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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?

The upper leg length of 20 - to 29 -year-old males is normally distributed with a mean length of \(43.7 \mathrm{~cm}\) and a standard deviation of \(4.2 \mathrm{~cm} .\) Source: "Anthropometric Reference Data for Children and Adults: U.S. Population, 1999-2002"; Volume 361, July 7, 2005. (a) What is the probability that a randomly selected 20 - to 29 -yearold male has an upper leg length that is less than \(40 \mathrm{~cm} ?\) (b) A random sample of 9 males who are 20 to 29 years old is obtained. What is the probability that the mean upper leg length is less than \(40 \mathrm{~cm} ?\) (c) What is the probability that a random sample of 12 males who are \(20-29\) years old results in a mean upper leg length that is less than \(40 \mathrm{~cm} ?\) (d) What effect does increasing the sample size have on the probability? Provide an explanation for this result. (e) A random sample of 15 males who are \(20-29\) years old results in a mean upper leg length of \(46 \mathrm{~cm} .\) Do you find this result unusual? Why?

The reading speed of second grade students is approximately normal, with a mean of 90 words per minute (wpm) and a standard deviation of 10 wpm. (a) What is the probability a randomly selected student will read more than 95 words per minute? (b) What is the probability that a random sample of 12 second grade students results in a mean reading rate of more than 95 words per minute? (c) What is the probability that a random sample of 24 second grade students results in a mean reading rate of more than 95 words per minute? (d) What effect does increasing the sample size have on the probability? Provide an explanation for this result. (e) A teacher instituted a new reading program at school. After 10 weeks in the program, it was found that the mean reading speed of a random sample of 20 second grade students was 92.8 wpm. What might you conclude based on this result? (f) There is a \(5 \%\) chance that the mean reading speed of a random sample of 20 second grade students will exceed what value?

Election Prediction Exit polling is a popular technique used to determine the outcome of an election prior to results being tallied. Suppose a referendum to increase funding for education is on the ballot in a large town (voting population over 100,000 ). An exit poll of 310 voters finds that 164 voted for the referendum. How likely are the results of your sample if the population proportion of voters in the town in favor of the referendum is 0.49 ? Based on your result, comment on the dangers of using exit polling to call elections. Include a discussion of the potential nonsampling error that could disrupt your findings.

The shape of the distribution of the time required to get an oil change at a 10 -minute oil-change facility is unknown. However, records indicate that the mean time for an oil change is 11.4 minutes, and the standard deviation for oilchange time is 3.2 minutes. (a) To compute probabilities regarding the sample mean using the normal model, what size sample would be required? (b) What is the probability that a random sample of \(n=40\) oil changes results in a sample mean time of less than 10 minutes? (c) Suppose the manager agrees to pay each employee a \(\$ 50\) bonus if they meet a certain goal. On a typical Saturday, the oil-change facility will perform 40 oil changes between 10 A.M. and 12 P.M. Treating this as a random sample, what mean oil-change time would there be a \(10 \%\) chance of being at or below? This will be the goal established by the manager.

Reincarnation Suppose \(21 \%\) of all American teens (age 13-17 years) believe in reincarnation. (a) Bob and Alicia both obtain a random sample of 100 American teens and ask each participant to disclose whether they believe in reincarnation or not. Is "belief in reincarnation" qualitative or quantitative? Explain. (b) Explain why Bob's sample of 100 randomly selected American teens might result in 18 who believe in reincarnation, while Alicia's independent sample of 100 randomly selected American teens might result in 22 who believe in reincarnation. (c) Why is it important to randomly select American teens to estimate the population proportion who believe in reincarnation? (d) In a survey of 100 American teens, how many would you expect to believe in reincarnation? (e) Below is the histogram of the sample proportion of 1000 different surveys in which \(n=20\) American teens were asked to disclose whether they believed in reincarnation. Explain why the normal model should not be used to describe the distribution of the sample proportion. (f) What minimum sample size would you require in order for the distribution of the sample proportion to be modeled by the normal distribution?

The quality-control manager of a Long John Silver's restaurant wants to analyze the length of time that a car spends at the drive-through window waiting for an order. It is determined that the mean time spent at the window is 59.3 seconds with a standard deviation of 13.1 seconds. The distribution of time at the window is skewed right (data based on information provided by Danica Williams, student at Joliet Junior College). (a) To obtain probabilities regarding a sample mean using the normal model, what size sample is required? (b) The quality-control manager wishes to use a new delivery system designed to get cars through the drive-through system faster. A random sample of 40 cars results in a sample mean time spent at the window of 56.8 seconds. What is the probability of obtaining a sample mean of 56.8 seconds or less, assuming that the population mean is 59.3 seconds? Do you think that the new system is effective? (c) Treat the next 50 cars that arrive as a simple random sample. There is a \(15 \%\) chance the mean time will be at or below ____ seconds

The amount of time Americans spend watching television is closely monitored by firms such as \(\mathrm{AC}\) Nielsen because this helps determine advertising pricing for commercials. (a) Do you think the variable "weekly time spent watching television" would be normally distributed? If not, what shape would you expect the variable to have? (b) According to the American Time Use Survey, adult Americans spend 2.35 hours per day watching television on a weekday. Assume that the standard deviation for "time spent watching television on a weekday" is 1.93 hours. If a random sample of 40 adult Americans is obtained, describe the sampling distribution of \(\bar{x},\) the mean amount of time spent watching television on a weekday. (c) Determine the probability that a random sample of 40 adult Americans results in a mean time watching television on a weekday of between 2 and 3 hours. (d) One consequence of the popularity of the Internet is that it is thought to reduce television watching. Suppose that a random sample of 35 individuals who consider themselves to be avid Internet users results in a mean time of 1.89 hours watching television on a weekday. Determine the likelihood of obtaining a sample mean of 1.89 hours or less from a population whose mean is presumed to be 2.35 hours. Based on the result obtained, do you think avid Internet users watch less television?

State the Central Limit Theorem

We assume that we are obtaining simple random samples from infinite populations when obtaining sampling distributions. If the size of the population is finite, we technically need a finite population correction factor. However, if the sample size is small relative to the size of the population, this factor can be ignored. Explain what an "infinite population" is. What is the finite population correction factor? How small must the sample size be relative to the size of the population so that we can ignore the factor? Finally, explain why the factor can be ignored for such samples.