91Ó°ÊÓ

True or False: The population proportion and sample proportion always have the same value.

Suppose a simple random sample of size \(n=40\) is obtained from a population with \(\mu=50\) and \(\sigma=4 .\) Does the population need to be normally distributed for the sampling distribution of \(\bar{x}\) to be approximately normally distributed? Why? What is the sampling distribution of \(\bar{x} ?\)

Suppose a simple random sample of size \(n=75\) is obtained from a population whose size is \(N=10,000\) and whose population proportion with a specified characteristic is \(p=0.8\) (a) Describe the sampling distribution of \(\hat{p}\) (b) What is the probability of obtaining \(x=63\) or more individuals with the characteristic? That is, what is \(P(\hat{p} \geq 0.84) ?\) (c) What is the probability of obtaining \(x=51\) or fewer individuals with the characteristic? That is, what is \(P(\hat{p} \leq 0.68) ?\)

Suppose a simple random sample of size \(n=200\) is obtained from a population whose size is \(N=25,000\) and whose population proportion with a specified characteristic is \(p=0.65\) (a) Describe the sampling distribution of \(\hat{p}\) (b) What is the probability of obtaining \(x=136\) or more individuals with the characteristic? That is, what is \(P(\hat{p} \geq 0.68) ?\) (c) What is the probability of obtaining \(x=118\) or fewer individuals with the characteristic? That is, what is \(P(\hat{p} \leq 0.59) ?\)

Suppose a simple random sample of size \(n=1460\) is obtained from a population whose size is \(N=1,500,000\) and whose population proportion with a specified characteristic is \(p=0.42\) (a) Describe the sampling distribution of \(\hat{p}\) (b) What is the probability of obtaining \(x=657\) or more individuals with the characteristic? (c) What is the probability of obtaining \(x=584\) or fewer individuals with the characteristic?

Suppose a simple random sample of size \(n=12\) is obtained from a population with \(\mu=64\) and \(\sigma=17\) (a) What must be true regarding the distribution of the population in order to use the normal model to compute probabilities regarding the sample mean? Assuming this condition is true, describe the sampling distribution of \(\bar{x}\) (b) Assuming the requirements described in part (a) are satisfied, determine \(P(\bar{x}<67.3)\) (c) Assuming the requirements described in part (a) are satisfied, determine \(P(\bar{x} \geq 65.2)\)

According to a CNN report, \(7 \%\) of the population do not have traditional phones and instead rely on only cell phones. Suppose a random sample of 750 telephone users is obtained. (a) Describe the sampling distribution of \(\hat{p},\) the sample proportion that is "cell-phone only." (b) In a random sample of 750 telephone users, what is the probability that more than \(8 \%\) are "cell-phone only"? (c) Would it be unusual if a random sample of 750 adults results in 40 or fewer being "cell-phone only"?

A report released in May 2005 by First Data Corp. indicated that \(43 \%\) of adults had received a "phishing" contact (a bogus e-mail that replicates an authentic site for the purpose of stealing personal information such as account numbers and passwords). Suppose a random sample of 800 adults is obtained. (a) In a random sample of 800 adults, what is the probability that no more than \(40 \%\) have received a phishing contact? (b) Would it be unusual if a random sample of 800 adults resulted in \(45 \%\) or more who had received a phishing contact?

The most famous geyser in the world, Old Faithful in Yellowstone National Park, has a mean time between eruptions of 85 minutes. If the interval of time between eruptions is normally distributed with standard deviation 21.25 minutes, answer the following questions: (Source: www.unmuseum.org) (a) What is the probability that a randomly selected time interval between eruptions is longer than 95 minutes? (b) What is the probability that a random sample of 20 time intervals between eruptions has a mean longer than 95 minutes? (c) What is the probability that a random sample of 30 time intervals between eruptions has a mean longer than 95 minutes? (d) What effect does increasing the sample size have on the probability? Provide an explanation for this result. (e) What might you conclude if a random sample of 30 time intervals between eruptions has a mean longer than 95 minutes?

The S\&P 500 is a collection of 500 stocks of publicly traded companies. Using data obtained from Yahoo!Finance, the monthly rates of return of the S\&P 500 since 1950 are normally distributed. The mean rate of return is \(0.007233(0.7233 \%),\) and the standard deviation for rate of return is \(0.04135(4.135 \%)\) \\(a) What is the probability that a randomly selected month has a positive rate of return? That is, what is \(P(x>0) ?\) (b) Treating the next 12 months as a simple random sample, what is the probability that the mean monthly rate of return will be positive? That is, with \(n=12,\) what is \(P(\bar{x}>0) ?\) (c) Treating the next 24 months as a simple random sample, what is the probability that the mean monthly rate of return will be positive? (d) Treating the next 36 months as a simple random sample, what is the probability that the mean monthly rate of return will be positive? (e) Use the results of parts (b)-(d) to describe the likelihood of earning a positive rate of return on stocks as the investment time horizon increases.