Answer :
Certainly! Let's go through the solution step-by-step.
### Step-by-Step Solution
#### Step 1: Formulate the Hypotheses
- Null Hypothesis ([tex]\(H_0\)[/tex]): Gender is independent of the education level.
- Alternative Hypothesis ([tex]\(H_a\)[/tex]): Gender is dependent on the education level.
#### Step 2: Observed Data
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline & \text{High School} & \text{Bachelors} & \text{Masters} & \text{Ph.D.} & \text{Total} \\ \hline \text{Female} & 60 & 54 & 46 & 41 & 201 \\ \hline \text{Male} & 40 & 44 & 53 & 57 & 194 \\ \hline \text{Total} & 100 & 98 & 99 & 98 & 395 \\ \hline \end{array} \][/tex]
#### Step 3: Calculate the Expected Frequencies
The expected frequency for each cell in the contingency table is calculated using the formula:
[tex]\[ E_{ij} = \frac{(\text{Row Total})_i \times (\text{Column Total})_j}{\text{Grand Total}} \][/tex]
Using this formula, we compute the expected frequencies as presented:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline & \text{High School} & \text{Bachelors} & \text{Masters} & \text{Ph.D.} \\ \hline \text{Female} & 50.89 & 49.87 & 50.38 & 49.87 \\ \hline \text{Male} & 49.11 & 48.13 & 48.62 & 48.13 \\ \hline \end{array} \][/tex]
#### Step 4: Calculating the Chi-Square Statistic
The Chi-Square statistic is calculated using the formula:
[tex]\[ \chi^2 = \sum \frac{(O_{ij} - E_{ij})^2}{E_{ij}} \][/tex]
Where [tex]\(O_{ij}\)[/tex] are the observed frequencies and [tex]\(E_{ij}\)[/tex] are the expected frequencies.
From the calculations, we get:
[tex]\[ \chi^2 = 8.006066246262538 \][/tex]
#### Step 5: Determine the Degrees of Freedom
The degrees of freedom (dof) for a contingency table is calculated using:
[tex]\[ \text{dof} = (r-1) \times (c-1) \][/tex]
Where [tex]\(r\)[/tex] is the number of rows and [tex]\(c\)[/tex] is the number of columns. Here,
[tex]\[ \text{dof} = (2-1) \times (4-1) = 1 \times 3 = 3 \][/tex]
#### Step 6: Determine the p-value and Conclusion
The p-value associated with the Chi-Square statistic is:
[tex]\[ p = 0.045886500891747214 \][/tex]
We compare the p-value with the significance level ([tex]\(\alpha = 0.05\)[/tex]).
Since [tex]\(p < 0.05\)[/tex], we reject the null hypothesis.
#### Conclusion
At a 5% level of statistical significance, we reject the null hypothesis. Therefore, we have sufficient evidence to conclude that gender is dependent on the education level.
### Step-by-Step Solution
#### Step 1: Formulate the Hypotheses
- Null Hypothesis ([tex]\(H_0\)[/tex]): Gender is independent of the education level.
- Alternative Hypothesis ([tex]\(H_a\)[/tex]): Gender is dependent on the education level.
#### Step 2: Observed Data
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline & \text{High School} & \text{Bachelors} & \text{Masters} & \text{Ph.D.} & \text{Total} \\ \hline \text{Female} & 60 & 54 & 46 & 41 & 201 \\ \hline \text{Male} & 40 & 44 & 53 & 57 & 194 \\ \hline \text{Total} & 100 & 98 & 99 & 98 & 395 \\ \hline \end{array} \][/tex]
#### Step 3: Calculate the Expected Frequencies
The expected frequency for each cell in the contingency table is calculated using the formula:
[tex]\[ E_{ij} = \frac{(\text{Row Total})_i \times (\text{Column Total})_j}{\text{Grand Total}} \][/tex]
Using this formula, we compute the expected frequencies as presented:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline & \text{High School} & \text{Bachelors} & \text{Masters} & \text{Ph.D.} \\ \hline \text{Female} & 50.89 & 49.87 & 50.38 & 49.87 \\ \hline \text{Male} & 49.11 & 48.13 & 48.62 & 48.13 \\ \hline \end{array} \][/tex]
#### Step 4: Calculating the Chi-Square Statistic
The Chi-Square statistic is calculated using the formula:
[tex]\[ \chi^2 = \sum \frac{(O_{ij} - E_{ij})^2}{E_{ij}} \][/tex]
Where [tex]\(O_{ij}\)[/tex] are the observed frequencies and [tex]\(E_{ij}\)[/tex] are the expected frequencies.
From the calculations, we get:
[tex]\[ \chi^2 = 8.006066246262538 \][/tex]
#### Step 5: Determine the Degrees of Freedom
The degrees of freedom (dof) for a contingency table is calculated using:
[tex]\[ \text{dof} = (r-1) \times (c-1) \][/tex]
Where [tex]\(r\)[/tex] is the number of rows and [tex]\(c\)[/tex] is the number of columns. Here,
[tex]\[ \text{dof} = (2-1) \times (4-1) = 1 \times 3 = 3 \][/tex]
#### Step 6: Determine the p-value and Conclusion
The p-value associated with the Chi-Square statistic is:
[tex]\[ p = 0.045886500891747214 \][/tex]
We compare the p-value with the significance level ([tex]\(\alpha = 0.05\)[/tex]).
Since [tex]\(p < 0.05\)[/tex], we reject the null hypothesis.
#### Conclusion
At a 5% level of statistical significance, we reject the null hypothesis. Therefore, we have sufficient evidence to conclude that gender is dependent on the education level.
