The proportion of adult women in the United States is approximately $51 \%$. A marketing survey telephones 490 people at random. Complete parts a through c below.
a) What proportion of the sample of 490 would you expect to be women?
$\square$ (Type an integer or a decimal Do not round)
b) What would the standard deviation of the sampling distribution be?
$\mathrm{SD}(\hat{\mathrm{p}})=\square$ (Round to three decimal places as needed)
c) How many women, on average, would you expect to find in a sample of that size?
women (Round to the nearest whole number as needed)
Answer
The average number of women expected in the sample would be the sample size multiplied by the proportion of women. Therefore, the average number of women expected in the sample is \(490 * 0.51 = 249.9\). Rounding this to the nearest whole number, we get \(\boxed{250}\).
Step 1 ：The proportion of adult women in the United States is approximately 51%. This proportion is expected to be the same in a sample of 490 people. Therefore, the expected proportion of women in the sample is \(\boxed{0.51}\).
Step 2 ：The standard deviation of the sampling distribution can be calculated using the formula for the standard deviation of a proportion, which is \(\sqrt{p(1-p)/n}\), where p is the proportion and n is the sample size. Substituting the given values, we get \(\sqrt{0.51(1-0.51)/490} = 0.02258317958127243\). Rounding this to three decimal places, we get \(\boxed{0.023}\).
Step 3 ：The average number of women expected in the sample would be the sample size multiplied by the proportion of women. Therefore, the average number of women expected in the sample is \(490 * 0.51 = 249.9\). Rounding this to the nearest whole number, we get \(\boxed{250}\).
