Question

according to a report published by the pew research center in 2010, 61% of millennials think that their generation has a unique and disincentive identity (n = 527). calculate the 95% confidence interval to estimate the percentage of millennials who believe that their generation has a distinctive identity as compared with the other generations (generation x, baby boomers, etc.) this 95% confidence interval is 56.83 to 65.17

Explanation:

Step1: Calculate the standard error

The formula for the standard error \(SE\) of a proportion is \(SE=\sqrt{\frac{p(1 - p)}{n}}\), where \(p = 0.61\) (the sample proportion) and \(n=527\) (the sample size).

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Step2: Find the critical value

For a 95% confidence interval, the critical value \(z\) (from the standard normal distribution) is \(z = 1.96\).

Step3: Calculate the margin of error

The margin of error \(ME=z\times SE\).

$$ ME = 1.96\times0.0212\approx0.0416 $$

Step4: Calculate the confidence interval

The lower bound \(=p - ME\) and the upper bound \(=p+ME\).

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Answer:

The 95% confidence interval is from \(56.84\%\) to \(65.16\%\)