To determine whether college major depends on being an athlete or not, we perform a chi-square Test of Independence at the 5% significance level, given the data in the contingency table and the provided chi-square values. Here are the steps and conclusions:
### Step 1: State the hypotheses:
- Null Hypothesis ([tex]$H_0$[/tex]): The two variables (college major and being an athlete) are independent. That means being an athlete does not affect the choice of college major.
- Alternative Hypothesis ([tex]$H_a$[/tex]): The two variables (college major and being an athlete) are dependent. That means being an athlete does affect the choice of college major.
### Step 2: Calculate the chi-square statistic
- Given in the problem: [tex]\(\chi_0^2 = 3.5\)[/tex]
### Step 3: Determine the critical value
- The critical value for the chi-square distribution with the appropriate degrees of freedom at a significance level of 0.05 is given as:
[tex]\[
\chi^2_{0.05} = 5.991
\][/tex]
### Step 4: Compare the test statistic to the critical value
- Compare the observed chi-square value ([tex]\(\chi_0^2 = 3.5\)[/tex]) with the critical value ([tex]\(5.991\)[/tex]).
### Step 5: Make the decision
- If the observed chi-square value is less than the critical value ([tex]\( \chi_0^2 < \chi^2_{0.05} \)[/tex]), we fail to reject the null hypothesis.
- If the observed chi-square value is greater than the critical value ([tex]\( \chi_0^2 > \chi^2_{0.05} \)[/tex]), we reject the null hypothesis.
In our case:
[tex]\[
3.5 < 5.991
\][/tex]
Therefore, we fail to reject the null hypothesis.
### Step 6: State the conclusion
- Since we fail to reject the null hypothesis, we conclude that there is not enough evidence to suggest that there is a dependency between being an athlete and college major. In other words, we cannot conclude that being an athlete has an effect on the choice of college major.
### Final Conclusion
The conclusions that can be made based on the chi-square Test of Independence result are:
- Fail to reject the null hypothesis. There is not enough evidence to suggest that the variables are dependent.
Thus, we conclude that there is not enough evidence to suggest that college major depends on whether a student is an athlete or not.
