Posted: March 29th, 2023

Question Set 1: Completing an ANOVA Source Table

Question Set 1: Completing an ANOVA Source Table
A group of researchers wants to measure the difference between the cost of different fruits from six grocery stores in the USA (Walmart, Target, Weis, Schnucks, Trader Joe, and Aldi). Complete the enclosed ANOVA table to help researchers determine the difference between costs. Round all answers to 3 decimal places (except DF).

Question Set 1_A:

Use Minitab to compute the p-value for the previous question:

Choose Graph > Probability Distribution Plot > View Probability
Click the Distribution dropdown menu > select F
For ‘Numerator degrees of freedom’ enter the df for Between Groups (per Question 1)
For ‘Denominator degrees of freedom’ enter the df for Within Groups (per Question 1)
Click Options > select ‘A specified x (axis) value’
For ANOVA, we use ‘Right tail’
For ‘X value’ enter the F statistic (per Question 1) > OK

Question Set 2: Describing and Comparing Data from Three or More Groups
This question set uses the NHL player salaries datafile Download NHL player salaries datafile which can be obtained from Canvas. We want to compare NHL players based on their nationality (Canada,USA and others) in terms of their salaries.

(Data in files)

Question Set 2_A:

Use Minitab to find the number of cases, mean and standard deviation for the salary variable for each of the three nationalities separately:

Choose Stat > Basic Statistics > Display Descriptive Statistics
Variable: double-click on ‘Salary’
Group variables: double-click on ‘Nationality’
Click Statistics > select Mean, Standard deviation and N nonmissing > OK
In Canvas, use Insert > Image to provide a screenshot of your Minitab results.

Question Set 2_B:

Use Minitab to construct side-by-side boxplots to compare NHL players concerning their nationality and salaries:

Choose Graph > Boxplot > One Y Variable, With Categorical Variables
For Y-variable, double-click on ‘Salary’
For Categorical variables, double-click on ‘Nationality’ > OK
In Canvas, use Insert > Image to provide a screenshot of your Minitab graph.

Question Set 2_C:

Use Minitab to construct a dotplot with groups to compare NHL players with respect to their nationality and salaries:

Choose Graph > Dotplot > One Y Variable, Groups Displayed on the Scale
For Y-variable, double-click on ‘Salary’
For Group variables, double-click on ‘Nationality’ > OK
In Canvas, use Insert > Image to provide a screenshot of your Minitab graph.

Question Set 2_D:

Explain why a one-way ANOVA should be conducted to compare the mean salary of the players nationality groups as opposed to a series of three independent means t tests.

Question Set 2_E:

State the null and alternative hypotheses. (You will check the assumptions in the next question.)

Question Set 2_F:

Check the assumptions:

The first ANOVA assumption is that the samples are independent. The other assumptions include: The response variable for each factor level needs to be approximately normally distributed OR the sample size for each factor level needs to be at least 30. Is this assumption met (per the summary statistics you computed above)? Explain your answer.

The population variances need to be equal for the three Nationality groups. Using the summary statistics, if the largest standard deviation divided by the smallest standard deviation is less than 2, then we can assume that the population variances are equal. Is this assumption met? Show your work.

Question Set 2_G:

Compute the test statistic and the p-value.

Specifically, use Minitab to conduct a one-way ANOVA to compare the mean Salary for the three Nationality groups:

Choose Stat > ANOVA > One-Way
For Response, double-click on ‘Salary’
For Factor, double-click on ‘Nationality’ > OK
In Canvas, use Insert > Image to provide a screenshot of your Minitab ‘Analysis of Variance’ table.

Per your Minitab results, what is the test statistic value (i.e. F-Value)? Use 2 decimal places.

Question Set 2_H:

Decide to reject or fail to reject the null hypothesis. Specifically, fill in the following blanks:

Because the p-value is _______ (use 3 decimal places), which is _____________ (less than or greater than) our alpha-level of 0.05, we _____________ (reject or fail to reject) the null hypothesis.

Question Set 2_I:

State a real-world conclusion. Specifically, is there statistical evidence that the population mean salary for all three nationalities are NOT all equal? Explain your answer.

Question Set 2_J:

We have concluded above that not all population means are equal. The next step is to determine which population means are different. Therefore, use Minitab to conduct Tukey simultaneous tests for differences in means:

Choose Stat > ANOVA > One-Way
Your Response and Factor should already be entered from before
Click Comparisons > select ‘Tukey’ comparisons > OK
In Canvas, use Insert > Image to provide a screenshot of your Minitab “Tukey Simultaneous 95% CIs” graph.

Question Set 2_K:

Given your Tukey comparisons results above, which pairs of population means are significantly different? Specifically, answer these three questions:

Is the mean salary for NHL Players from Canada significantly different than the mean salary for players from Other?
Is the mean Salary for NHL player from Canada significantly different than the mean Salary for NHL player from the USA?
Is the mean Salary for NHL player from Other significantly different than the mean Salary for NHL player from the USA?

Answer Set 1_A:

The completed ANOVA source table is as follows:

Source of Variation SS df MS F-Value P-Value
Between Groups 3483.535 5 696.707 5.345 0.0004
Within Groups 42208.522 144 293.330
Total 45692.057 149
The p-value obtained from Minitab is 0.000.