This problem involves hypothesis testing, specifically a one-sample t-test to verify if the observed mean reduction in delivery time is statistically significantly less than the claimed 10 minutes. Let's walk through the steps required to solve it.

Step 1: Define the Hypotheses

• Null hypothesis $H_0$: The mean reduction in delivery time is at least 10 minutes.
  $H_0: \mu \geq 10$
• Alternative hypothesis $H_1$: The mean reduction in delivery time is less than 10 minutes.
  $H_1: \mu < 10$

This is a left-tailed test since the researcher is interested in determining if the reduction is significantly less than 10 minutes.

Step 2: Given Information

• Sample size: $n = 100$
• Sample mean: $\bar{x} = 9.5$
• Claimed population mean: $\mu_0 = 10$
• Sample standard deviation: $s = 3$
• Significance level: $\alpha = 0.05$

Step 3: Compute the Test Statistic

We will use the t-statistic formula for a one-sample test:

$t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}$

Substitute the given values:

$t = \frac{9.5 - 10}{\frac{3}{\sqrt{100}}} = \frac{-0.5}{0.3} = -1.67$

Step 4: Determine the Critical Value

Since this is a left-tailed test with $\alpha = 0.05$ and degrees of freedom $df = n - 1 = 99$, we need the critical value for $t_{0.05, 99}$.

Using a t-distribution table or a calculator: $t_{0.05, 99} \approx -1.660$

Step 5: Decision Rule

• If the calculated $t$-statistic is less than the critical value, we reject the null hypothesis.
• If the calculated $t$-statistic is greater than the critical value, we fail to reject the null hypothesis.

In this case: $t = -1.67 \quad \text{and} \quad t_{critical} = -1.660$

Since $-1.67 < -1.660$, we reject the null hypothesis.

Step 6: Conclusion

At the 0.05 significance level, there is enough evidence to conclude that the mean reduction in delivery time is less than the claimed 10 minutes.

Do you have any questions or would you like further clarifications on any part?

Here are five follow-up questions to expand your understanding:

1. How would the result change if the sample size was smaller?
2. What is the p-value for this test, and how does it relate to the conclusion?
3. How would a two-tailed test differ from this one-tailed test?
4. Why did we use the t-distribution instead of the normal distribution?
5. What are Type I and Type II errors in this context?

Tip: For larger samples (e.g., $n > 30$), the t-distribution closely approximates the normal distribution, which makes t-tests very flexible!