Use the two-proportion plus-four z-interval procedure to find the required confidence interval.
-A survey of randomly chosen adults found that 27 of the 86 women and 37 of the 87 men follow regular exercise programs. Construct a 95% confidence interval for the difference in the
Proportions of women and men who have regular exercise programs.
A)-0.277 to 0.486
B)0.150 to 0.486
C)-0.250 to 0.033
D)0.177 to 0.460

Question

Use the two-proportion plus-four z-interval procedure to find the required confidence interval.
-A survey of randomly chosen adults found that 27 of the 86 women and 37 of the 87 men follow regular exercise programs. Construct a 95% confidence interval for the difference in the
Proportions of women and men who have regular exercise programs. 
A)-0.277 to 0.486
B)0.150 to 0.486
C)-0.250 to 0.033
D)0.177 to 0.460

Quizplus · Accepted Answer

The two-proportion plus-four z-interval procedure involves adjusting the sample sizes and successes for both groups by adding two successes and two failures to each. This adjustment is used to calculate the confidence interval for the difference in proportions when sample sizes are small or when the observed proportions are close to 0 or 1. The formula for the confidence interval is:$$ \left( \hat{p}_1 - \hat{p}_2 \right) \pm z^* \sqrt{ \frac{\hat{p}_1(1 - \hat{p}_1)}{n_1} + \frac{\hat{p}_2(1 - \hat{p}_2)}{n_2} } $$where $\hat{p}_1$ and $\hat{p}_2$ are the adjusted proportions of successes in each group, $n_1$ and $n_2$ are the adjusted sample sizes, and $z^*$ is the critical value from the standard normal distribution for the desired level of confidence (approximately 1.96 for 95% confidence).Given:- Original counts for women: 27 successes out of 86 (exercise), adjusted to 29 out of 90.- Original counts for men: 37 successes out of 87 (exercise), adjusted to 39 out of 91.Adjusted proportions:- Women: $\hat{p}_1 = \frac{29}{90}$- Men: $\hat{p}_2 = \frac{39}{91}$The calculation involves substituting these values and the critical value into the formula. The correct interval, based on these calculations, is the one that includes the difference in proportions between women and men, accounting for the variability in those proportions. The correct answer reflects the interval that would result from these calculations, which is option C, indicating a confidence interval for the difference in proportions that includes both negative and positive values, suggesting that the true difference could favor either group or be negligible.

Use the Two-Proportion Plus-Four Z-Interval Procedure to Find the Required