Question

the figure to the right shows the results of a survey in which 1010 adults from country a, 1012 adults from country b, 1002 adults from country c, 996 adults from country d, and

national identity and birthplace
people from different countries who believe national identity is strongly tied to birthplace

the 95% confidence interval for the proportion of adults from country a who say national identity is strongly tied to birthplace is (0.291, 0.349)
(round to three decimal places as needed.)
the 95% confidence interval for the proportion of adults from country b who say national identity is strongly tied to birthplace is (0.175, 0.225)
(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: Find the sample proportion \( \hat{p} \) and sample size \( n \)

For Country D, \( \hat{p}=0.52 \), \( n = 996 \).

Step2: Calculate the standard error \( SE \)

The formula for the standard error of a proportion is \( SE=\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \).
Substitute \( \hat{p}=0.52 \) and \( n = 996 \):
\( SE=\sqrt{\frac{0.52\times(1 - 0.52)}{996}}=\sqrt{\frac{0.52\times0.48}{996}}\approx\sqrt{\frac{0.2496}{996}}\approx\sqrt{0.0002506}\approx0.0158 \)

Step3: Find the margin of error \( ME \)

For a 95% confidence interval, the critical value \( z = 1.96 \).
The margin of error \( ME=z\times SE \).
Substitute \( z = 1.96 \) and \( SE\approx0.0158 \):
\( ME=1.96\times0.0158\approx0.031 \)

Step4: Calculate the confidence interval

The confidence interval is \( \hat{p}-MESubstitute \( \hat{p}=0.52 \) and \( ME = 0.031 \):
\( 0.52-0.031 = 0.489 \), \( 0.52 + 0.031=0.551 \)

Answer:

\( (0.489,0.551) \)