Question

a countrys tax collection agency reported that 92% of individual tax returns were filed electronically in 2017. a random sample of 210 tax returns from 2018 was selected. from this sample, 181 were filed electronically. complete parts a through c.

a. construct a 90% confidence interval to estimate the actual proportion of taxpayers who filed electronically 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 sample proportion

The sample proportion $\hat{p}=\frac{x}{n}$, where $x = 181$ and $n=210$. So, $\hat{p}=\frac{181}{210}\approx0.862$.

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 $z_{\alpha/2}=z_{0.05}\approx1.645$ (from standard normal distribution table).

Step3: Calculate margin of error

The margin of error $E = z_{\alpha/2}\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$. Substitute $\hat{p}=0.862$, $n = 210$, and $z_{\alpha/2}=1.645$.

First, calculate $\hat{p}(1-\hat{p})=0.862\times(1 - 0.862)=0.862\times0.138 = 0.119$.

Then, $\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}=\sqrt{\frac{0.119}{210}}\approx\sqrt{0.000567}\approx0.024$.

$E=1.645\times0.024\approx0.040$.

Step4: Calculate confidence interval

The lower limit $= \hat{p}-E=0.862 - 0.040=0.822$.

The upper limit $=\hat{p}+E=0.862+0.040 = 0.902$.

Answer:

The confidence interval has a lower limit of $0.822$ and an upper limit of $0.902$.