A survey of 2323 adults in a certain large country aged 18 and older conducted by a reputable polling organization found that 423 have donated blood in the past two years. Complete parts (a) through (c) below.
Click here to view the standard normal distribution table (page 1).
Click here to view the standard normal distribution table (page 2).
(Round to three decimal places as needed.)
(c) Construct and interpret a $90 \%$ confidence interval for the population proportion of adults in the country who have donated blood in the past two years. Select the correct choice below and fill in any answer boxes within your choice.
(Type integers or decimals rounded to three decimal places as needed. Use ascending order.)
A. There is a $\%$ probability the proportion of adults in the country aged 18 and older who have donated blood in the past two years is between and
B. We are $\%$ confident the proportion of adults in the country aged 18 and older who have donated blood in the past two years is between and
Answer
\(\boxed{\text{We are 90% confident the proportion of adults in the country aged 18 and older who have donated blood in the past two years is between 0.169 and 0.195.}}\)
Step 1 ：Given that the sample size (n) is 2323 and the number of adults who have donated blood in the past two years (x) is 423, we can calculate the sample proportion (\(\hat{p}\)) as \(\frac{x}{n}\) = \(\frac{423}{2323}\) = 0.182.
Step 2 ：The z-score corresponding to a 90% confidence level is 1.645.
Step 3 ：We can calculate the standard error (se) using the formula \(se = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\) = \sqrt{\frac{0.182(1-0.182)}{2323}} = 0.008.
Step 4 ：Using the formula for a confidence interval for a proportion, we can calculate the lower and upper bounds of the confidence interval as \(\hat{p} \pm Z_{\alpha/2} \times se\).
Step 5 ：The lower bound of the confidence interval is \(\hat{p} - Z_{\alpha/2} \times se\) = 0.182 - 1.645 \times 0.008 = 0.169.
Step 6 ：The upper bound of the confidence interval is \(\hat{p} + Z_{\alpha/2} \times se\) = 0.182 + 1.645 \times 0.008 = 0.195.
Step 7 ：\(\boxed{\text{We are 90% confident the proportion of adults in the country aged 18 and older who have donated blood in the past two years is between 0.169 and 0.195.}}\)
