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Formulate the hypothesis and test if gender and education level are dependent at [tex][tex]$5\%$[/tex][/tex] level of statistical significance.

A random sample of 395 people in a country were surveyed to find out if gender is independent of education level. Each person was asked to report the highest education level they had obtained. The data from the survey is summarized in the following table:

\begin{tabular}{|l|r|r|r|r|r|}
\hline & High School & Bachelors & Masters & Ph.D. & Total \\
\hline Female & 60 & 54 & 46 & 41 & 201 \\
\hline Male & 40 & 44 & 53 & 57 & 194 \\
\hline Total & 100 & 98 & 99 & 98 & 395 \\
\hline
\end{tabular}

Formulate the hypothesis and test if gender and education level are dependent at [tex]5\%[/tex] level of statistical significance.

Sagot :

GinnyAnswer
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.
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