Question 1:

According to https://www.sba.gov/sites/default/files/files/Size_Standards_Table.pdf, the size limit for commercial bakeries to still be considered a small business is 1,000 employees. Now suppose that someone claims that the average small commercial bakery has 400 employees. In order to test that claim, you collect data on 100 small bakeries and find that the mean number of employees in the sample is 385 employees. The standard deviation is 78 employees.

Part (a)

Given that the average in the sample is not equal to what is claimed in the null hypothesis, explain why it is still necessary to carry out a hypothesis test to test the claim.

Part (b)

Using a level of significance α of 0.05, test the null hypothesis H₀: µ=400 using the 6-step procedure.

Part (c)

Using a level of significance α of 0.05, test the null hypothesis H₀: µ=400 using the p-value.

(Reminder: Your decision whether to reject or not reject H₀ will be the same with the 6-step procedure and the p-value, but you must show how you arrive at that conclusion.)

Question 2:

Part (a)

Suppose that someone claims that adult koalas eat an average of 392 grams of eucalyptus leaves per day. Using information from a sample of 50 koalas (sample mean = 395 grams, sample standard deviation = 53 grams), carry out a hypothesis test (via the 6-step procedure or using the p-value) with a level of significance α = 0.01 to test H₀: µ = 392 and show your work.

Part (b)

Does your result in part (a) prove that the population mean is 392 grams? Explain your answer.

Part (c)

Suppose that someone claims that adult koalas eat an average of 398 grams of eucalyptus leaves per day. Using information from a sample of 50 koalas (sample mean = 395 grams, sample standard deviation = 53 grams), carry out a hypothesis test (via the 6-step procedure or using the p-value) with a level of significance α = 0.01 to test H0: µ = 398 and show your work.

Part (d)

Does your result in part (c) prove that the population mean is 398 grams? Explain your answer.

Part (e)

Use your results from above (and the notes and the textbook, if you wish), to explain why we use the term "do not reject H₀" and not "accept H₀". (Hint: You may also wish to go back to your answers to parts (b) and (d) to make sure that they are consistent with your answer to this part.)

Question 3:

Part (a)

The following information is given:

- The sample consists of 119 cars driving down Hamburg Turnpike. Their mean speed is 42 miles per hour.

- α = 0.05

- The test statistic has a standard normal distribution (if H₀ is true).

If it is possible to determine the critical value(s) based on this information, determine it/them. If it is not possible, explain why it is not possible.

Part (b)

Part (b) is independent from Part (a), i.e., do not use information from Part (a) to answer Part (b).
The following information is given:

- The null hypothesis is that the mean thickness of a certain kind of wire is 0.5 millimeters.

- The test statistic has a standard normal distribution (if H₀ is true).

- The sample mean is 0.49 millimeters and the sample standard deviation is 0.02 millimeters.

If it is possible to determine the critical value(s) based on this information, determine it/them. If it is not possible, explain why it is not possible.