Question

A random sample of 12 shearing pins is taken in a study of the Rockwell hardness of the head on the pin. Measurements on the Rockwell hardness were made for each of the 12, yielding an average value of 48.5. Assuming the population standard deviation is 1.5, construct a 90% confidence interval for the mean Rockwell hardness. Test the hypothesis that the average Rockwell hardness is 48 at a 0.01 level of significance.

Answer 1

sample mean, xbar = 48.5
sample standard deviation, σ = 1.5
sample size, n = 12

Given CI level is 90%, hence α = 1 - 0.9 = 0.1
α/2 = 0.1/2 = 0.05, Zc = Z(α/2) = 1.64

CI = (xbar - Zc * s/sqrt(n) , xbar + Zc * s/sqrt(n))
CI = (48.5 - 1.64 * 1.5/sqrt(12) , 48.5 + 1.64 * 1.5/sqrt(12))
CI = (47.79 , 49.21)

B)
Below are the null and alternative Hypothesis,
Null Hypothesis, H0: μ = 48
Alternative Hypothesis, Ha: μ ≠ 48

Test statistic,
z = (xbar - mu)/(sigma/sqrt(n))
z = (48.5 - 48)/(1.5/sqrt(12))
z = 1.15

P-value Approach
P-value = 0.2501
As P-value >= 0.01, fail to reject null hypothesis.

The harness is not different than 48