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Does a college major depend on whether you are an athlete or not? A survey asks a group of college students about their majors and groups them by whether they are athletes or not. The data are recorded in the contingency table below, and a chi-square Test of Independence at the [tex]$5 \%$[/tex] significance level is performed.

\begin{tabular}{|c|c|c|c|c|}
\hline & Arts & Humanities & Sciences & Row Total \\
\hline Athlete & 17 & 24 & 15 & 56 \\
\hline Non-Athlete & 19 & 11 & 9 & 39 \\
\hline Column Total & 36 & 35 & 24 & 95 \\
\hline
\end{tabular}

(a) Select the correct null and alternative hypotheses for this test.

Select all that apply:

A. [tex]$H_a$[/tex] : The two variables are dependent, so being an athlete does affect college major.
B. [tex]$H_0$[/tex] : The two variables are independent, so being an athlete does not affect college major.
C. [tex]$H_0$[/tex] : The two variables are dependent, so being an athlete does affect college major.
D. [tex]$H_a$[/tex] : The two variables are independent, so being an athlete does not affect college major.



Answer :

GinnyAnswer
To determine whether college major depends on whether a student is an athlete or not, we perform a chi-square Test of Independence. Before we proceed with the test, we need to establish our hypotheses.

In hypothesis testing, the null and alternative hypotheses play crucial roles:

1. Null Hypothesis ([tex]\( H_0 \)[/tex]):
The null hypothesis states that there is no association between the two categorical variables in question. In this context, the null hypothesis would state that being an athlete does not affect a student's college major. In other words, the variables "being an athlete" and "college major" are independent.

2. Alternative Hypothesis ([tex]\( H_a \)[/tex]):
The alternative hypothesis suggests that there is an association between the two variables. In this case, it would mean that being an athlete does affect a student's college major. Therefore, the variables "being an athlete" and "college major" are dependent.

Given the options:
1. [tex]\( H_a \)[/tex]: The two variables are dependent, so being an athlete does affect college major.
2. [tex]\( H_0 \)[/tex]: The two variables are independent, so being an athlete does not affect college major.
3. [tex]\( H_0 \)[/tex]: The two variables are dependent, so being an athlete does affect college major.
4. [tex]\( H_a \)[/tex]: The two variables are independent, so being an athlete does not affect college major.

The correct hypotheses align with the descriptions provided:

- The correct null hypothesis ([tex]\( H_0 \)[/tex]) states that "The two variables are independent, so being an athlete does not affect college major."
- The correct alternative hypothesis ([tex]\( H_a \)[/tex]) states that "The two variables are dependent, so being an athlete does affect college major."

Thus, the correct selections are:

Null Hypothesis ([tex]\( H_0 \)[/tex]):
[tex]\[ H_0: \text{The two variables are independent, so being an athlete does not affect college major.} \][/tex]

Alternative Hypothesis ([tex]\( H_a \)[/tex]):
[tex]\[ H_a: \text{The two variables are dependent, so being an athlete does affect college major.} \][/tex]

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