Final Answer:
A one-tailed right-tailed test should be performed to determine if there is sufficient evidence to support the technician's claim that the mean rod length is more than 18 inches. The test statistic is the z-score, which is calculated to be 2.78. At the α=0.01 significance level, the p-value is 0.0027, which is less than 0.01. Therefore, there is sufficient evidence to reject the null hypothesis and support the technician's claim that the mean rod length is more than 18 inches.
Step-by-step explanation:
Hypothesis Test:
Since the technician is claiming that the mean rod length is more than 18 inches, this is a one-tailed right-tailed test. This means that we will only reject the null hypothesis if the sample mean is significantly greater than 18 inches.
Test Statistic:
The test statistic for this test is the z-score, which is calculated as follows:
z = (xbar - μ₀) / σ
where:
xbar is the sample mean (18.08 inches)
μ₀ is the hypothesized population mean (18 inches)
σ is the sample standard deviation (0.23 inches)
Plugging in the values, we get:
z = (18.08 - 18) / 0.23 = 2.78
Significance Level and P-value:
The significance level (α) is the probability of rejecting the null hypothesis when it is actually true. In this case, α = 0.01. The p-value is the probability of getting a test statistic as extreme or more extreme than the one we observed, assuming the null hypothesis is true. In this case, the p-value is 0.0027.
Since the p-value is less than the significance level, we reject the null hypothesis and conclude that there is sufficient evidence to support the technician's claim that the mean rod length is more than 18 inches.