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A sociologist is studying religious identity among college students. She samples

1300 new college graduates who consider themselves to have had at least a

"moderate" level of faith in a religion when entering college. Of these 1300

students, 612 grew up in homes where reading fiction was discouraged, and 688

grew up in homes where it was met with a neutral or positive view. Of those who

were discouraged from reading fiction, 188 still maintained at least a moderate

level of faith in their initial religion by their senior year. Of those who were not

discouraged, 272 still maintained at least a moderate level of faith in their initial

religion by their senior year. She plans to test at 1% significance if discouragement

from reading fiction results in lower percentages of religious freshman retaining

their faith by the time they graduate.

At 1% significance, can she conclude that religious freshman who had been

discouraged from reading fiction remain faithful at lower rates?

Yes

No

Verified Answer :

To determine whether the sociologist can conclude that religious freshmen who had been discouraged from reading fiction remain faithful at lower rates, we can perform a hypothesis test for the difference in proportions.

Let's define the following:

\( p_1 \) = Proportion of students discouraged from reading fiction who maintained at least a moderate level of faith in their initial religion by senior year.

\( p_2 \) = Proportion of students not discouraged from reading fiction who maintained at least a moderate level of faith in their initial religion by senior year.

We want to test the null hypothesis (\( H_0 \)) that there is no difference in the proportions of students maintaining faith regardless of whether they were discouraged from reading fiction. The alternative hypothesis (\( H_1 \)) is that students discouraged from reading fiction have a lower proportion of maintaining faith compared to those not discouraged.

\[ H_0: p_1 = p_2 \]

\[ H_1: p_1 < p_2 \]

We'll conduct a one-tailed test because we want to determine if the proportion of students discouraged from reading fiction who maintained faith is significantly lower than those not discouraged.

We can use the Z-test for proportions to test this hypothesis. The test statistic is given by:

\[ Z = \frac{(p_1 - p_2) - 0}{\sqrt{p(1-p)\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}} \]

Where:

- \( p = \frac{n_1p_1 + n_2p_2}{n_1 + n_2} \) is the pooled sample proportion.

- \( n_1 \) and \( n_2 \) are the sample sizes.

- \( p_1 \) and \( p_2 \) are the sample proportions.

Let's calculate the test statistic and determine if it falls in the critical region for a one-tailed test at 1% significance level. We'll reject the null hypothesis if the test statistic falls in the critical region. If it doesn't, we fail to reject the null hypothesis.

First, let's calculate the sample proportions \( p_1 \) and \( p_2 \):

\[ p_1 = \frac{188}{612} \approx 0.3072 \]

\[ p_2 = \frac{272}{688} \approx 0.3953 \]

Next, let's calculate the pooled sample proportion \( p \):

\[ p = \frac{612 \times 0.3072 + 688 \times 0.3953}{1300} \approx 0.3520 \]

Now, let's calculate the standard error:

\[ \text{Standard Error} = \sqrt{p(1-p)\left(\frac{1}{n_1} + \frac{1}{n_2}\right)} \]

\[ \text{Standard Error} = \sqrt{0.3520(1-0.3520)\left(\frac{1}{612} + \frac{1}{688}\right)} \]

\[ \text{Standard Error} \approx \sqrt{0.3520(0.6480)\left(0.0016 + 0.0015\right)} \]

\[ \text{Standard Error} \approx \sqrt{0.3520(0.6480) \times 0.0031} \]

\[ \text{Standard Error} \approx \sqrt{0.3520(0.0020)} \]

\[ \text{Standard Error} \approx \sqrt{0.000704} \]

\[ \text{Standard Error} \approx 0.0265 \]

Now, let's calculate the test statistic \( Z \):

\[ Z = \frac{(0.3072 - 0.3953) - 0}{0.0265} \]

\[ Z \approx \frac{-0.0881}{0.0265} \]

\[ Z \approx -3.32 \]

Now, we need to find the critical value for a one-tailed test at 1% significance level. Since this is a lower-tailed test, we're interested in finding the critical value that corresponds to 1% in the lower tail of the standard normal distribution.

Using a standard normal distribution table or a calculator, we find that the critical value for a one-tailed test at 1% significance level is approximately -2.33.

Since our calculated test statistic \( Z = -3.32 \) is less than the critical value -2.33, it falls in the critical region. Therefore, we reject the null hypothesis \( H_0 \) in favor of the alternative hypothesis \( H_1 \).

So, at 1% significance level, the sociologist can conclude that religious freshmen who had been discouraged from reading fiction remain faithful at lower rates.

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