Using the data below, and assuming that the standard deviation is unknown and mu

Question

Using the data below, and assuming that the standard deviation is unknown and must be estimated from the data, develop a 98% confidence interval for the true mean tool life. Based on this confidence interval, comment on Kingwood Tool's claim. Also, compute the confidence interval for the standard deviation at 95% confidence level.

"The Kingwood Tool Company claims that the life of a new grade of cutting tool is 70 minutes on average. Previous studies have shown that the distribution of tool lives is normal with a standard deviation of 8 minutes. A random sample of six cutting tools is tested, producing the following results: 55.0, 70.1, 58.4, 53.2, 62.0 and 67.9 minutes. Risk level 0.02"

Explanation / Answer

n=6, xbar = 61.1 (based on the data), s=8

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Given a=0.02, Z(0.01)=2.33 (check standard normal table)
So 98% CI is
xbar ± Z*s/n

--> 61.1 ± 2.33*8/sqrt(6)

--> ( 53.49, 68.71)

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Given a=0.05, Z(0.025)=1.96 (check standard normal table)
So 95% CI is
xbar ± Z*s/n

--> 61.1 ± 1.96*8/sqrt(6)

--> ( 54.698, 67.501)

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