a) The null hypothesis H₀ and the alternative hypothesis H₁ can be stated as follows:
Null Hypothesis (H₀): The mean completion time under new management is equal to or greater than 12.2 minutes, i.e., H₀: μ ≥ 12.2
Alternative Hypothesis (H₁): The mean completion time under new management is less than 12.2 minutes, i.e., H₁: μ < 12.2
b) The type of test statistic is a one-tailed Z-test because we know the population standard deviation and we are testing for a value less than a given value.
c) The computed Z-test statistic is -0.008.
d) The critical value for a one-tailed test at the 0.05 level of significance from the Z-table is approximately -1.645.
e) The computed Z-test statistic -0.008 is greater than the critical value -1.645, we do not reject the null hypothesis.
How the hypotheses are determined:
Scheduled mean completion time = 12.2 minutes
The standard deviation = 1.8 minutes
The null hypothesis H₀ and the alternative hypothesis H₁ can be stated as follows:
Null Hypothesis (H₀): The mean completion time under new management is equal to or greater than 12.2 minutes, i.e., H₀: μ ≥ 12.2
Alternative Hypothesis (H₁): The mean completion time under new management is less than 12.2 minutes, i.e., H₁: μ < 12.2
c) Computing the value of the test statistic:
The formula for the Z-test statistic is Z =
Here,
=12.1
Sample mean, μ = 12.2
Population mean,
= 1.8
(population standard deviation), and n = 48
(sample size).
Substituting these values into the formula, we get the Z-test statistic.
=
= -0.008
d) The critical value for a one-tailed test at the 0.05 level of significance from the Z-table is approximately -1.645.
e) Supporting the claim:
Thus, since -0.008 is greater than -1.645, we do not reject the null hypothesis. This means we cannot support the claim that the mean completion time under new management is less than 12.2 minutes based on the given sample data.