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 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 *round to two decimals

Explanation:

Step1: Identify the formula

The formula for the confidence interval for a proportion is $\hat{p}\pm z\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$, where $\hat{p}$ is the sample proportion, $z$ is the z - score, and $n$ is the sample size.

Step2: Determine the values

Given $\hat{p}=0.61$, $n = 527$. For a 99% confidence interval, the z - score $z = 2.576$.

Step3: Calculate the standard error

First, calculate the standard error $SE=\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}=\sqrt{\frac{0.61\times(1 - 0.61)}{527}}=\sqrt{\frac{0.61\times0.39}{527}}\approx\sqrt{\frac{0.2379}{527}}\approx\sqrt{0.0004514}\approx0.0213$.

Step4: Calculate the margin of error

The margin of error $ME=z\times SE=2.576\times0.0213\approx0.055$.

Step5: Calculate the confidence interval

The lower limit is $\hat{p}-ME=0.61 - 0.055 = 0.555$. The upper limit is $\hat{p}+ME=0.61+0.055 = 0.665$.

Answer:

The 99% confidence interval is $(0.56, 0.67)$ (rounded to two decimals).