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.