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A researcher wishes to test if the variance of scores on a particular exam is greater for college students who work full time than for those who don't work at all. He samples the scores of 55 students who work full time (Group A) and 44 students who do not work (Group B). The results are as follows:

**Group A:**

[tex]
\[
\begin{array}{l}
87.6, 112, 95.1, 179.8, 186.7, 183.4, 66.6, 122.5, 160.6, 173.3, 130.5, 175.9, 119.7, 63.5, 137, \\
85.1, 139.2, 40.8, 166.4, 210.9, 70.6, 89.6, 192.8, 146.6, 38.5, 123.8, 170.8, 58.5, 205.7, 166, \\
149.8, 88.5, 102.2, 181.4, 148.6, 202.1, 199.4, 187.9, 97.3, 99.5, 128.9, 171.1, 182.4, 63.4, \\
162.4, 105.1, 56.9, 124.8, 52.5, 182.8, 119.6, 105.9, 117.1, 198.2, 225.1
\end{array}
\]
[/tex]

**Group B:**

[tex]
\[
\begin{array}{l}
232, 145, 142.7, 119.2, 135.9, 187.7, 95.8, 45.8, 137.8, 151.7, 77.1, 135.6, 138.3, 157.4, 83.2, \\
148.7, 123.1, 164.5, 153.8, 118.7, 181.9, 155.3, 141.8, 52.8, 85.4, 273.6, 199.7, 90.5, 181.8, \\
141.1, 37.5, 67.9, 132.6, 98.6, 140.2, 119.1, 80.9, 172.3, 94.7, 260.3, 158, 151, 196.4, 158.4
\end{array}
\]
[/tex]

Perform a hypothesis test using an 8% level of significance.

**Step 1:** State the null and alternative hypotheses.

[tex]
\[
\begin{array}{l}
H_0: \frac{\sigma_A^2}{\sigma_B^2} = 1 \\
H_a: \frac{\sigma_A^2}{\sigma_B^2} > 1
\end{array}
\]
[/tex]

(We will be performing a one-tailed F-test.)

Answer :

To solve this problem, we need to conduct a hypothesis test to determine if the variance of scores for college students who work full time (Group A) is greater than for those who do not work (Group B). We'll perform an F-test for equality of variances. Here's a step-by-step guide to the process:

### Step 1: State the Hypotheses
- Null Hypothesis (H0): The variances are equal. [tex]\( \frac{\sigma_A^2}{\sigma_B^2} = 1 \)[/tex]
- Alternative Hypothesis (Ha): The variance of Group A is greater than Group B. [tex]\( \frac{\sigma_A^2}{\sigma_B^2} > 1 \)[/tex]

### Step 2: Gather the Data
We have the scores for both groups:

- Group A: 55 scores
- Group B: 44 scores

### Step 3: Calculate the Sample Variances
Calculate the sample variances for both groups:
- Variance for Group A: 2495.05
- Variance for Group B: 2609.92

### Step 4: Calculate the F-Statistic
The F-statistic is calculated by dividing the variance of Group A by the variance of Group B:

[tex]\[ F = \frac{\text{Variance of Group A}}{\text{Variance of Group B}} = \frac{2495.05}{2609.92} \approx 0.956 \][/tex]

### Step 5: Determine the Critical Value
To find the critical value for the F-test, we need:
- Significance Level ([tex]\(\alpha\)[/tex]): 0.08
- Degrees of Freedom for Group A (dfA): 54 (since it’s 55 scores minus 1)
- Degrees of Freedom for Group B (dfB): 43 (since it’s 44 scores minus 1)

The critical F-value at 8% significance, for dfA = 54 and dfB = 43, is approximately 1.516.

### Step 6: Make a Decision
Compare the calculated F-statistic to the critical F-value:

- F-statistic: 0.956
- Critical F-value: 1.516

Since the F-statistic (0.956) is less than the critical F-value (1.516), we do not reject the null hypothesis.

### Conclusion
Based on this test, we do not have sufficient evidence to claim that the variance of scores for students who work full time is greater than for students who do not work.

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