Question 5.4: The article “A Non-Local Approach to Model the Combined Effe......
The article “A Non-Local Approach to Model the Combined Effects of Frequency Defects and Shot-Peening on the Fatigue Strength of a Pearlitic Steel” (B. Gerin, E. Pessard, et al., Theoretical and Applied Fracture Mechanics, 2018:19–32) reports that in a sample of 70 steel connecting rods subject to fatigue testing, the average fatigue strength, in MPa, was 408.2 with a standard deviation of 72.9. Find a 95% confidence interval for the mean fatigue strength under these conditions.
First let’s translate the problem into statistical language. We have a simple random sam-ple X_{1}, …, X_{70} of strengths. The sample mean and standard deviation are \overline{X} = 408.20 and s = 72.9. The population mean is unknown, and denoted by 𝜇.
The confidence interval has the form \overline{X} ± z_{\alpha ∕ 2}\sigma_{\overline{X}}, as specified in expression (5.1). Since we want a 95% confidence interval, the confidence level 1 − 𝛼 is equal to 0.95. Thus 𝛼 = 0.05, and z_{\alpha ∕ 2} = z_{.025} = 1.96. We approximate 𝜎 with s = 72.9, and obtain \sigma_{X} ≈ 72.9 ∕ \sqrt{70} = 8.7132. Thus the 95% confidence interval is 408.2 ± (1.96)(8.7132). This can be written as 408.20 ± 17.08, or as (391.12, 425.28).