The problem you provided is related to hypothesis testing for the population mean $\mu$ using a one-tailed Z-test. Here's the solution step by step:

(a) State the null hypothesis $H_0$ and the alternative hypothesis $H_1$:

• Null Hypothesis $H_0$: $\mu = 15.5$ (The mean completion time under new management is not less than 15.5 minutes.)
• Alternative Hypothesis $H_1$: $\mu < 15.5$ (The mean completion time under new management is less than 15.5 minutes.)

(b) Determine the type of test statistic to use:

Since the population standard deviation ($\sigma = 1.6$) is known and the sample size is $n = 19$, the appropriate test statistic is the Z-test.

(c) Find the value of the test statistic (rounded to 3 decimal places):

The formula for the Z-test statistic is:

$Z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}$

Where:

• $\bar{x} = 15.2$ (sample mean)
• $\mu = 15.5$ (population mean under $H_0$)
• $\sigma = 1.6$ (population standard deviation)
• $n = 19$ (sample size)

Substitute the values:

$Z = \frac{15.2 - 15.5}{\frac{1.6}{\sqrt{19}}}$

$Z = \frac{-0.3}{\frac{1.6}{4.3589}} = \frac{-0.3}{0.3671} \approx -0.817$

Z = -0.817

(d) Find the p-value (rounded to 3 or more decimal places):

For a one-tailed Z-test, the p-value corresponds to the area to the left of $Z = -0.817$ in the standard normal distribution.

Using a Z-table or statistical software, the p-value for $Z = -0.817$ is approximately:

$p \text{-value} \approx 0.206$

(e) Can we support the claim at $\alpha = 0.10$ level of significance?

Compare the p-value ($0.206$) with the significance level ($\alpha = 0.10$):

• $p > \alpha$ ($0.206 > 0.10$), so we fail to reject the null hypothesis.

Conclusion: At the 0.10 level of significance, there is not enough evidence to support the claim that the population mean completion time under new management is less than 15.5 minutes.

Let me know if you need further clarification or additional details! Here are related questions you might find helpful:

1. What would happen if the significance level was $\alpha = 0.05$?
2. How does the sample size affect the test statistic?
3. Why do we use the Z-test instead of the t-test in this scenario?
4. Can we increase the power of the test by altering the sample size?
5. How would the decision change if $\bar{x}$ were 15.1 instead of 15.2?

Tip: Always double-check