Question

suppose a population proportion is 0.7 and a sample of size 300 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 ±0.03 of the population proportion. b. calculate the probability that the sample proportion will fall within ±0.05 of the population proportion. click here to expand the distribution tables. attempt 1: 2 attempts remaining. submit answer next item

Explanation:

Step1: Calculate the standard deviation of the sample proportion

The formula for the standard deviation of the sample proportion $\sigma_{\hat{p}}$ is $\sqrt{\frac{p(1 - p)}{n}}$, where $p = 0.7$ and $n=300$. So, $\sigma_{\hat{p}}=\sqrt{\frac{0.7\times(1 - 0.7)}{300}}=\sqrt{\frac{0.7\times0.3}{300}}=\sqrt{\frac{0.21}{300}}\approx\sqrt{0.0007}\approx0.0265$.

Step2: Calculate the z - scores for part a

For the interval $\hat{p}$ within $\pm0.03$ of the population proportion $p = 0.7$. The lower limit $\hat{p}_1=0.7 - 0.03=0.67$ and the upper limit $\hat{p}_2=0.7 + 0.03 = 0.73$. The z - score is calculated as $z=\frac{\hat{p}-p}{\sigma_{\hat{p}}}$. For $\hat{p}_1$, $z_1=\frac{0.67 - 0.7}{0.0265}=\frac{- 0.03}{0.0265}\approx - 1.13$. For $\hat{p}_2$, $z_2=\frac{0.73 - 0.7}{0.0265}=\frac{0.03}{0.0265}\approx1.13$. Using the standard normal distribution table, $P(-1.13

Step3: Calculate the z - scores for part b

For the interval $\hat{p}$ within $\pm0.05$ of the population proportion $p = 0.7$. The lower limit $\hat{p}_3=0.7 - 0.05 = 0.65$ and the upper limit $\hat{p}_4=0.7+0.05 = 0.75$. For $\hat{p}_3$, $z_3=\frac{0.65 - 0.7}{0.0265}=\frac{-0.05}{0.0265}\approx - 1.89$. For $\hat{p}_4$, $z_4=\frac{0.75 - 0.7}{0.0265}=\frac{0.05}{0.0265}\approx1.89$. Using the standard normal distribution table, $P(-1.89

Answer:

a. $0.7416$
b. $0.9412$