Question

Construct a​ 95% confidence interval to estimate the population proportion using the data below.     xequals23 nequals95 Nequals500 The​ 95% confidence interval for the population proportion is (nothing comma nothing ). ​(Round to three decimal places as​ needed.)

Answer 1

Answer:

95% Confidence interval:  (0.1645,0.3197)

Step-by-step explanation:

We are given the following in the question:

Sample size, n = 95

N = 500

x = 23

Finite population correction =

$=\sqrt{\dfrac{N-n}{N-1}} = \sqrt{\dfrac{500-95}{500-1}} = 0.9009$

$\hat{p} = \dfrac{x}{n} = \dfrac{23}{95} = 0.2421$

95% Confidence interval:

$\hat{p}\pm z_{stat}\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}\times fpc$

$z_{critical}\text{ at}~\alpha_{0.05} = 1.96$

Putting the values, we get:

$0.2421\pm 1.96(\sqrt{\dfrac{0.2421(1-0.2421)}{95}})\times 0.9009 \\\\= 0.2421\pm 0.0776\\=(0.1645,0.3197)$