Question

Jake Yum's new computer program "Learn to be a Great Cook" is selling off the shelves, and Jake wants to know whether there is a gender difference in the cooking quality of individuals who use the software. So Jake asked 10 males and 20 females to spend 20 hours over a month to go through the program. Jake then asked each participant to cook a dinner for him that includes a meat, 2 veggies, a bread, a dessert, and an appropriate wine. All meals were subsequently ranked, and the sum of ranks for males equaled 80, while the sum of ranks for females equaled 385. Use a Mann-Whitney U test, an alpha of .05, and two tailed. (Note: include hypotheses, detailed calculations, critical value, and outcome statement in sentence form).

In the box below, provide the following information:

Null Hypothesis in sentence form (1 point):

Alternative Hypothesis in sentence form (1 point):  
Critical Value(s) (2 points):

Calculations (4 points): Note: the more detail you provide, the more partial credit that I can give you if you make a mistake.

Outcome (determination of significance or not, effect size if appropriate, and what this reflects in everyday language, 2 points)

Answer 1

Mann Whitney U test is a non parametric test that is used to compare outcomes the means between two independent groups.

Null hypothesis:

H₀ : The cooking quality of males using the software and the cooking quality of female using the software is same i.e.

where = mean of cooking quality of males using the software

and = mean of cooking quality of females using the software

Alternative hypothesis:

H₀ : The cooking quality of males using the software and the cooking quality of female using the software is different i.e.

Calculations:

The test statistic for the Mann Whitney U Test is denoted U and is the smaller of U_Male and U_Female defined below:

U_Male = N_M * N_F + N_M * (N_M + 1) / 2 - R_M

U_{Female =} N_M * N_F + N_F * (N_F + 1) / 2 - R_F

Terms used:

N_M = Number of males

N_F = Number of females

R_M = Sum of Ranks of males

R_F = Sum of Ranks of females

Now

U_Male = 10 * 20 + 10 * (10 + 1) / 2 - 80 = 175

U_Female = 10 * 20 + 20 * (20 + 1) / 2 - 385 = 25

Therefore U = Min(175,25) = 25

Outcome

The appropriate critical value can be found in the U tables. To determine the appropriate critical value we need sample sizes (N_M=10 and N_F=20) and our two-sided level of significance (α=0.05).

The critical value for this test with N_M=10, N_F=20 and α =0.05 is 55 and the decision rule is as follows: Reject H₀ if U < 55

Now 25 < 55 which is true. Hence we reject H₀ at 5% level of significance and conclude that the cooking quality of males using the software and the cooking quality of female using the software is different.

So in everyday language it simply means there is a gender difference in the cooking quality of individuals who use the software.