There are 4 responses. Read each response. Write a 75-100 word response for each. 

Response 1.) Deborah 

The two-way analysis of variance test is an extension of the one-way ANOVA test that examines the influence of different categorical independent variables on one dependent variable. While the one-way ANOVA measures the significant effect of one independent variable , the two-way ANOVA is used when there is more than one independent variable and multiple observations for each independent variable. The two-way ANOVA can not only determine the main effect of contributions of each Independent variable but also identifies if there is a significant interaction effect between the independent variables.

We define a factorial design as having fully replicated measures on two or more crossed factors. In a factorial design multiple independent effects are tested.  Each level of one factor is tested in combination with each level of the other, so the design is rectangular. The two way analysis of variance aims to investigate both the independent and combined effect of each factor on the response variable. The combined effect is investigated by assessing whether there is a significant interaction between the factors.

An example of using a two-way ANOVA F test you might want to find out if there is an interaction between income and gender for anxiety level at job interviews. The anxiety level is the outcome, or the variable that can be measured. Gender and Income are the two categorical variables. These categorical variables are also the independent variables, which are called factors in a Two Way ANOVA.  The factors can be split into levels. In the above example, income level could be split into three levels: low, middle and high income. Gender could be split into three levels: male, female, and transgender. Treatment groups are all possible combinations of the factors. In this example there would be 3 x 3 = 9 treatment groups.

An example of not using a two-way ANOVA F test would be if a professor wants to know if three different studying techniques lead to different exam scores. The professor performs a one-way ANOVA F test instead of two-way ANOVA.

Reference:

Zach. (2021) One-Way vs Two-way ANOVA:  When to Use Each

Response 2.) Michael 

One should use the two-way ANOVA F test to examine possible differences among the means of each factor in a factorial design when there are two factors (Levine et al., 2021). A factor is a characteristic of comparison populations measured with a numerical variable. For example, a health care consultant would like to compare the revenue of physicians by specialty (factor A) and the impact of employment type (e.g., self-employed versus hospital-employed) (factor B) for the physician as it relates to the revenue by specialty.  

If the study included cardiology, gastroenterology, orthopedics, and family practice for factor A, it would have four levels of this factor. The ANOVA F test would test the following hypotheses:

Hypothesis Factor A: H0: There is no difference in physician revenue due to specialty.

Hypothesis Factor B: H0: There is no difference in physician revenue due to employment type.

Hypothesis Interaction of A and B: H0: There is no interaction between factors A and B.

The health care consultant may choose to conduct another study, but this second study would not require a two-way ANOVA test. In this second study, the consultant is interested in evaluating the impact of one factor on physician revenue. This one factor could be office location. The levels of this factor could be the Main Street office, the Oak Street office, the Elm Street office, and the Church Street office. The consultant would use the one-way ANOVA test for this study.

Levine, D. M., Stephan, D. F., & Szabat, K. A. (2021). Statistics for managers using Microsoft Excel (9th ed.). Pearson.

Response 3.) Bryana 

Chi-square test:

When we presented the binomial random variable, we made inferences about the binomial parameter p using large sample methods based on the z statistic. In this chapter, we extend this idea to make inferences about the multinomial parameters p1   p2 ………….pk, using a different type of statistic. This statistic, who approximate sampling distribution was derived by a British statistician named Karl Pearson in 1900, is called the chi-square statistic.

The expected frequencies can be calculated using the formula.

Ei = np1 for any of the cells I =1,2,….k.

The Pearson ch-square statistic uses the differences (Oi-Ei) by first squaring these differences to eliminate negative contributions, and then forming a weighted average of the squared differences. Pearson’s CHI – Square Test statistic:

X2 = E (Oi -Ei)2/Ei

summed over all k cells, with Ei = np1

The pearson’s chi-square test are always upper tailed test.

Response 4.) Mary 

Under what conditions should you use the chi-squared test to determine whether there is a difference among the proportions of more than two independent populations? Provide two examples.

A t-test for difference between population means in two independent variables.

X 2 t-test for relationship between two qualitative variables.

Z test for difference between population proportion in two independent samples

Reference

Levine, D. M., Szabat, K. A., & Stephan, D. (2021).  Statistics for managers using Microsoft Excel.  Pearson.  ISBN:-13: 9780135969854