Question

A microwave manufacturing company has just switched to a new automated production system. Unfortunately, the new machinery has been frequently failing and requiring repairs and service. The company has been able to provide its customers with a completion time of 6 days or less. To analyze whether the completion time has increased, the production manager took a sample of 36 jobs and found that the sample mean completion time was 6.5 days with a sample standard deviation of 1.5 days.

a) At a significance level of 0.10, test whether the completion time has increased.

b) Find and interpret a 99% confidence interval for the mean completion time.

Answer 1

a)

Below are the null and alternative Hypothesis,
Null Hypothesis: μ = 6
Alternative Hypothesis: μ > 6

Rejection Region
This is right tailed test, for α = 0.1 and df = 35
Critical value of t is 1.306.
Hence reject H0 if t > 1.306

Test statistic,
t = (xbar - mu)/(s/sqrt(n))
t = (6.5 - 6)/(1.5/sqrt(36))
t = 2

P-value Approach
P-value = 0.027
As P-value < 0.1, reject the null hypothesis.

b)

sample mean, xbar = 6.5
sample standard deviation, s = 1.5
sample size, n = 36
degrees of freedom, df = n - 1 = 35

Given CI level is 99%, hence α = 1 - 0.99 = 0.01
α/2 = 0.01/2 = 0.005, tc = t(α/2, df) = 2.724

ME = tc * s/sqrt(n)
ME = 2.724 * 1.5/sqrt(36)
ME = 0.681

CI = (xbar - tc * s/sqrt(n) , xbar + tc * s/sqrt(n))
CI = (6.5 - 2.724 * 1.5/sqrt(36) , 6.5 + 2.724 * 1.5/sqrt(36))
CI = (5.819 , 7.181)
Therefore, based on the data provided, the 99% confidence interval for the population mean is 5.819 < μ < 7.181 which indicates that we are 99% confident that the true population mean μ is contained by the interval (5.819 , 7.181)