Question

the homeownership rates in a certain country have been declining. the countrys homeownership rate was 48.6% in 2017 (compared to 85.2% in another country). suppose, from a random sample of 269 households in the country in 2018, 115 were occupied by the owners of the residence. complete parts a through c.
a. construct a 98% confidence interval to estimate the actual proportion of households in the country that are occupied by their owners in 2018.
the confidence interval has a lower limit of (square) and an upper limit of (square).
(round to three decimal places as needed.)

Explanation:

Step1: Calculate the sample proportion

The sample proportion $\hat{p}$ is calculated as $\hat{p}=\frac{x}{n}$, where $x = 115$ (number of successes) and $n=269$ (sample size).
$\hat{p}=\frac{115}{269}\approx0.4275$

Step2: Find the z - value

For a 98% confidence interval, the significance level $\alpha=1 - 0.98=0.02$. Then $\frac{\alpha}{2}=0.01$. The z - value $z_{\frac{\alpha}{2}}$ is the value such that $P(Z>z_{\frac{\alpha}{2}})=0.01$. From the standard normal table, $z_{\frac{\alpha}{2}} = 2.33$.

Step3: Calculate the margin of error

The margin of error $E$ for a proportion is given by $E = z_{\frac{\alpha}{2}}\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$.
Substitute $\hat{p}=0.4275$, $n = 269$, and $z_{\frac{\alpha}{2}}=2.33$ into the formula:

$$
LATEXBLOCK0
$$

Step4: Calculate the confidence interval

The confidence interval for the population proportion $p$ is $\hat{p}-ESubstitute $\hat{p}=0.4275$ and $E = 0.0704$ into the formula:
Lower limit: $0.4275-0.0704=0.357$
Upper limit: $0.4275 + 0.0704=0.498$

Answer:

The confidence interval has a lower limit of $0.357$ and an upper limit of $0.498$.