Answer: The p-value of the test is less than 0.001, indicating strong evidence against the null hypothesis. This suggests that the delivery company's claim that their delivery people deliver more than 70 packages per hour is justified.
Explanation:
To determine whether the delivery company's claim that their delivery people deliver more than 70 packages per hour is justified, we can perform a hypothesis test.
The null hypothesis (H0) is that the average number of packages delivered by the delivery people is equal to or less than 70 per hour. The alternative hypothesis (Ha) is that the average number of packages delivered is greater than 70 per hour.
Let's assume a significance level of α = 0.05.
To perform the hypothesis test, we can calculate the test statistic using the formula:
test statistic = (sample mean - population mean) / (sample standard deviation / sqrt(sample size))
In this case, the sample mean is 72.5, the population mean (claimed average) is 70, the sample standard deviation is 20, and the sample size is 400.
Calculating the test statistic:
test statistic = (72.5 - 70) / (20 / sqrt(400)) = 2.5 / (20 / 20) = 2.5
Next, we need to determine the critical value for our one-tailed test at α = 0.05. Since our alternative hypothesis is that the average number of packages delivered is greater than 70, we will use the right-tailed test.
Using a t-table or a statistical calculator, we can find the critical value at α = 0.05 and degrees of freedom (n-1) = 399. Let's assume the critical value is 1.645.
Finally, we can compare the test statistic to the critical value to make our decision. If the test statistic is greater than the critical value, we reject the null hypothesis; otherwise, we fail to reject the null hypothesis.
In this case, the test statistic (2.5) is greater than the critical value (1.645), so we reject the null hypothesis.
The p-value of the test is the probability of obtaining a test statistic as extreme as the one observed (or more extreme) if the null hypothesis is true. In this case, the p-value is the probability of observing a test statistic greater than 2.5, assuming the null hypothesis is true.
To calculate the p-value, we can use a t-table or a statistical calculator with the degrees of freedom (n-1) = 399.
Using the t-table or calculator, we find that the p-value is less than 0.001.
Therefore, the p-value of the test is less than 0.001, indicating strong evidence against the null hypothesis. This suggests that the delivery company's claim that their delivery people deliver more than 70 packages per hour is justified.
Hope this helps!