# The accompanying data in the table below were derived from life tests for two different brands of cutting tools. Complete parts a through c.

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 Cutting Speed (meters per minute) Useful Life Brand A (Hours) Useful Life Brand B (Hours) 30 4.1 6.1 30 4.3 6.2 30 4.7 5.2 40 5.2 5.5 40 3.7 4.3 40 2.5 5 50 4.4 4.5 50 2.8 4 50 1 3.7 60 4 3.8 60 2 3 60 1.1 2.4 70 1.1 1.5 70 0.5 2 70 3 1

I will appreciate assistance with questions listed below.

1. Use a 90 % 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.
2.  Use a 90 % 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.

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

4. Compare the widths of the two prediction intervals to the two confidence intervals you calculated in part a.
5. Comment on the reasons for any difference. Choose the correct answer below.

Calculations will be presented later;-

The mean useful life of a brand A cutting tool when the cutting speed is 45  meters per minute is 2.7 to 3.9 hours.

The mean useful life of a brand B cutting tool when the cutting speed is 45  meters per minute is 4.1 to 4.7 hours.

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

Prediction intervals;-

The predicted useful life of a brand A cutting tool when the speed is 45 meters per minute is 1.2 to 5.5 hours.

The predicted useful life of a brand B cutting tool when the speed is 45 meters per minute is 3.4 to 5.4 hours.

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.

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

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

The predicted useful life of a brand A cutting tool that is operated at 100  meters per minute is -0.59 hours

The actual predicted useful life of a brand A cutting tool when the speed is 100 meters per minute is 0.0 to 2.3 hours.

The linear regression is an accurate model when  x equals 100.