91Ó°ÊÓ

What happens to the standard deviation of \(\hat{p}\) as the sample size increases? If the sample size is increased by a factor of 4 what happens to the standard deviation of \(\hat{p} ?\)

Determine \(\mu_{\bar{x}}\) and \(\sigma_{\bar{x}}\) from the given parameters of the population and the sample size. \(\mu=80, \sigma=14, n=49\)

Determine \(\mu_{\bar{x}}\) and \(\sigma_{\bar{x}}\) from the given parameters of the population and the sample size. \(\mu=52, \sigma=10, n=21\)

Determine \(\mu_{\bar{x}}\) and \(\sigma_{\bar{x}}\) from the given parameters of the population and the sample size. \(\mu=27, \sigma=6, n=15\)

Foreign Language According to a study done by Wakefield Research, the proportion of Americans who can order a meal in a foreign language is \(0.47 .\) (a) Suppose a random sample of 200 Americans is asked to disclose whether they can order a meal in a foreign language. Is the response to this question qualitative or quantitative? Explain. (b) Explain why the sample proportion, \(\hat{p},\) is a random variable. What is the source of the variability? (c) Describe the sampling distribution of \(\hat{p},\) the proportion of Americans who can order a meal in a foreign language. Be sure to verify the model requirements. (d) In the sample obtained in part (a), what is the probability the proportion of Americans who can order a meal in a foreign language is greater than \(0.5 ?\) (e) Would it be unusual that, in a survey of 200 Americans, 80 or fewer Americans can order a meal in a foreign language? Why?

Are You Satisfied? According to a study done by the Gallup organization, the proportion of Americans who are satisfied with the way things are going in their lives is 0.82 . (a) Suppose a random sample of 100 Americans is asked, "Are you satisfied with the way things are going in your life?" Is the response to this question qualitative or quantitative? Explain. (b) Explain why the sample proportion, \(\hat{p}\), is a random variable. What is the source of the variability? (c) Describe the sampling distribution of \(\hat{p},\) the proportion of Americans who are satisfied with the way things are going in their life. Be sure to verify the model requirements. (d) In the sample obtained in part (a), what is the probability the proportion who are satisfied with the way things are going in their life exceeds \(0.85 ?\) (e) Would it be unusual for a survey of 100 Americans to reveal that 75 or fewer are satisfied with the way things are going in their life? Why?

A simple random sample of size \(n=36\) is obtained from a population with \(\mu=64\) and \(\sigma=18\). (a) Describe the sampling distribution of \(\bar{x}\). (b) What is \(P(\bar{x}<62.6) ?\) (c) What is \(P(\bar{x} \geq 68.7) ?\) (d) What is \(P(59.8<\bar{x}<65.9) ?\)

Marriage Obsolete? According to a study done by the Pew Research Center, \(39 \%\) of adult Americans believe that marriage is now obsolete. (a) Suppose a random sample of 500 adult Americans is asked whether marriage is obsolete. Describe the sampling distribution of \(\hat{p}\), the proportion of adult Americans who believe marriage is obsolete. (b) What is the probability that in a random sample of 500 adult Americans less than \(38 \%\) believe that marriage is obsolete? (c) What is the probability that in a random sample of 500 adult Americans between \(40 \%\) and \(45 \%\) believe that marriage is obsolete? (d) Would it be unusual for a random sample of 500 adult Americans to result in 210 or more who believe marriage is obsolete?

Credit Cards According to creditcard.com, \(29 \%\) of adults do not own a credit card. (a) Suppose a random sample of 500 adults is asked, "Do you own a credit card?" Describe the sampling distribution of \(\hat{p}\), the proportion of adults who do not own a credit card. (b) What is the probability that in a random sample of 500 adults more than \(30 \%\) do not own a credit card? (c) What is the probability that in a random sample of 500 adults between \(25 \%\) and \(30 \%\) do not own a credit card? (d) Would it be unusual for a random sample of 500 adults to result in 125 or fewer who do not own a credit card? Why?

The length of human pregnancies is approximately normally distributed with mean \(\mu=266\) days and standard deviation \(\sigma=16\) days. (a) What is the probability a randomly selected pregnancy lasts less than 260 days? (b) Suppose a random sample of 20 pregnancies is obtained. Describe the sampling distribution of the sample mean length of human pregnancies. (c) What is the probability that a random sample of 20 pregnancies has a mean gestation period of 260 days or less? (d) What is the probability that a random sample of 50 pregnancies has a mean gestation period of 260 days or less? (e) What might you conclude if a random sample of 50 pregnancies resulted in a mean gestation period of 260 days or less? (f) What is the probability a random sample of size 15 will have a mean gestation period within 10 days of the mean?