A survey of 2307 adults in a certain large country aged 18 and older conducted by a reputable polling organization found that 403 have donated blood in the past two years. Complete parts (a) through (c) below. donated blood in the past two years. \[ \hat{\mathrm{p}}=0.175 \] (Round to three decimal places as needed.) (b) Verify that the requirements for constructing a confidence interval about $p$ are satisfied. The sample can be assumed to be a simple random sample, the value of $\hat{n p}(1-\hat{p})$ is 332.60 , which is greater than or equal to 10 , and the sample size can be assumed to be less than or equal to $5 \%$ of the population size. (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. We are $\square \%$ confident the proportion of adults in the country aged 18 and older who have donated blood in the past two years is between $\square$ and $\square$. B. There is a $\square$ chance the proportion of adults in the country aged 18 and older who have donated blood in the past two years is between $\square$ and $\square$.
Solution
Step 1 ：A survey of 2307 adults in a certain large country aged 18 and older conducted by a reputable polling organization found that 403 have donated blood in the past two years. The sample proportion of adults who have donated blood in the past two years is given as \( \hat{p} = 0.175 \).
Step 2 ：We need to verify that the requirements for constructing a confidence interval about \( p \) are satisfied. The sample can be assumed to be a simple random sample, the value of \( \hat{n p}(1-\hat{p}) \) is 332.60 , which is greater than or equal to 10 , and the sample size can be assumed to be less than or equal to 5% of the population size.
Step 3 ：We are asked to construct a 90% confidence interval for the population proportion of adults in the country who have donated blood in the past two years. We can use the formula for the confidence interval for a proportion, which is \( \hat{p} \pm Z \sqrt{(\hat{p}(1-\hat{p}))/n} \), where \( \hat{p} \) is the sample proportion, \( n \) is the sample size, and \( Z \) is the Z-score for the desired confidence level. For a 90% confidence level, the Z-score is approximately 1.645.
Step 4 ：Substituting the given values into the formula, we get the lower and upper bounds of the confidence interval as 0.162 and 0.188 respectively.
Step 5 ：\(\boxed{\text{Final Answer: We are 90% confident the proportion of adults in the country aged 18 and older who have donated blood in the past two years is between 0.162 and 0.188.}}\)
