To test the claim that the population mean completion time under new management is less than 15 minutes, we'll perform a one-tailed hypothesis test using the following steps:

Step 1: State the Hypotheses

• Null Hypothesis ($H_0$): The population mean completion time under new management is 15 minutes. $H_0: \mu = 15$
• Alternative Hypothesis ($H_1$): The population mean completion time under new management is less than 15 minutes. $H_1: \mu < 15$

Step 2: Identify the Type of Test Statistic

Since the population standard deviation is known (1.3 minutes), and the sample size is large ($n = 42$), we will use a z-test.

Step 3: Calculate the Test Statistic

The test statistic for a z-test is given by:

$z = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}}$

Where:

• $\bar{x} = 14.6$ (sample mean)
• $\mu_0 = 15$ (population mean under the null hypothesis)
• $\sigma = 1.3$ (population standard deviation)
• $n = 42$ (sample size)

Let's plug in the values:

$z = \frac{14.6 - 15}{\frac{1.3}{\sqrt{42}}}$

I'll calculate this now.The calculated test statistic is approximately $z = -1.994$.

Step 4: Find the P-value

The P-value for a one-tailed z-test can be found by looking up the cumulative probability corresponding to the calculated z-value in the standard normal distribution table.

Since this is a left-tailed test, we are interested in $P(Z < -1.994)$.

Let's calculate the P-value.The P-value is approximately $0.023$.

Step 5: Conclusion

Compare the P-value to the significance level $\alpha = 0.05$:

• If $\text{P-value} < \alpha$, reject the null hypothesis $H_0$.
• If $\text{P-value} \geq \alpha$, do not reject the null hypothesis.

In this case, $\text{P-value} = 0.023$ is less than $\alpha = 0.05$, so we reject the null hypothesis.

Conclusion

At the 0.05 level of significance, there is sufficient evidence to support the claim that the population mean completion time under new management is less than 15 minutes.

Would you like more details or have any questions?

Relative Questions:

1. What is a one-tailed test, and when is it used?
2. How does the P-value help in hypothesis testing?
3. What assumptions must be met to use a z-test?
4. How would the test change if the sample size were smaller?
5. What is the importance of the significance level in hypothesis testing?
6. How do you interpret a rejection of the null hypothesis?
7. What would happen if the population standard deviation were unknown?
8. Can the results of this test be generalized to all such situations?

Tip:

Always ensure that the conditions for using a specific statistical test are met before applying it.