Question 5.4: The article "Study on the Life Distribution of Microdrill s"...
The article “Study on the Life Distribution of Microdrill s” (Z. Yang, Y. Chen, and Y. Yang, Journal of Engineering Manufacture, 2002:301- 305) reports that in a sample of 50 microdrills drilling a low-carbon alloy steel, the average lifetime (expressed as the number of holes drilled before failure) was 12.68 with a standard deviation of 6.83. Find a 95% confidence interval for the mean lifetime of microdrills under these conditions.
First let’s translate the problem into statistical language. We have a simple random sample X_{1}, … , X_{50} of lifetimes. The sample mean and standard deviation are \bar{X} = 12.68 and s = 6.83. The population mean is unknown, and denoted by μ. The confidence interval has the form \bar{X} ± z_{α/2}σ_{\bar{X}}, as specified in expression (5.4).
\bar{X} ± z_{α/2}σ_{\bar{X}} (5.4)
Since we want a 95% confidence interval, the confidence level 1 – α is equal to 0.95. Thus α = 0.05, and the critical value is z_{α/2} = z_{.025} = 1.96. We approximate σ with s = 6.83, and obtain the standard error σ_{\bar{X}} ≈ 6.83/\sqrt{50} = 0.9659. Therefore the 95% confidence interval is 12.68 ± (1.96)(0.9659). This can be written as 12.68 ± 1.89, or as (10.79, 14.57).