Solved: The ACT and SAT are two college entrance exams.

Problem 1AYU A random variable Z that is normally distributed with mean ? = 0 and standard deviation ? = 1 is said to have the ____ ____ _____.

Problem 2AYU The notation z? is the z-score that the area under the standard normal curve to the right of z? is _____.

Problem 3AYU If X is a normal random variable with mean 40 and standard deviation 10 and P(X > 45) = 0.3085, then P(X<35) = ______.

If X is a normal random variable with mean 40 and standard deviation 10 and P(X < 38) = 0.4207, then \(P(X \le 38) =\) ______.

In Problem 5-12, find the indicated areas. For each problem, be sure to draw a standard normal curve and shade the area that is to be found. Determine the area under the standard normal curve that lies to the left of (a) z = ?2.45 (b) z = ?0.43 (c) z = 1.35 (d) z = 3.49

Find the indicated areas. For each problem, be sure to draw a standard normal curve and shade the area that is to be found. Determine the area under the standard normal curve that lies to the left of (a) z = ?3.49 (b) z = ?1.99 (c) z = 0.92 (d) z = 2.90

Problem 7AYU Find the indicated areas. For each problem, be sure to draw a standard normal curve and shade the area that is to be found. Determine the area under the standard normal curve that lies to the right of (a) z = ?3.01 ________________ (b) z = ?1.59 ________________ (c) z = 1.78 ________________ (d) z = 3.11

In Problem 5-12, find the indicated areas. For each problem, be sure to draw a standard normal curve and shade the area that is to be found. Determine the area under the standard normal curve that lies to the right of (a) z = ?3.49 (b) z = ?0.55 (c) z = 2.23 (d) z = 3.45

In Problem 5-12, find the indicated areas. For each problem, be sure to draw a standard normal curve and shade the area that is to be found. Determine the area under the standard normal curve that lies between (a) z = ?2.04 and z = 2.04 (b) z = ?0.55 and z = 0 (c) z = ?1.04 and z = 2.76

Find the indicated areas. For each problem, be sure to draw a standard normal curve and shade the area that is to be found. Determine the area under the standard normal curve that lies between (a) z = ?2.55 and z = 2.55 (b) z = ?1.67 and z = 0 (c) z = ?3.03 and z = 1.98

In Problem 5-12, find the indicated areas. For each problem, be sure to draw a standard normal curve and shade the area that is to be found. Determine the total area under the standard normal curve (a) to the left of z = ?2 or to the right of z = 2 (b) to the left of z = ?1.56 or to the right of z = 2.56 (c) to the left of z = ?0.24 or to the right of z = 1.20

In Problem 5-12, find the indicated areas. For each problem, be sure to draw a standard normal curve and shade the area that is to be found. Determine the total area under the standard normal curve (a) to the left of z = ?2.94 or to the right of z = 2.94 (b) to the left of z = ?1.68 or to the right of z = 3.05 (c) to the left of z = ?0.88 or to the right of z = 1.23

Find the indicated z-score. Be sure to draw a standard normal curve that depicts the solution. Find the z-score such that the area under the standard normal curve to its left is 0.1.

Find the indicated z-score. Be sure to draw a standard normal curve that depicts the solution. Find the z-score such that the area under the standard normal curve to its left is 0.2.

Find the indicated z-score. Be sure to draw a standard normal curve that depicts the solution. Find the z-score such that the area under the standard normal curve to its right is 0.25.

In Problems 13-18, find the indicated z-score. Be sure to draw a standard normal curve that depicts the solution. Find the z-score such that the area under the standard normal curve to its right is 0.35.

Find the indicated z-score. Be sure to draw a standard normal curve that depicts the solution. Find z-scores that separate the middle 99% of the distribution from the area in the tails of the standard normal distribution.

Problem 18AYU Find the indicated z-score. Be sure to draw a standard normal curve that depicts the solution. Find the z-scores that separate the middle 94% of the distribution from the area in the tails of the standard normal distribution.

Problem 19AYU Find the value of z z0.01

Find the value of \(z_\alpha\). \(z_{0.02}\)

Find the value of \(z_\alpha\) \(z_{0.025}\)

In Problems 19-22, find the value of \(z_{\alpha}\). \(z_{0.15}\)

Problem 23AYU Assume that the random variable X is normally distributed, with mean ? = 50 and standard deviation ? = 7. Compute the following probabilities. Be sure to draw a normal curve with the area corresponding to the probability shaded. P(X > 35)

In Problems 23-32, assume that the random variable X is normally distributed, with mean \(\mu=50\) and standard deviation \(\sigma=7\). Compute the following probabilities. Be sure to draw a normal curve with the area corresponding to the probability shaded. \(P(X>65)\)

Assume that the random variable X is normally distributed, with mean \(\mu=50\) and standard deviation \(\sigma=7\). Compute the following probabilities. Be sure to draw a normal curve with the area corresponding to the probability shaded. \(P(X \le 45)\)

In Problems 23-32, assume that the random variable X is normally distributed, with mean \(\mu=50\) and standard deviation \(\sigma=7\). Compute the following probabilities. Be sure to draw a normal curve with the area corresponding to the probability shaded. \(P(X \leq 58)\)

Assume that the random variable X is normally distributed, with mean \(\mu=50\) and standard deviation \(\sigma=7\). Compute the following probabilities. Be sure to draw a normal curve with the area corresponding to the probability shaded. P(40<x<65)

Assume that the random variable X is normally distributed, with mean \(\mu=50\) and standard deviation \(\sigma=7\). Compute the following probabilities. Be sure to draw a normal curve with the area corresponding to the probability shaded. P(56<X<68)

Assume that the random variable X is normally distributed, with mean \(\mu=50\) and standard deviation \(\sigma=7\). Compute the following probabilities. Be sure to draw a normal curve with the area corresponding to the probability shaded. \(P(55 \leq X \leq 70)\)

Assume that the random variable X is normally distributed, with mean \(\mu=50\) and standard deviation \(\sigma=7\). Compute the following probabilities. Be sure to draw a normal curve with the area corresponding to the probability shaded. P(40 ? X ? 49)

Assume that the random variable X is normally distributed, with mean \(\mu = 50\) and standard deviation \(\sigma = 7\). Compute the following probabilities. Be sure to draw a normal curve with the area corresponding to the probability shaded. \(P(38<X \le 55)\)

In Problems 23-32, assume that the random variable X is normally distributed, with mean \(\mu\) = 50 and standard deviation \(\sigma\) = 7. Compute the following probabilities. Be sure to draw a normal curve with the area corresponding to the probability shaded. \(P(56\ \leq\ X\ <\ 66)\)

Assume that the random variable X is normally distributed, with mean \(\mu=50\) and standard deviation \(\sigma = 7\). Find each indicated percentile for X. The 9th percentile

In Problems 33-36, assume that the random variable X is normally distributed, with mean \(\mu\) = 50 and standard deviation \(\sigma\) = 7. Find each indicated percentile for X. The 90th percentile

Assume that the random variable X is normally distributed, with mean \(\mu = 50\) and standard deviation \(\sigma= 7\). Find each indicated percentile for X. The 81st percentile

In Problems 33-36, assume that the random variable X is normally distributed, with mean \(\mu\) = 50 and standard deviation \(\sigma\) = 7. Find each indicated percentile for X. The 38th percentile

Problem 37AYU Egg Incubation Times The mean incubation time of fertilized chicken eggs kept at 100.5°F in a still-air incubator is 21 days. Suppose that the incubation times are approximately normally distributed with a standard deviation of 1 day. Source: University of Illinois Extension (a) What is the probability that a randomly selected fertilized chicken egg hatches in less than 20 days? ________________ (b) What is the probability that a randomly selected fertilized chicken egg takes over 22 days to hatch? ________________ (c) What is the probability that a randomly selected fertilized chicken egg hatches between 19 and 21 days? ________________ (d) Would it be unusual for an egg to hatch in less than 18 days? Why?

Reading Rates The reading speed of sixth-grade students is approximately normal, with a mean speed of 125 words per minute and a standard deviation of 24 words per minute. (a) What is the probability that a randomly selected sixth-grade student reads less than 100 words per minute? (b) What is the probability that a randomly selected sixth-grade student reads more than 140 words per minute (c) What is the probability that a randomly selected sixth-grade student reads between 110 and 130 words per minute? (d) Would it be unusual for a sixth-grader to read more than 200 words per minute? Why?

Problem 39AYU Chips Ahoy! Cookies The number of chocolate chips in an 18-ounce bag of Chips Ahoy! chocolate chip cookies is approximately normally distributed with a mean of 1262 chips and standard deviation 118 chips according to a study by cadets of the U.S. Air Force Academy. Source: Brad Warner and Jim Rutledge, Chance 12(1): 10-14, 1999 (a) What is the probability that a randomly selected 18-ounce bag of Chips Ahoy! contains between 1000 and 1400 chocolate chips, inclusive? ________________ (b) What is the probability that a randomly selected 18-ounce bag of Chips Ahoy! contains fewer than 1000 chocolate chips? ________________ (c) What proportion of 18-ounce bags of Chips Ahoy! contains more than 1200 chocolate chips? ________________ (d) What proportion of 18-ounce bags of Chips Ahoy! contains fewer than 1125 chocolate chips? ________________ (e) What is the percentile rank of an 18-ounce bag of Chips Ahoy! that contains 1475 chocolate chips? ________________ (f) What is the percentile rank of an 18-ounce bag of Chips Ahoy! that contains 1050 chocolate chips?

Wendy's Drive-Through Fast-food restaurants spend quite a bit of time studying the amount of time cars spend in their drive throughs. Certainly, the faster the cars get through the drive-through, the more the opportunity for making money. QSR Magazine studied drive-through times for fast-food restaurants and found Wendy's had the best time, with a mean time spent in the drive-through of 138.5 seconds. Assuming drive-through times are normally distributed with a standard deviation of 29 seconds, answer the following. (a) What is the probability that a randomly selected car will get through Wendy's drive-through in less than 100 seconds? (b) What is the probability that a randomly selected car will spend more than 160 seconds in Wendy's drive-through? (c) What proportion of cars spend between 2 and 3 minutes in Wendy's drive-through? (d) Would it be unusual for a car to spend more than 3 minutes in Wendy's drive-through? Why?

Problem 41AYU Gestation Period The lengths of human pregnancies are approximately normally distributed, with mean ? = 266 days and standard deviation ? = 16 days. (a) What proportion of pregnancies lasts more than 270 days? ________________ (b) What proportion of pregnancies lasts less than 250 days? ________________ (c) What proportion of pregnancies lasts between 240 and 280 days? ________________ (d) What is the probability that a randomly selected pregnancy lasts more than 280 days? ________________ (e) What is the probability that a randomly selected pregnancy lasts no more than 245 days? ________________ (f) A “very preterm” baby is one whose gestation period is less than 224 days. Are very preterm babies unusual?

Problem 42AYU Light Bulbs General Electric manufactures a decorative Crystal Clear 60-watt light bulb that it advertises will last 1500 hours. Suppose that the lifetimes of the light bulbs are approximately normally distributed, with a mean of 1550 hours and a standard deviation of 57 hours. (a) What proportion of the light bulbs will last less than the advertised time? ________________ (b) What proportion of the light bulbs will last more than 1650 hours? ________________ (c) What is the probability that a randomly selected GE Crystal Clear 60-watt light bulb will last between 1625 and 1725 hours? ________________ (d) What is the probability that a randomly selected GE Crystal Clear 60-watt light bulb will last longer than 1400 hours?

Manufacturing Steel rods are manufactured with a mean length of 25 centimeters (cm). Because of variability in the manufacturing process, the lengths of the rods are approximately normally distributed, with a standard deviation of 0.07 cm. (a) What proportion of rods has a length less than 24.9 cm? (b) Any rods that are shorter than 24.85 cm or longer than 25.15 cm are discarded. What proportion of rods will be discarded? (c) Using the results of part (b), if 5000 rods are manufactured in a day, how many should the plant manager expect to discard? (d) If an order comes in for 10,000 steel rods, how many rods should the plant manager manufacture if the order states that all rods must be between 24.9 cm and 25.1 cm?

Problem 44AYU Manufacturing Ball bearings are manufactured with a mean diameter of 5 millimeters (mm). Because of variability in the manufacturing process, the diameters of the ball bearings are approximately normally distributed, with a standard deviation of 0.02 mm. (a) What proportion of ball bearings has a diameter more than 5.3 mm? ________________ (b) Any ball bearings that have a diameter less than 4.95 mm or greater than 5.05 mm are discarded. What proportion of ball bearings will be discarded? ________________ (c) Using the results of part (b), if 30,000 ball bearings are manufactured in a day, how many should the plant manager expect to discard? ________________ (d) If an order comes in for 50,000 ball bearings, how many bearings should the plant manager manufacture if the order states that all ball bearings must be between 4.97 mm and 5.3 mm?

Problem 45AYU NCAA Basketball Point Spreads In sports betting, Las Vegas sports books establish winning margins for a team that is favored to win a game. An individual can place a wager on the game and will win if the team bet upon wins after accounting for the spread. For example, if Team A is favored by 5 points, and wins the game by 7 points, then a bet on Team A is a winning bet. However, if Team A wins the game by only 3 points, then a bet on Team A is a losing bet. In games where a team is favored by 12 or fewer points, the margin of victory for the favored team relative to the spread is approximately normally distributed with a mean of 0 points and a standard deviation of 10.9 points. Source: Justin Wolfers, “Point Shaving: Corruption in NCAA Basketball” (a) In games where a team is favored by 12 or fewer points, what is the probability that the favored team wins by 5 or more points relative to the spread? ________________ (b) In games where a team is favored by 12 or fewer points, what is the probability that the favored team loses by 2 or more points relative to the spread? ________________ (c) Explain the meaning of “the margin of victory relative to the spread has a mean of 0 points.” Does this imply that the spreads are accurate for games in which a team is favored by 12 or fewer points?

Problem 46AYU NCAA Basketball Point Spreads Revisited See Problem 45. In games where a team is favored by more than 12 points, the margin of victory for the favored team relative to the spread is normally distributed with a mean of ?1.0 point and a standard deviation of 10.9 points. Source: Justin Wolfers, “Point Shaving: Corruption in NCAA Basketball” (a) In games where a team is favored by more than 12 points, what is the probability that the favored team wins by 5 or more points relative to the spread? ________________ (b) In games where a team is favored by more than 12 points, what is the probability that the favored team loses by 2 or more points relative to the spread? ________________ (c) In games where a team is favored by more than 12 points, what is the probability that the favored team “beats the spread”? Does this imply that the possible point shaving spreads are accurate for games in which a team is favored by more than 12 points?

Egg Incubation Times The mean incubation time of fertilized chicken eggs kept at 100.5°F in a still-air incubator is 21 days. Suppose that the incubation times are approximately normally distributed with a standard deviation of 1 day. Source: University of Illinois Extension (a) Determine the 17th percentile for incubation times of fertilized chicken eggs. (b) Determine the incubation times that make up the middle 95% of fertilized chicken eggs.

Reading Rates The reading speed of sixth-grade students is approximately normal, with a mean speed of 125 words per minute and a standard deviation of 24 words per minute. (a) What is the reading speed of a sixth-grader whose reading speed is at the 90th percentile? (b) A school psychologist wants to determine reading rates for unusual students (both slow and fast). Determine the reading rates of the middle 95% of all sixth-grade students. What are the cutoff points for unusual readers?

Chips Ahoy! Cookies The number of chocolate chips in an 18-ounce bag of Chips Ahoy! chocolate chip cookies is approximately normally distributed, with a mean of 1262 chips and a standard deviation of 118 chips, according to a study by cadets of the U.S. Air Force Academy. Source: Brad Warner and Jim Rutledge, Chance 12(1): 10-14,1999 (a) Determine the 30 th percentile for the number of chocolate chips in an 18 -ounce bag of Chips Ahoy! cookies. (b) Determine the number of chocolate chips in a bag of Chips Ahoy! that make up the middle \(99 \%\) of bags.

Wendy’s Drive-Through Fast-food restaurants spend quite a bit of time studying the amount of time cars spend in their drive-through. Certainly, the faster the cars get through the drive-through, the more the opportunity for making money. QSR Magazine studied drive-through times for fast-food restaurants and found Wendy’s had the best time, with a mean time a car spent in the drive-through equal to 138.5 seconds. Assume that drive-through times are normally distributed, with a standard deviation of 29 seconds. Suppose that Wendy’s wants to institute a policy at its restaurants that it will not charge any patron that must wait more than a certain amount of time for an order. Management does not want to give away free meals to more than 1% of the patrons. What time would you recommend Wendy’s advertise as the maximum wait time before a free meal is awarded?

Problem 51AYU Speedy Lube The time required for Speedy Lube to complete an oil change service on an automobile approximately follows a normal distribution, with a mean of 17 minutes and a standard deviation of 2.5 minutes. (a) Speedy Lube guarantees customers that the service will take no longer than 20 minutes. If it does take longer, the customer will receive the service for half-price. What percent of customers receive the service for half price? ________________ (b) If Speedy Lube does not want to give the discount to more than 3% of its customers, how long should it make the guaranteed time limit?

Putting It Together: Birth Weights The following data represent the distribution of birth weights (in grams) for babies in which the pregnancy went full term (37 to 41 weeks). (a) Construct a relative frequency distribution for birth weight. (b) Draw a relative frequency histogram for birth weight. Describe the shape of the distribution. (c) Determine the mean and standard deviation birth weight. (d) Use the normal model to determine the proportion of babies in each class. (e) Compare the proportions predicted by the normal model to the relative frequencies found in part (a). Do you believe that the normal model is effective in describing the birth weights of babies?

Give three interpretations for the area under a normal curve.

Explain why P (X < 30) should be reported as < 0.0001 if X is a normal random variable with mean 100 and standard deviation 15.

Problem 55AYU Explain why P (X ? 220) should be reported as > 0.9999 if X is a normal random variable with mean 100 and standard deviation 15.

Problem 56AYU The ACT and SAT are two college entrance exams. The composite score on the ACT is approximately normally distributed with mean 21.1 and standard deviation 5.1. The composite score on the SAT is approximately normally distributed with mean 1026 with standard deviation 210. Suppose you scored 26 on the ACT and 1240 on the SAT. Which exam did you score better on? Justify your reasoning using the normal model