Question

suppose a population proportion is 0.7 and a sample of size of 400 will be collected. answer the following questions assuming that the sample proportion p̂ will be used to estimate the population proportion. (round your answers to four decimal places.) a. calculate the probability that the sample proportion will fall within ±0.02 of the population proportion. b. calculate the probability that the sample proportion will fall within ±0.04 of the population proportion.

Explanation:

Step1: Calculate the standard error

The formula for the standard error of a proportion is $\sigma_{\hat{p}}=\sqrt{\frac{p(1 - p)}{n}}$, where $p = 0.7$ and $n=400$. So, $\sigma_{\hat{p}}=\sqrt{\frac{0.7\times(1 - 0.7)}{400}}=\sqrt{\frac{0.7\times0.3}{400}}=\sqrt{\frac{0.21}{400}}=\sqrt{0.000525}\approx0.0229$.

Step2: Calculate z - scores for part a

The margin of error $E = 0.02$. The z - score is calculated using $z=\frac{E}{\sigma_{\hat{p}}}$. So, $z=\frac{0.02}{0.0229}\approx0.87$. We want $P(- 0.87

Step3: Calculate z - scores for part b

The margin of error $E = 0.04$. The z - score is $z=\frac{E}{\sigma_{\hat{p}}}=\frac{0.04}{0.0229}\approx1.75$. We want $P(-1.75

Answer:

a. 0.6156
b. 0.9198