Question

a survey of 2293 adults in a certain large country aged 18 and older conducted by a reputable polling organization found that 404 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) donated blood in the past two years. \\( \hat { p } = 0.176 \\) (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 \\( n \hat { p } ( 1 - \hat { p } ) \\) is 332.820, 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 ) oa. 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 ob. 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

Explanation:

Step1: Calculate the confidence interval

The formula for the confidence interval for a proportion is \(\hat{p}\pm z_{\alpha/2}\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\)
Given \(\hat{p}=0.176\), \(n = 2293\), for a \(90\%\) confidence interval, \(\alpha=1 - 0.90=0.10\), \(\alpha/2=0.05\), and \(z_{\alpha/2}=z_{0.05}\approx1.645\)
First, calculate \(\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}=\sqrt{\frac{0.176\times(1 - 0.176)}{2293}}\approx\sqrt{\frac{0.176\times0.824}{2293}}\approx\sqrt{\frac{0.145}{2293}}\approx\sqrt{0.0000632}\approx0.00795\)
Then, \(z_{\alpha/2}\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}=1.645\times0.00795\approx0.0131\)
The lower bound is \(\hat{p}-z_{\alpha/2}\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}=0.176-0.0131 = 0.163\)
The upper bound is \(\hat{p}+z_{\alpha/2}\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}=0.176 + 0.0131=0.189\)

Step2: Interpret the confidence interval

A confidence interval gives a range of values within which we are confident the population parameter lies. A \(90\%\) confidence interval means that if we were to construct many such intervals from different samples, about \(90\%\) of them would contain the true population proportion.

Answer:

B. 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 \(0.163\) and \(0.189\)