Question

consider two independent binomial experiments. in the first one, 75 trials had 24 successes. in the second one, 60 trials had 30 successes.
answer the following questions. use a confidence level of 83%.
use 4 decimal places for each answer. do not round from one part to the next when performing the calculations, though.
a. find the confidence interval.
( < p_{1}-p_{2} < )
b. which of the following does your interval contain?
both positive and negative values
only positive values
only negative values
c. what does this mean in terms of the population proportions?
we are 83% confident that the first population proportion is greater than the second population proportion.
we are 83% confident that the first population proportion might be equal to the second population proportion.
we are 83% confident that the first population proportion is less than the second population proportion.

Explanation:

Step1: Calculate sample proportions

$\hat{p}_1 = \frac{24}{75} = 0.32$, $\hat{p}_2 = \frac{30}{60} = 0.5$

Step2: Find z-score for 83% confidence

Confidence level = 83% → $\alpha = 0.17$, $\alpha/2 = 0.085$. $z_{0.085} \approx 1.37$ (from standard normal table)

Step3: Compute standard error

$SE = \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}} = \sqrt{\frac{0.32×0.68}{75} + \frac{0.5×0.5}{60}} \approx \sqrt{0.002869 + 0.004167} \approx 0.0839$

Step4: Calculate margin of error

$ME = z×SE = 1.37×0.0839 \approx 0.1149$

Step5: Compute confidence interval

$\hat{p}_1 - \hat{p}_2 = 0.32 - 0.5 = -0.18$. Interval: $-0.18 - 0.1149 = -0.2949$ to $-0.18 + 0.1149 = -0.0651$

Answer:

a. -0.2949 < p₁ - p₂ < -0.0651
b. only negative values
c. We are 83% confident that the first population proportion is less than the second population proportion.