Question

The historical mean yield of soybeans from farms in Iowa is 32.9 bushels per acre. A hypothesis test is run to test the claim that the true mean yield of a dry summer was less than the historical mean. If the farmers believe they have evidence of a dry summer (smaller true mean yield), then they will spend a lot of money on irrigation system.

Which of the following describes the likely consequence of making a Type I error for this problem is?

	1.	Farmers do not spend a lot of money on an irrigation system that is needed.
	2.	Farmers do not spend a lot of money on an irrigation system that is not needed.
	3.	Farmers spend a lot of money on an irrigation system that is needed.
	4.	Farmers spend a lot of money on an irrigation system that is not needed.

A company claims that a new manufacturing process decreases the mean amount of aluminum needed for cans and therefore decreases the weight. Independent random samples of aluminum cans made by the old process and the new process are taken. Output from running hypothesis tests are given below. Is there evidence at the 5% significance level to support the claim that the mean weight for all old cans is greater than the mean weight for all new cans? Select the correct conclusion.

Matched Pairs Design                         Welch’s (Independent) T Test

Test Statistic = 1.866                           Test Statistic = 1.466

p-value = 0.037                                   p-value = 0.075

	1.	Using the Matched Pairs design, there is not sufficient evidence at the 5% significance level to support the claim that the mean weight for all old cans is more than the mean weight for all new cans.
	2.	Using the Matched Pairs design, there is sufficient evidence at the 5% significance level to support the claim that the mean weight for all old cans is more than the mean weight for all new cans.
	3.	Using the Welch’s design, there is sufficient evidence at the 5% significance level to support the claim that the mean weight for all old cans is more than the mean weight for all new cans.
	4.	Using the Welch’s design, there is not sufficient evidence at the 5% significance level to support the claim that the mean weight for all old cans is more than the mean weight for all new cans.

Answer 1

In this case null hypothesis is that the true mean yield of a dry summer is greater than or equal to 32.9 bushels per acre. If null hypothesis is true then farmer does not need to spend money on irrigation system.

Type I error is rejecting the null hypothesis when it is true. In this case, type I error is concluding that the true mean yield of a dry summer is less than 32.9 bushels per acre even when the actual mean is greater than or equal to 32.9 bushels per acre. So, the correct answer is as follows:

4. Farmers spend a lot of money on an irrigation system that is not needed.

In this case samples are independent so Matched pair test cannot be used. So, the correct answer is:

4. Using the Welch’s design, there is not sufficient evidence at the 5% significance level to support the claim that the mean weight for all old cans is more than the mean weight for all new cans.