Questions: Before changes to its management staff, an automobile assembly line operation had a scheduled mean completion time of 13.2 minutes. The standard deviation of completion times was 1.8 minutes. An analyst at the company suspects that, under new management, the mean completion time, μ, is now less than 13.2 minutes. To test this claim, a random sample of 28 completion times under new management was taken by the analyst. The sample had a mean of 12.9 minutes. Assume that the population is normally distributed. Can we support, at the 0.10 level of significance, the claim that the population mean completion time under new management is less than 13.2 minutes? Assume that the population standard deviation of completion times has not changed under new management. Perform a one-tailed test. Then complete the parts below. Carry your intermediate computations to three or more decimal places, and round your responses as specified below. (If necessary, consult a list of formulas.) (a) State the null hypothesis H0 and the alternative hypothesis H1. H0: H1: (b) Determine the type of test statistic to use. (c) Find the value of the test statistic. (Round to three or more decimal places.) (d) Find the p-value. (Round to three or more decimal places.) (e) Can we support the claim that the population mean completion time under new management is less than 13.2 minutes? Yes No

Before changes to its management staff, an automobile assembly line operation had a scheduled mean completion time of 13.2 minutes. The standard deviation of completion times was 1.8 minutes. An analyst at the company suspects that, under new management, the mean completion time, μ, is now less than 13.2 minutes. To test this claim, a random sample of 28 completion times under new management was taken by the analyst. The sample had a mean of 12.9 minutes. Assume that the population is normally distributed. Can we support, at the 0.10 level of significance, the claim that the population mean completion time under new management is less than 13.2 minutes? Assume that the population standard deviation of completion times has not changed under new management. Perform a one-tailed test. Then complete the parts below. Carry your intermediate computations to three or more decimal places, and round your responses as specified below. (If necessary, consult a list of formulas.) (a) State the null hypothesis H0 and the alternative hypothesis H1. H0: H1: (b) Determine the type of test statistic to use. (c) Find the value of the test statistic. (Round to three or more decimal places.) (d) Find the p-value. (Round to three or more decimal places.) (e) Can we support the claim that the population mean completion time under new management is less than 13.2 minutes? Yes No
Transcript text: Before changes to its management staff, an automobile assembly line operation had a scheduled mean completion time of 13.2 minutes. The standard deviation of completion times was 1.8 minutes. An analyst at the company suspects that, under new management, the mean completion time, $\mu$, is now less than 13.2 minutes. To test this claim, a random sample of 28 completion times under new management was taken by the analyst. The sample had a mean of 12.9 minutes. Assume that the population is normally distributed. Can we support, at the 0.10 level of significance, the claim that the population mean completion time under new management is less than 13.2 minutes? Assume that the population standard deviation of completion times has not changed under new management. Perform a one-tailed test. Then complete the parts below. Carry your intermediate computations to three or more decimal places, and round your responses as specified below. (If necessary, consult a list of formulas.) (a) State the null hypothesis $H_{0}$ and the alternative hypothesis $H_{1}$. \[ \begin{array}{l} H_{0}: \square \\ H_{1}: \square \end{array} \] (b) Determine the type of test statistic to use. $\square$ (c) Find the value of the test statistic. (Round to three or more decimal places.) $\square$ (d) Find the $p$-value. (Round to three or more decimal places.) $\square$ (e) Can we support the claim that the population mean completion time under new management is less than 13.2 minutes? Yes No
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Solution

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Solution Steps

Step 1: State the Hypotheses

We begin by stating the null and alternative hypotheses for the test: \[ \begin{align_} H_{0}: & \quad \mu = 13.2 \\ H_{1}: & \quad \mu < 13.2 \end{align_} \]

Step 2: Determine the Type of Test Statistic

Since the population standard deviation is known, we will use a Z-test for this hypothesis test.

Step 3: Calculate the Test Statistic

The test statistic \( z \) is calculated using the formula: \[ z = \frac{\bar{X} - \mu_0}{\frac{\sigma}{\sqrt{n}}} \] Substituting the values: \[ z = \frac{12.9 - 13.2}{\frac{1.8}{\sqrt{28}}} = \frac{-0.3}{0.3401680257083045} \approx -0.882 \]

Step 4: Calculate the p-value

The p-value is determined using the cumulative distribution function (CDF) for the standard normal distribution: \[ p\text{-value} = P(Z < -0.882) \approx 0.1889 \]

Step 5: Make a Decision

To determine whether to reject the null hypothesis, we compare the p-value to the significance level \( \alpha = 0.10 \):

  • If \( p\text{-value} < \alpha \), we reject \( H_0 \).
  • If \( p\text{-value} \geq \alpha \), we fail to reject \( H_0 \).

In this case, since \( 0.1889 \geq 0.10 \), we fail to reject the null hypothesis.

Step 6: Conclusion

Based on the results of the hypothesis test, we do not have sufficient evidence to support the claim that the population mean completion time under new management is less than 13.2 minutes.

Final Answer

(a) \[ \begin{array}{l} H_{0}: \mu = 13.2 \\ H_{1}: \mu < 13.2 \end{array} \]

(b) Z-test

(c) \( z \approx -0.882 \)

(d) \( p\text{-value} \approx 0.1889 \)

(e) No

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