Question

the figure to the right shows the results of a survey in which 1008 adults from country a, 1018 adults from country b, 1007 adults from country c, 1019 adults from country d, and 1003 adults from country e were asked whether national identity is strongly tied to birthplace.
construct a 95% confidence interval for the population proportion of adults who say national identity is strongly tied to birthplace for each country listed.
(0.166, 0.214)
(round to three decimal places as needed.)
the 95% confidence interval for the proportion of adults from country c who say national identity is strongly tied to birthplace is (0.262, 0.318)
(round to three decimal places as needed.)
the 95% confidence interval for the proportion of adults from country d who say national identity is strongly tied to birthplace is (□□)
(round to three decimal places as needed.)

Explanation:

Step1: Recall the formula for confidence interval of proportion

The formula for a \(95\%\) confidence interval for a proportion \(p\) is \(\hat{p}\pm z\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\), where for a \(95\%\) confidence interval \(z = 1.96\), \(\hat{p}\) is the sample proportion, and \(n\) is the sample size. For Country \(D\), \(\hat{p}=0.47\) and \(n = 1019\).

Step2: Calculate the margin of error \(E\)

First, calculate \(\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}=\sqrt{\frac{0.47\times(1 - 0.47)}{1019}}=\sqrt{\frac{0.47\times0.53}{1019}}\approx\sqrt{\frac{0.2491}{1019}}\approx\sqrt{0.0002444}\approx0.0156\). Then \(E=z\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}=1.96\times0.0156\approx0.0306\).

Step3: Calculate the lower and upper bounds of the confidence interval

The lower bound is \(\hat{p}-E=0.47 - 0.0306=0.439\). The upper bound is \(\hat{p}+E=0.47+ 0.0306=0.501\).

Answer:

\((0.439,0.501)\)