To address this problem, we'll perform a one-tailed t-test to determine if the mean completion time of the new process is statistically less than 70 minutes.

(a) State the Hypotheses

• Null Hypothesis (H₀): The population mean completion time is equal to 70 minutes. $H_0: \mu = 70$

• Alternative Hypothesis (H₁): The population mean completion time is less than 70 minutes. $H_1: \mu < 70$

(b) Determine the Type of Test Statistic

Since the sample size is small (n = 19), and the population standard deviation is unknown, we will use the t-statistic for this hypothesis test.

(c) Find the Value of the Test Statistic

The test statistic is calculated using the formula: $t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$ where:

• $\bar{x}$ = sample mean = 65 minutes
• $\mu_0$ = hypothesized population mean = 70 minutes
• $s$ = sample standard deviation = 12 minutes
• $n$ = sample size = 19

Substituting the values: $t = \frac{65 - 70}{12/\sqrt{19}} = \frac{-5}{2.753}$

Calculating this: $t \approx -1.816$

(d) Find the Critical Value

For a one-tailed test at the 0.10 significance level with $df = n - 1 = 19 - 1 = 18$ degrees of freedom, we need to find the critical value $t_{0.10, 18}$.

Using a t-distribution table or calculator, the critical value is approximately: $t_{0.10, 18} \approx -1.330$

(e) Conclusion

Compare the calculated test statistic to the critical value:

• If $t \leq t_{0.10, 18}$, reject the null hypothesis $H_0$.
• If $t > t_{0.10, 18}$, fail to reject the null hypothesis $H_0$.

Here, $t \approx -1.816$ is less than the critical value of $-1.330$, so we reject the null hypothesis.

Conclusion: Yes, at the 0.10 level of significance, there is sufficient evidence to conclude that the mean completion time using the new process is less than 70 minutes.

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Would you like a detailed explanation of any step, or do you have any further questions?

Here are some related questions to consider:

1. How would the conclusion change if the significance level were 0.05 instead of 0.10?
2. What assumptions underlie the use of the t-test in this context?
3. How does the sample size affect the reliability of the test?
4. What would the test statistic be if the sample mean was 68 minutes instead of 65?
5. What is the difference between a one-tailed and two-tailed t-test?
6. Why is the critical value different for different degrees of freedom?
7. How would the test result differ if the standard deviation were 8 minutes instead of 12?
8. What steps would you take to ensure that the process improvements are statistically significant over time?

Tip: Understanding the context of the significance level ($\alpha$) is crucial. A lower $\alpha$ level (e.g., 0.01) makes it harder to reject the null hypothesis, reducing the risk of a Type I error but increasing the chance of a Type II error.