Solved: In Exercise, find the mean and standard deviation

Problem 1CQ Find each probability using the standard normal distribution. (a) P(z > -2.54) (b) P(z<3.09) (c) P(-0.88<z<0.88) (d) P(z<-1.445 or z > -0.715)

Problem 1CT The mean amount of money that U.S. adults spend on food in a week is $151 and the standard deviation is $49. Random samples of size 50 are drawn from this population and the mean of each sample is determined. (a) Find the mean and standard deviation of the sampling distribution of sample means. (b) What is the probability that the mean amount spent on food in a week for a certain sample is more than $160? (c) What is the probability that the mean amount spent on food in a week for a certain sample is between $135 and $150?

Problem 70E In Exercise, determine whether you can use a normal distribution to approximate the binomial distribution. If you can, use the normal distribution to approximate the indicated probabilities and sketch their graphs. If you cannot, explain why and use a binomial distribution to find the indicated probabilities. Thirty-one percent of people in the United States have type A+ blood. You randomly select 40 people in the United States and ask them whether their blood type is A+. Find the probability that the number of people who have A+ blood is (a) exactly 15, (b) less than 10, and (c) between 20 and 35, and (d) identify any unusual events. Explain.

Problem 2CQ The random variable x is normally distributed with the given parameters. Find each probability. (a) µ = 9.2, ? ? 1.62, P(x<5.97) (b) µ = 87, ? ? 19, P(x > 40.5) (c) µ = 5.5, ? ? 0.08, P(5.36<x<5.64) (d) µ = 18.5, ? ? 4.25, P(19.6<x<26.1)

Problem 2CT In Exercise, the random variable x is normally distributed with mean µ = 18 and standard deviation ? = 7.6. Find each probability. (a) P(x > 20) (b) P(0<x<5) (c) P(x<9 or x > 27)

In Exercises 1 and 2, use the normal curve to estimate the mean and standard deviation.

Problem 3CT In Exercise, the random variable x is normally distributed with mean µ = 18 and standard deviation ? = 7.6. Find the value of x that has 88.3% of the distribution’s area to its left.

In Exercises 3 and 4, use the normal curves shown at the left. Which normal curve has the greatest mean? Explain your reasoning.

Problem 4CQ In Exercise, use the following information. In a standardized IQ test, scores were normally distributed, with a mean score of 100 and a standardized deviation of 15. Find the probability that a randomly selected person has an IQ score between 95 and 105. Is this an unusual event? Explain.

In Exercises 1 and 2, use the normal curve to estimate the mean and standard deviation.

Problem 3CQ In Exercise, use the following information. In a standardized IQ test, scores were normally distributed, with a mean score of 100 and a standardized deviation of 15. Find the probability that a randomly selected person has an IQ score higher than 125. Is this an unusual event? Explain.

In Exercises 3 and 4, use the normal curves shown at the left. Which normal curve has the greatest standard deviation? Explain your reasoning.

Problem 4CT In Exercise, the random variable x is normally distributed with mean µ = 18 and standard deviation ? = 7.6. Find the value of x that has 64.8% of the distribution’s area to its right.

In Exercises 5 and 6, find the area of the indicated region under the standard normal curve. If convenient, use technology to find the area.

Problem 6CQ In Exercise, use the following information. In a standardized IQ test, scores were normally distributed, with a mean score of 100 and a standardized deviation of 15. Out of 2000 randomly selected people, about how many would you expect to have IQ scores less than 90?

Problem 5CT In Exercise, determine whether you can use a normal distribution to approximate the binomial distribution. If you can, use the normal distribution to approximate the indicated probabilities and sketch their graphs. If you cannot, explain why and use a binomial distribution to find the indicated probabilities. A survey of U.S. adults found that 64% watch NFL football. You randomly select 20 U.S. adults and ask them whether they watch NFL football. Find the probability that the number who watch NFL football is (a) exactly 10, (b) less than 7, and (c) at least 15, and (d) identify any unusual events. Explain.

Problem 5CQ In Exercise, use the following information. In a standardized IQ test, scores were normally distributed, with a mean score of 100 and a standardized deviation of 15. What percent of the IQ scores are greater than 112?

Problem 6CT In Exercise, determine whether you can use a normal distribution to approximate the binomial distribution. If you can, use the normal distribution to approximate the indicated probabilities and sketch their graphs. If you cannot, explain why and use a binomial distribution to find the indicated probabilities. A survey of U.S. adults ages 25 and older found that 86% have a high school diploma. You randomly select 30 U.S. adults ages 25 and older. Find the probability that the number who have a high school diploma is (a) exactly 25, (b) more than 25, and (c) less than 25, and (d) identify any unusual events. Explain.

In Exercises 5 and 6, find the area of the indicated region under the standard normal curve. If convenient, use technology to find the area.

Problem 7CT In Exercise, use the following information. The amounts of time Facebook users spend on the website each month are normally distributed, with a mean of 6.7 hours and a standard deviation of 1.8 hours. Find the probability that a Facebook user spends less than four hours on the website in a month. Is this an unusual event? Explain.

Problem 7CQ In Exercise, use the following information. In a standardized IQ test, scores were normally distributed, with a mean score of 100 and a standardized deviation of 15. What is the lowest score that would still place a person in the top 5% of the scores?

Problem 8CQ In Exercise, use the following information. In a standardized IQ test, scores were normally distributed, with a mean score of 100 and a standardized deviation of 15. What is the highest score that would still place a person in the bottom 10% of the scores?

Problem 7E In Exercise, find the indicated area under the standard normal curve. If convenient, use technology to find the area. To the left of z = 0.33

Problem 8CT In Exercise, use the following information. The amounts of time Facebook users spend on the website each month are normally distributed, with a mean of 6.7 hours and a standard deviation of 1.8 hours. Find the probability that a Facebook user spends more than 10 hours on the website in a month. Is this an unusual event? Explain.

Problem 8E In Exercise, find the indicated area under the standard normal curve. If convenient, use technology to find the area. To the left of z = -1.95

Problem 9CQ In Exercise, use the following information. In a standardized IQ test, scores were normally distributed, with a mean score of 100 and a standardized deviation of 15. A random sample of 60 people is selected from this population. What is the probability that the mean IQ score of the sample is greater than 105? Interpret the result.

Problem 9CT In Exercise, use the following information. The amounts of time Facebook users spend on the website each month are normally distributed, with a mean of 6.7 hours and a standard deviation of 1.8 hours. Out of 800 Facebook users, about how many would you expect to spend between 2 and 3 hours on the website in a month?

Problem 10CT In Exercise, use the following information. The amounts of time Facebook users spend on the website each month are normally distributed, with a mean of 6.7 hours and a standard deviation of 1.8 hours. What is the lowest amount of time spent on Facebook in a month that would still place a user in the top 15% of times?

Problem 9E In Exercise, find the indicated area under the standard normal curve. If convenient, use technology to find the area. To the right of z = -0.57

Problem 10E In Exercise, find the indicated area under the standard normal curve. If convenient, use technology to find the area. To the right of z = 3.22

Problem 10CQ In Exercise, use the following information. In a standardized IQ test, scores were normally distributed, with a mean score of 100 and a standardized deviation of 15. Are you more likely to randomly select one person with an IQ score greater than 105 or are you more likely to randomly select a sample of 15 people with a mean IQ score greater than 105? Explain.

Problem 11CT In Exercise, use the following information. The amounts of time Facebook users spend on the website each month are normally distributed, with a mean of 6.7 hours and a standard deviation of 1.8 hours. Between what two values does the middle 60% of the times lie?

Problem 11CQ In Exercise, use the following information. In a survey of U.S. adults, 88% say they are at least somewhat concerned that their personal online data is being used without their knowledge. You randomly select 45 U.S. adults and ask them whether they are at least somewhat concerned that their online data is being used without their knowledge. Determine whether you can use a normal distribution to approximate the binomial distribution. If you can, find the mean and standard deviation. If you cannot, explain why.

Problem 11E In Exercise, find the indicated area under the standard normal curve. If convenient, use technology to find the area. To the left of z = -2.825

Problem 12CQ In Exercise, use the following information. In a survey of U.S. adults, 88% say they are at least somewhat concerned that their personal online data is being used without their knowledge. You randomly select 45 U.S. adults and ask them whether they are at least somewhat concerned that their online data is being used without their knowledge. Find the probability that the number of U.S. adults who say they are at least somewhat concerned that their personal online data is being used without their knowledge is (a) at most 35, (b) less than 40, and (c) exactly 43, and (d) identify any unusual events. Explain.

Problem 12E In Exercise, find the indicated area under the standard normal curve. If convenient, use technology to find the area. To the right of z = 0.015

Problem 12CT In Exercise, use the following information. The amounts of time Facebook users spend on the website each month are normally distributed, with a mean of 6.7 hours and a standard deviation of 1.8 hours. Random samples of size 8 are drawn from this population and the mean of each sample is determined. Is the sampling distribution of sample means normally distributed? Explain.

Problem 13E In Exercise, find the indicated area under the standard normal curve. If convenient, use technology to find the area. Between z = -1.64 and z = 0

Problem 14E In Exercise, find the indicated area under the standard normal curve. If convenient, use technology to find the area. Between z = -1.55 and z = 1.04

Problem 15E In Exercise, find the indicated area under the standard normal curve. If convenient, use technology to find the area. Between z = 0.05 and z = 1.71

Problem 16E In Exercise, find the indicated area under the standard normal curve. If convenient, use technology to find the area. Between z = -2.68 and z = 2.68

Problem 17E In Exercise, find the indicated area under the standard normal curve. If convenient, use technology to find the area. To the left of z = -1.5 and to the right of z = 1.5

Problem 18E In Exercise, find the indicated area under the standard normal curve. If convenient, use technology to find the area. To the left of z = 0.64 and to the right of z = 3.415

Problem 19E In Exercise, use the following information. The scores for the science reasoning portion of the ACT test are normally distributed. In a recent year, the mean test score was 20.9 and the standard deviation was 5.2. The test scores of four students selected at random are 17, 29, 8, and 23. Find the z-score that corresponds to each value.

Problem 20E In Exercise, use the following information. The scores for the science reasoning portion of the ACT test are normally distributed. In a recent year, the mean test score was 20.9 and the standard deviation was 5.2. The test scores of four students selected at random are 17, 29, 8, and 23. Determine whether any of the values are unusual.

Problem 21E In Exercise, find the indicated probability using the standard normal distribution. If convenient, use technology to find the probability. P(z < 1.28)

Problem 22E In Exercise, find the indicated probability using the standard normal distribution. If convenient, use technology to find the probability. P(z > -0.74)

Problem 23E In Exercise, find the indicated probability using the standard normal distribution. If convenient, use technology to find the probability. P(-2.15<z < 1.55)

Problem 24E In Exercise, find the indicated probability using the standard normal distribution. If convenient, use technology to find the probability. P(0.42<z < 3.15)

Problem 25E In Exercise, find the indicated probability using the standard normal distribution. If convenient, use technology to find the probability. P(z < -2.50 or z > 2.50)

Problem 26E In Exercise, find the indicated probability using the standard normal distribution. If convenient, use technology to find the probability. P(z < 0 or z > 1.68)

Problem 27E In Exercise, the random variable x is normally distributed with mean µ = 74 and standard deviation ? = 8. Find the indicated probability. P(x < 84)

Problem 28E In Exercise, the random variable x is normally distributed with mean µ = 74 and standard deviation ? = 8. Find the indicated probability. P(x < 55)

Problem 29E In Exercise, the random variable x is normally distributed with mean µ = 74 and standard deviation ? = 8. Find the indicated probability. P(x > 80)

Problem 31E In Exercise, the random variable x is normally distributed with mean µ = 74 and standard deviation ? = 8. Find the indicated probability. P(60<x < 70)

Problem 30E In Exercise, the random variable x is normally distributed with mean µ = 74 and standard deviation ? = 8. Find the indicated probability. P(x > 71.6)

Problem 33E In Exercise find the indicated probabilities. If convenient, use technology to find the probabilities. In a study of migrating Sandhill Cranes, the distances traveled in a day were normally distributed, with a mean of 267 kilometers and a standard deviation of 86 kilometers. Find the probability that the distance traveled in a day by a randomly selected Sandhill Crane from the study is (a) less than 200 kilometers. (b) between 250 and 350 kilometers. (c) greater than 500 kilometers.

Problem 32E In Exercise, the random variable x is normally distributed with mean µ = 74 and standard deviation ? = 8. Find the indicated probability. P(72<x < 82)

Problem 36E Determine whether any of the events in Exercise are unusual. Explain your reasoning. Exercise In a study of bumblebee bats, one of the world’s smallest mammals, the weights were normally distributed, with a mean of 2.0 grams and a standard deviation of 0.25 gram. Find the probability that a randomly selected bat from the study weighs (a) between 1.8 grams and 2.2 grams. (b) between 2.1 grams and 2.7 grams. (c) more than 2.3 grams.

Problem 37E In Exercise, use the Standard Normal Table to find the z-score that corresponds to the cumulative area or percentile. If the area is not in the table, use the entry closest to the area. If the area is halfway between two entries, use the z-score halfway between the corresponding z-scores. If convenient, use technology to find the z-score. 0.4721

Problem 35E Determine whether any of the events in Exercise are unusual. Explain your reasoning. Exercise In a study of migrating Sandhill Cranes, the distances traveled in a day were normally distributed, with a mean of 267 kilometers and a standard deviation of 86 kilometers. Find the probability that the distance traveled in a day by a randomly selected Sandhill Crane from the study is (a) less than 200 kilometers. (b) between 250 and 350 kilometers. (c) greater than 500 kilometers.

Problem 38E In Exercise, use the Standard Normal Table to find the z-score that corresponds to the cumulative area or percentile. If the area is not in the table, use the entry closest to the area. If the area is halfway between two entries, use the z-score halfway between the corresponding z-scores. If convenient, use technology to find the z-score. 0.1

Problem 40E In Exercise, use the Standard Normal Table to find the z-score that corresponds to the cumulative area or percentile. If the area is not in the table, use the entry closest to the area. If the area is halfway between two entries, use the z-score halfway between the corresponding z-scores. If convenient, use technology to find the z-score. P2

Problem 42E In Exercise, use the Standard Normal Table to find the z-score that corresponds to the cumulative area or percentile. If the area is not in the table, use the entry closest to the area. If the area is halfway between two entries, use the z-score halfway between the corresponding z-scores. If convenient, use technology to find the z-score. P46

Problem 34E In Exercise find the indicated probabilities. If convenient, use technology to find the probabilities. In a study of bumblebee bats, one of the world’s smallest mammals, the weights were normally distributed, with a mean of 2.0 grams and a standard deviation of 0.25 gram. Find the probability that a randomly selected bat from the study weighs (a) between 1.8 grams and 2.2 grams. (b) between 2.1 grams and 2.7 grams. (c) more than 2.3 grams.

Problem 43E Find the z-score that has 30.5% of the distribution’s area to its right.

Problem 44E Find the z-score for which 94% of the distribution’s area lies between -z and z.

Problem 39E In Exercise, use the Standard Normal Table to find the z-score that corresponds to the cumulative area or percentile. If the area is not in the table, use the entry closest to the area. If the area is halfway between two entries, use the z-score halfway between the corresponding z-scores. If convenient, use technology to find the z-score. 0.993

In Exercises 45–50, use the following information. On a dry surface, the braking distances (in feet), from 60 miles per hour to a complete stop, of a sedan can be approximated by a normal distribution, as shown in the figure at the left. (Adapted from Consumer Reports) Find the braking distance of a sedan that corresponds to \(z=-2.5\). ________________ Equation Transcription: Text Transcription: mu=127 ft sigma=3.81 ft z=-2.5

In Exercises 45–50, use the following information. On a dry surface, the braking distances (in feet), from 60 miles per hour to a complete stop, of a sedan can be approximated by a normal distribution, as shown in the figure at the left. (Adapted from Consumer Reports) Find the braking distance of a sedan that corresponds to \(z=1.2\). ________________ Equation Transcription: Text Transcription: mu=127 ft sigma=3.81 ft z=1.2

In Exercises 45–50, use the following information. On a dry surface, the braking distances (in feet), from 60 miles per hour to a complete stop, of a sedan can be approximated by a normal distribution, as shown in the figure at the left. (Adapted from Consumer Reports) What braking distance of a sedan represents the third quartile? ________________ Equation Transcription: Text Transcription: mu=127 ft sigma=3.81 ft

In Exercises 45–50, use the following information. On a dry surface, the braking distances (in feet), from 60 miles per hour to a complete stop, of a sedan can be approximated by a normal distribution, as shown in the figure at the left. (Adapted from Consumer Reports) What braking distance of a sedan represents the 95th percentile? ________________ Equation Transcription: Text Transcription: mu=127 ft sigma=3.81 ft

In Exercises 45–50, use the following information. On a dry surface, the braking distances (in feet), from 60 miles per hour to a complete stop, of a sedan can be approximated by a normal distribution, as shown in the figure at the left. (Adapted from Consumer Reports) What is the shortest braking distance of a sedan that can be in the top 10% of braking distances? ________________ Equation Transcription: Text Transcription: mu=127 ft sigma=3.81 ft

In Exercises 45–50, use the following information. On a dry surface, the braking distances (in feet), from 60 miles per hour to a complete stop, of a sedan can be approximated by a normal distribution, as shown in the figure at the left. (Adapted from Consumer Reports) What is the longest braking distance of a sedan that can be in the bottom 5% of braking distances? ________________ Equation Transcription: Text Transcription: mu=127 ft sigma=3.81 ft

Problem 51E In Exercise, find the mean and standard deviation of the population. List all samples (with replacement) of the given size from that population and find the mean of each. Find the mean and standard deviation of the sampling distribution of sample means and compare them with the mean and standard deviation of the population. The goals scored in a season by the four starting defenders on a soccer team are 1, 2, 0, and 3. Use a sample size of 2.

Problem 52E In Exercise, find the mean and standard deviation of the population. List all samples (with replacement) of the given size from that population and find the mean of each. Find the mean and standard deviation of the sampling distribution of sample means and compare them with the mean and standard deviation of the population. The minutes of overtime reported by each of the three executives at a corporation are 90, 120, and 210. Use a sample size of 3.

Problem 54E In Exercise, use the Central Limit Theorem to find the mean and standard deviation of the indicated sampling distribution of sample means. Then sketch a graph of the sampling distribution. The annual per capita consumption of red meat by people in the United States is normally distributed, with a mean of 107.9 pounds and a standard deviation of 35.1 pounds. Random samples of size 40 are drawn from this population, and the mean of each sample is determined.

Problem 53E In Exercise, use the Central Limit Theorem to find the mean and standard deviation of the indicated sampling distribution of sample means. Then sketch a graph of the sampling distribution. The annual per capita consumption of citrus fruits by people in the United States is normally distributed, with a mean of 85.6 pounds and a standard deviation of 20.5 pounds. Random samples of size 35 are drawn from this population, and the mean of each sample is determined.

Problem 55E In Exercise, find the indicated probabilities and interpret the results. If convenient, use technology to find the probabilities. Refer to Exercise. A random sample of 12 Sandhill Cranes is selected from the study. Find the probability that the mean distance traveled of the sample is (a) less than 200 kilometers, (b) between 250 and 350 kilometers, and (c) greater than 500 kilometers, and (d) compare your answers with those in the Exercise below. Exercise In a study of migrating Sandhill Cranes, the distances traveled in a day were normally distributed, with a mean of 267 kilometers and a standard deviation of 86 kilometers. Find the probability that the distance traveled in a day by a randomly selected Sandhill Crane from the study is (a) less than 200 kilometers. (b) between 250 and 350 kilometers. (c) greater than 500 kilometers.

Problem 56E In Exercise, find the indicated probabilities and interpret the results. If convenient, use technology to find the probabilities. Refer to Exercise. A random sample of seven bumblebee bats is selected from the study. Find the probability that the mean weight of the sample is (a) between 1.8 grams and 2.2 grams, (b) between 2.1 grams and 2.7 grams, and (c) more than 2.3 grams, and (d) compare your answers with those in Exercise. Exercise In a study of bumblebee bats, one of the world’s smallest mammals, the weights were normally distributed, with a mean of 2.0 grams and a standard deviation of 0.25 gram. Find the probability that a randomly selected bat from the study weighs (a) between 1.8 grams and 2.2 grams. (b) between 2.1 grams and 2.7 grams. (c) more than 2.3 grams.

Problem 41E In Exercise, use the Standard Normal Table to find the z-score that corresponds to the cumulative area or percentile. If the area is not in the table, use the entry closest to the area. If the area is halfway between two entries, use the z-score halfway between the corresponding z-scores. If convenient, use technology to find the z-score. P85

Problem 57E In Exercise, find the indicated probabilities and interpret the results. If convenient, use technology to find the probabilities. The mean value of land and buildings per acre for farms in Illinois is $6700. A random sample of 36 Illinois farms is selected. What is the probability that the mean value of land and buildings per acre for the sample is (a) less than $7200, (b) more than $6500, and (c) between $7000 and $7400? Assume ? = $1250.

Problem 58E In Exercise, find the indicated probabilities and interpret the results. If convenient, use technology to find the probabilities. The mean value of land and buildings per acre for farms in Colorado is $1170. A random sample of 32 Colorado farms is selected. What is the probability that the mean value of land and buildings per acre for the sample is (a) less than $1200, (b) more than $1275, and (c) between $1100 and 1250? Assume ? = $200.

Problem 60E In Exercise, find the indicated probabilities and interpret the results. If convenient, use technology to find the probabilities. The mean annual salary for parole officers is about $50,830. A random sample of 50 parole officers is selected. What is the probability that the mean annual salary of the sample is (a) less than $50,000 and (b) more than $53,500? Assume ? = $8520.

Problem 59E In Exercise, find the indicated probabilities and interpret the results. If convenient, use technology to find the probabilities. The mean annual salary for chauffeurs is about $30,800. A random sample of 45 chauffeurs is selected. What is the probability that the mean annual salary of the sample is (a) less than $30,000 and (b) more than $34,000? Assume ? = $5600.

Problem 63E In Exercise, write the binomial probability in words. Then, use a continuity correction to convert the binomial probability to a normal distribution probability. P(x ? 25)

Problem 67E In Exercise, write the binomial probability in words. Then, use a continuity correction to convert the binomial probability to a normal distribution probability. P(x < 60)

Problem 64E In Exercise, write the binomial probability in words. Then, use a continuity correction to convert the binomial probability to a normal distribution probability. P(x ? 36)

Problem 66E In Exercise, write the binomial probability in words. Then, use a continuity correction to convert the binomial probability to a normal distribution probability. P(x > 14)

Problem 61E In Exercise, a binomial experiment is given. Determine whether you can use a normal distribution to approximate the binomial distribution. If you can, find the mean and standard deviation. If you cannot, explain why. A survey of U.S. adults found that 73% think the federal government should require that genetically modified food be labeled as such. You randomly select 12 U.S. adults and ask them whether they think the federal government should require that genetically modified food be labeled as such.

Problem 65E In Exercise, write the binomial probability in words. Then, use a continuity correction to convert the binomial probability to a normal distribution probability. P(x = 45)

Problem 68E In Exercise, write the binomial probability in words. Then, use a continuity correction to convert the binomial probability to a normal distribution probability. P(54<x < 64)

Problem 69E In Exercise, determine whether you can use a normal distribution to approximate the binomial distribution. If you can, use the normal distribution to approximate the indicated probabilities and sketch their graphs. If you cannot, explain why and use a binomial distribution to find the indicated probabilities. A survey found that 52% of U.S. teens ages 16 to 18 have a savings account. You randomly select 45 U.S. teens ages 16 to 18 and ask them whether they have a savings account. Find the probability that the number who have a savings account is (a) at most 15, (b) exactly 25, and (c) greater than 30, and (d) identify any unusual events. Explain.

Problem 62E In Exercise, a binomial experiment is given. Determine whether you can use a normal distribution to approximate the binomial distribution. If you can, find the mean and standard deviation. If you cannot, explain why. A survey of U.S. adults found that 41% would be comfortable using a cell phone scan as an airline, train, or other transportation ticket. You randomly select 20 U.S. adults and ask them whether they would be comfortable using a cell phone scan as an airline, train, or other transportation ticket.