Question

according to a report published by the pew research center in 2010, 61% of millennials think that their generation has a unique and disinctive identity (n = 527).
calculate the 99% 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 99% confidence interval is to

Explanation:

Step1: Determine the formula

For a proportion confidence interval: $\hat{p}\pm z\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$, where $\hat{p}$ is the sample proportion, $z$ is the z - score, and $n$ is the sample size.
Given $\hat{p}=0.61$, $n = 527$, and for a $99\%$ confidence interval, $z = 2.576$.

Step2: Calculate the margin of error

First, calculate $\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}=\sqrt{\frac{0.61\times(1 - 0.61)}{527}}=\sqrt{\frac{0.61\times0.39}{527}}\approx\sqrt{\frac{0.2379}{527}}\approx\sqrt{0.0004514}\approx0.02125$.
Then, the margin of error $E=z\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}=2.576\times0.02125\approx0.0547$.

Step3: Calculate the confidence interval

The lower limit is $\hat{p}-E=0.61 - 0.0547=0.5553 = 55.53\%$.
The upper limit is $\hat{p}+E=0.61+0.0547 = 0.6647=66.47\%$.

Answer:

The $99\%$ confidence interval is $55.53\%$ to $66.47\%$.