The company has a delivery target time of 20 minutes on average. If the company wishes to conduct a hypothesis test to assess whether the mean delivery time is longer than the target time, what would be the null and alternative hypotheses?
Null hypothesis: H0: μ = 20, H1: μ > 20. From the hypothesis test in Question 13, which of the following conclusions is TRUE?

Since the t-statistic is 35.5, there is enough evidence to suggest that the true average delivery time is longer than 20 minutes.

Since the p-value of the test is 41.6%, there is not enough evidence to suggest that the true average delivery time is longer than 20 minutes.

Since the p-value of the test is 0.31%, there is enough evidence to suggest that the true average delivery time is longer than 20 minutes.

Since the t-statistic is 0.21 there is not enough evidence to suggest that the true average delivery time is longer than 20 minutes.

For the hypothesis test with the null and alternative hypotheses defined as:

- **Null Hypothesis (H₀)**: μ = 20 (the mean delivery time is equal to 20 minutes)
- **Alternative Hypothesis (H₁)**: μ > 20 (the mean delivery time is longer than 20 minutes)

To determine the correct conclusion among the given options, we need to analyze the given t-statistic or p-value relative to the hypothesis test:

1. **Since the t-statistic is 35.5, there is enough evidence to suggest that the true average delivery time is longer than 20 minutes.**

   - A t-statistic of 35.5 would indicate a very extreme value far from the mean under the null hypothesis, suggesting a strong reason to reject H₀ in favor of H₁. This option seems correct because a high t-statistic would imply that the sample mean is significantly higher than the target time of 20 minutes.

2. **Since the p-value of the test is 41.6%, there is not enough evidence to suggest that the true average delivery time is longer than 20 minutes.**

   - A p-value of 41.6% is quite high, indicating that there is a 41.6% probability of observing a result at least as extreme as the test statistic under the null hypothesis. Therefore, there is not enough evidence to reject H₀.

3. **Since the p-value of the test is 0.31%, there is enough evidence to suggest that the true average delivery time is longer than 20 minutes.**

   - A p-value of 0.31% is very low, suggesting that the observed data is very unlikely under H₀. This would indicate enough evidence to reject H₀ and accept H₁, making this statement correct.

4. **Since the t-statistic is 0.21 there is not enough evidence to suggest that the true average delivery time is longer than 20 minutes.**

   - A t-statistic of 0.21 would be close to the mean of the distribution under the null hypothesis, indicating little to no evidence against H₀. Therefore, this option also indicates insufficient evidence to reject H₀.

**Conclusion:**
- The correct conclusion among the options is: **Since the p-value of the test is 0.31%, there is enough evidence to suggest that the true average delivery time is longer than 20 minutes.**

This is because a p-value of 0.31% (option 3) is the only one that suggests strong evidence against the null hypothesis, supporting the alternative hypothesis that the mean delivery time is longer than 20 minutes.

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Would you like to discuss this further, or do you have any questions? Here are some related questions:

1. What is the relationship between the p-value and the significance level (α) in hypothesis testing?
2. How is the t-statistic calculated in a one-sample t-test?
3. What are the assumptions required for a one-sample t-test?
4. How does the size of the sample affect the t-statistic and p-value?
5. What would be the conclusion if the p-value was exactly equal to the significance level (α)?

**Tip:** Always compare the p-value to the significance level (usually 0.05 or 5%) to determine if the evidence is strong enough to reject the null hypothesis.

Explore hypothesis testing to assess whether the mean delivery time of a company is longer than the target time of 20 minutes. Analyze t-statistic and p-value in statistical testing. Learn the significance of the test results and their implications.

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