prediction intervals are both smaller than the corresponding confidence intervals.

C.The two prediction intervals are the same size as the corresponding confidence intervals.

D.There is no difference in the widths of the four intervals.

Comment on the reasons for any difference. Choose the correct answer below.

A. The value of t α/2 for the estimated mean value of y is smaller than the value of t α/2 for the predicted value of y.

Answer: B. The standard error for the estimated mean value of y is smaller than the standard error for the predicted value of y.

C. The standard error for the estimated mean value of y is larger than the standard error for the predicted value of y.

D. The value of t α/2 for the estimated mean value of y is larger than the value of t α/2 for the predicted value of y.

c. Suppose you were asked to predict the useful life of a brand A cutting tool for a cutting speed of x=100 meters per minute. Because the given value of x is outside the range of the sample​ x-values, the prediction is an example of extrapolation. Predict the useful life of a brand A cutting tool that is operated at 100 meters per minute and construct a 95​% prediction interval for the actual useful life of the tool. What additional assumption do you have to make in order to ensure the validity of an​ extrapolation?

The predicted useful life of a brand A cutting tool that is operated at 100 meters per minute is -0.97 hours. ​(Round to two decimal places as​ needed.)

The actual predicted useful life of a brand A cutting tool when the speed is 100 meters per minute is -4.4 to 2.4 hours. (Round to one decimal place as​ needed.)

What additional assumption do you have to make in order to ensure the validity of an​ extrapolation?

Answer: A.The linear regression is an accurate model when x=100.

B.The value of t α/2 can be found for x=100.

C.There is no additional assumption required.

Excel Addon Megastat used.

Menu used: correlation/Regression ---- Regression Analysis

Regression Analysis
	r²	0.532	n	15
	r	-0.729	k	1
	Std. Error	1.140	Dep. Var.	Brand A
ANOVA table
Source	SS	df	MS	F	p-value
Regression	19.2000	1	19.2000	14.78	.0020
Residual	16.8893	13	1.2992
Total	36.0893	14
Regression output					confidence interval
variables	coefficients	std. error	t (df=13)	p-value	95% lower	95% upper
Intercept	7.0267	1.0813	6.498	2.01E-05	4.6906	9.3627
Speed	-0.0800	0.0208	-3.844	.0020	-0.1250	-0.0350
Predicted values for: Brand A
		95% Confidence Interval		95% Prediction Interval
Speed	Predicted	lower	upper	lower	upper	Leverage
45	3.4267	2.7523	4.1010	0.8736	5.9798	0.075
Regression Analysis
	r²	0.896	n	15
	r	-0.946	k	1
	Std. Error	0.573	Dep. Var.	Brand B
ANOVA table
Source	SS	df	MS	F	p-value
Regression	36.7413	1	36.7413	111.74	9.42E-08
Residual	4.2747	13	0.3288
Total	41.0160	14
Regression output					confidence interval
variables	coefficients	std. error	t (df=13)	p-value	95% lower	95% upper
Intercept	9.4933	0.5440	17.451	2.11E-10	8.3181	10.6686
Speed	-0.1107	0.0105	-10.571	9.42E-08	-0.1333	-0.0880
Predicted values for: Brand B
		95% Confidence Interval		95% Prediction Interval
Speed	Predicted	lower	upper	lower	upper	Leverage
45	4.5133	4.1741	4.8526	3.2289	5.7978	0.075

Regression Analysis
	r²	0.532	n	15
	r	-0.729	k	1
	Std. Error	1.140	Dep. Var.	Brand A
ANOVA table
Source	SS	df	MS	F	p-value
Regression	19.2000	1	19.2000	14.78	.0020
Residual	16.8893	13	1.2992
Total	36.0893	14
Regression output					confidence interval
variables	coefficients	std. error	t (df=13)	p-value	95% lower	95% upper
Intercept	7.0267	1.0813	6.498	2.01E-05	4.6906	9.3627
Speed	-0.0800	0.0208	-3.844	.0020	-0.1250	-0.0350
Predicted values for: Brand A
		95% Confidence Interval		95% Prediction Interval
Speed	Predicted	lower	upper	lower	upper	Leverage
100	-0.9733	-3.3094	1.3627	-4.3675	2.4209	0.900