Question

the homeownership rates in a certain country have been declining. the countrys homeownership rate was 48.5% in 2017 (compared to 65.2% in another country). suppose, from a random sample of 209 households in the country in 2018, 115 were occupied by the owners of the residence. complete parts a through c

a. construct a 90% confidence interval to estimate the actual proportion of households in the country that are occupied by their owners in 2018.

this confidence interval has a lower limit of 0.357 and an upper limit of 0.466

(round to three decimal places as needed.)

b. what is the margin of error for this sample?

the margin of error is 0.071

(round to three decimal places as needed.)

c. is there evidence that this proportion has changed since 2017 based on this sample?

this sample evidence that this proportion has changed since 2017, since the

Explanation:

Step1: Calculate sample proportion

The sample proportion \( \hat{p}=\frac{x}{n} \), where \( x = 115 \) and \( n=209 \). So \( \hat{p}=\frac{115}{209}\approx0.550 \)

Step2: Find critical value

For a \( 90\% \) confidence interval, the significance level \( \alpha=1 - 0.90=0.10 \), and \( \alpha/2=0.05 \). The critical value \( z_{\alpha/2}=z_{0.05} = 1.645 \)

Step3: Calculate margin of error (already given as \( E = 0.071 \))

The formula for margin of error for proportion is \( E=z_{\alpha/2}\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \)

Step4: Check if proportion has changed

The 2017 proportion \( p_0=0.485 \). The confidence interval is \( (0.357,0.466) \) (lower limit \( L = 0.357 \), upper limit \( U=0.466 \))

Since \( 0.485\) is not in the interval \( (0.357,0.466) \)

Answer:

This sample provides evidence that this proportion has changed since 2017, since the 2017 proportion \( 0.485 \) is not within the \( 90\% \) confidence interval \( (0.357,0.466) \)