Two Population Means Answer Guide
A tomato farmer with a very large farm of approximately 2200 acres had heard about a new type of rather expensive fertilizer which would supposedly significantly increase his production. The frugal farmer wanted to test the new fertilizer before committing the large investment required to fertilize a farm of his size. He therefore selected 15 parcels of land on his property and divided them each into two portions.
He bought just enough of the new fertilizer to spread over one half of each parcel and then spread the old fertilizer over the other half of each parcel. His yields in pounds per tomato plant were as follows:
What if you were the fertilizer sales representative and your job was to prove the superiority of the new product to the farmer?
(1) You should start by running the same test he did in which he came to the decision that he could not conclude a difference.
After conducting the same two-tailed test that the farmer performed himself, a conclusion to reject the null hypothesis was determined. Check out the full spreadsheet for details.
(2) Perform the test as it should have been done and find if you come to a different conclusion.
Successful application of the test requires us to find the paired sample data. Paired data is calculated by subtracting the value of the new fertilizer from the old fertilizer, then using zero as the expected mean baseline.
With this data calculated, the correct hypotheses for a one-tailed t-test are as follows:
H0 : paired sample mean <= 0
H1: paired sample mean > 0
with a 0.05 significance level, the decision rule is determined to reject H0 if the t-value is greater than or equal to 1.645. We found a test statistic of 4.84, therefore the optimal decision supports the null hypothesis. Also, the p-value was 1 which strengthens the support for the null hypothesis.
(3) Explain why the results were different and why your test was a stronger and more reliable test.
The test conducted by the farmer only provided evidence that the new fertilizer performed differently than the old fertilizer. This method is not optimal because he needs to find out specifically if the new fertilizer is superior, which requires a one-tail test. His method also treated each sample as isolated from the others, even though they were plotted on the same parcel. This situation can be improved by a paired sample mean. This method creates a more reliable value for the mean and standard deviation in the test.
Spreadsheet of Hypothesis Tests
Looking for the spreadsheet of the tomato farmer hypothesis tests here.