A marketing researcher decides to test whether coffee drinkers have a favorable view of coffee labeled as “Fair Trade.” He has 15 randomly selected coffee drinkers taste his company’s Morning Blend, then the exact same coffee labeled Fair Trade Blend. They are asked to rate the coffees on a 1 to 10 scale, 10 being best. The data is recorded in Minitab and a statistical test is done at α = 0.05 to determine if “Fair Trade” coffee is rated more highly.
a.) What is the correct hypothesis test for testing this data?
b.) What condition(s), if any, are necessary for the test to be valid?
c.) What are the hypotheses for this test?
a.) The correct hypothesis test for testing this data is a paired t-test.
b.) The paired t-test requires the following conditions to be valid:
1. The observations within each pair must be dependent or related to each other. In this case, the same individuals are being asked to rate the two coffees, so the observations are dependent.
2. The differences between the paired observations should be approximately normally distributed. This assumption can be checked using graphical methods or by checking the skewness and kurtosis of the differences.
c.) The hypotheses for this test are:
- Null hypothesis (H0): The mean rating for the "Morning Blend" coffee is equal to the mean rating for the "Fair Trade Blend" coffee.
- Alternative hypothesis (Ha): The mean rating for the "Fair Trade Blend" coffee is higher than the mean rating for the "Morning Blend" coffee.
a.) The correct hypothesis test for testing this data is a paired t-test. The reason for using a paired t-test is that each coffee drinker provides two ratings (for Morning Blend and Fair Trade Blend). By comparing the mean ratings for the two conditions, the researcher can determine if there is a significant difference in the perception of the coffee with the different labels.
b.) The paired t-test relies on the assumption that the differences between the pairs of observations follow a normal distribution. Additionally, the observations should be independent of each other. To ensure the validity of the test, it is important to check if these assumptions are satisfied. If the data violates these assumptions, an alternative method may need to be used.
c.) The hypotheses for this test are as follows:
- Null Hypothesis (H₀): The mean rating for Morning Blend is the same as the mean rating for Fair Trade Blend.
- Alternative Hypothesis (H₁): The mean rating for Fair Trade Blend is higher than the mean rating for Morning Blend.
In statistical notation, the hypotheses can be written as:
- H₀: μd = 0 (where μd represents the mean difference between the ratings)
- H₁: μd > 0