Questions: One hundred volunteers were divided into two equal-sized groups. Each volunteer took a math test that involved transforming strings of eight digits into a new string that fit a set of given rules, as well as a third, hidden rule. Prior to taking the test, one group received 8 hours of sleep, while the other group stayed awake all night. The scientists monitored the volunteers to determine whether and when they figured out the rule. Of the volunteers who slept, 38 discovered the rule; of the volunteers who stayed awake, 15 discovered the rule. What can you infer about the proportions of volunteers in the two groups who discover the rule? Support your answer with a 90% confidence interval. Let p̂1 be the proportion of volunteers who figured out the third rule in the group that slept and let p̂2 be the proportion of volunteers who figured out the third rule in the group that stayed awake all night. The 90% confidence interval for (p1-p2) is , (Round to the nearest thousandth as needed.)

Solution Steps

Let \( \hat{p}_1 \) be the proportion of volunteers who discovered the rule in the group that slept, and \( \hat{p}_2 \) be the proportion of volunteers who discovered the rule in the group that stayed awake. The sample proportions are calculated as follows:

\[ \hat{p}_1 = \frac{38}{50} = 0.76 \] \[ \hat{p}_2 = \frac{15}{50} = 0.3 \]

To find the 90% confidence interval for the difference in proportions \( (\hat{p}_1 - \hat{p}_2) \), we use the formula:

\[ (\hat{p}_1 - \hat{p}_2) \pm z \sqrt{\frac{\hat{p}_1(1 - \hat{p}_1)}{n_1} + \frac{\hat{p}_2(1 - \hat{p}_2)}{n_2}} \]

Where \( z \) is the z-score corresponding to a 90% confidence level, which is approximately \( 1.645 \).

First, we calculate the standard error:

\[ \text{SE} = \sqrt{\frac{0.76(1 - 0.76)}{50} + \frac{0.3(1 - 0.3)}{50}} = \sqrt{\frac{0.76 \cdot 0.24}{50} + \frac{0.3 \cdot 0.7}{50}} \]

Calculating each term:

\[ \frac{0.76 \cdot 0.24}{50} = 0.003648 \] \[ \frac{0.3 \cdot 0.7}{50} = 0.0042 \]

Thus,

\[ \text{SE} = \sqrt{0.003648 + 0.0042} = \sqrt{0.007848} \approx 0.0885 \]

Now we can calculate the confidence interval:

\[ 0.76 - 0.3 \pm 1.645 \cdot 0.0885 \]

Calculating the margin of error:

\[ 1.645 \cdot 0.0885 \approx 0.145 \]

Thus, the confidence interval is:

\[ (0.76 - 0.3 - 0.145, 0.76 - 0.3 + 0.145) = (0.314, 0.606) \]

Final Answer