Question

out of a sample of 760 people, 367 own their homes. construct a 95% confidence interval for the population mean of people in the world that own their homes.
o ci = (43.62%, 52.96%)
o ci = (46.87%, 52.56%)
o ci = (44.74%, 51.84%)
o ci = (45.31%, 51.27%)

Explanation:

Answer:

To construct a 95% confidence interval for a proportion, we use the formula $\hat{p}\pm z\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$, where $\hat{p}$ is the sample - proportion, $z$ is the z - value corresponding to the desired confidence level, and $n$ is the sample size.

First, calculate the sample proportion $\hat{p}$:
$\hat{p}=\frac{x}{n}$, where $x = 367$ and $n = 760$.

Step1: Calculate the sample proportion $\hat{p}$

$\hat{p}=\frac{367}{760}\approx0.4829$

For a 95% confidence interval, the z - value $z = 1.96$.

Step2: Calculate the margin of error $E$

$E=z\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}=1.96\sqrt{\frac{0.4829\times(1 - 0.4829)}{760}}$
$=1.96\sqrt{\frac{0.4829\times0.5171}{760}}$
$=1.96\sqrt{\frac{0.2494}{760}}$
$=1.96\sqrt{0.00032816}$
$=1.96\times0.0181$
$=0.0355$

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

The lower bound is $\hat{p}-E=0.4829 - 0.0355=0.4474$ or 44.74%
The upper bound is $\hat{p}+E=0.4829+0.0355 = 0.5184$ or 51.84%

So the 95% confidence interval is $(44.74\%,51.84\%)$. The answer is CI=(44.74%, 51.84%)