Chapter 5 Review The Normal Probability and Standardization

Transcription

1 Chapter 5 Review The Normal Probability and Standardization MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) Approximately % of the area under the normal curve is between μ - σ and μ +σ. A) 99.7 B) 95 C)68 D) 50 1) A random variable X is normally distributed with μ = 60. Convert the value of X to a z-score, if the standard deviation is as given. 2) X = 72; σ = 3 A) 3 B) 5 C)12 D) 4 2) SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Find the standard normal-curve area indicated. 3) Between z = -1.3 and z = ) 4) To the left of z = ) The number of ounces of soda that a vending machine dispenses per cup is normally distributed with a mean of 12 ounces and a standard deviation of 4 ounces. 5) Find the probability that more than 16 ounces is dispensed in a cup. 5) 6) Find the probability that between 15 and 18 ounces are dispensed in a cup. 6) 7) Find the number of ounces above which 80% of the dispensed sodas will fall. Round to the nearest tenth. 7) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. A random variable X is normally distributed with μ = 60. Convert the value of X to a z-score, if the standard deviation is as given. 8) X = 30; σ = 8 A) -8 B) C)3.75 D) 8 8) 1

2 Obtain the shaded area under the standard normal curve. 9) 9) z A) B) C) D) ) 10) z A) B) C) D) ) 11) z A) B) C) D) find the required z-score. 12) Find the z-score for which the area under the standard normal curve to its left is 0.96 A) 1.03 B) C) 1.75 D) ) 13) Find the z-score having area 0.86 to its right under the standard normal curve; that is, find z ) A) 1.08 B) C) D) Provide an appropriate response. 14) Scores on a standardized test are normally distributed with a mean of 96 and a standard deviation of 12. An individual's test score is found to be 128. Find the z-score corresponding to this value. 14) A) B) C) 0.38 D)

3 find the required z-score. 15) Determine the two z-scores that divide the area under the standard normal curve into a middle 0.96 area and two outside 0.02 areas. 15) A) 0 and 2.05 B) and 1.75 C) and 2.05 D) and 2.33 Provide an appropriate response. 16) Find the z-scores for which 98% of the distribution's area lies between -z and z. A) (-1.96, 1.96) B) (-0.99, 0.99) C)(-2.33, 2.33) D) (-1.645, 1.645) 16) 17) A physical fitness association is including the mile run in its secondary-school fitness test. The time for this event for boys in secondary school is known to possess a normal distribution with a mean of 440 seconds and a standard deviation of 60 seconds. Find the probability that a randomly selected boy in secondary school can run the mile in less than 302 seconds. 17) A) B) C) D) ) A physical fitness association is including the mile run in its secondary-school fitness test. The time for this event for boys in secondary school is known to possess a normal distribution with a mean of 470 seconds and a standard deviation of 50 seconds. Find the probability that a randomly selected boy in secondary school will take longer than 355 seconds to run the mile. 18) A) B) C) D) ) The length of time it takes college students to find a parking spot in the library parking lot follows a normal distribution with a mean of 6.5 minutes and a standard deviation of 1 minute. Find the probability that a randomly selected college student will find a parking spot in the library parking lot in less than 6.0 minutes. 19) A) B) C) D) ) The tread life of a particular brand of tire is a random variable best described by a normal distribution with a mean of 60,000 miles and a standard deviation of 2500 miles. What is the probability a particular tire of this brand will last longer than 57,500 miles? 20) A) B) C) D) SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 21) A firm believes the internal rate of return for its proposed investment can best be described by a normal distribution with mean 20% and standard deviation 3%. What is the probability that the internal rate of return for the investment will be at least 15.5%? 21) 3

4 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 22) A physical fitness association is including the mile run in its secondary-school fitness test. The time for this event for boys in secondary school is known to possess a normal distribution with a mean of 450 seconds and a standard deviation of 50 seconds. The fitness association wants to recognize the fastest 10% of the boys with certificates of recognition. What time would the boys need to beat in order to earn a certificate of recognition from the fitness association? 22) A) sec B) 386 sec C)514 sec D) sec 23) The amount of corn chips dispensed into a 16-ounce bag by the dispensing machine has been identified as possessing a normal distribution with a mean of 16.5 ounces and a standard deviation of 0.2 ounce. What chip amount represents the 67th percentile for the bag weight distribution? 23) A) oz B) oz C)16.09 oz D) oz 24) The length of time it takes college students to find a parking spot in the library parking lot follows a normal distribution with a mean of 4.5 minutes and a standard deviation of 1 minute. Find the cut-off time which 75.8% of the college students exceed when trying to find a parking spot in the library parking lot. 24) A) 5.3 min B) 5.0 min C)4.8 min D) 5.2 min Solve the problem. 25) A recent survey was conducted to compare the cost of solar energy to the cost of gas or electric energy. Results of the survey revealed that the distribution of the amount of the monthly utility bill of a 3-bedroom house using gas or electric energy had a mean of $115 and a standard deviation of $12. Three solar homes reported monthly utility bills of $75, $74, and $77. Which of the following statements is true? 25) A) Homes using solar power may have lower utility bills than homes using only gas and electricity. B) The utility bills for homes using solar power are about the same as those for homes using only gas and electricity. C)Homes using solar power always have lower utility bills than homes using only gas and electricity. D) Homes using solar power may actually have higher utility bills than homes using only gas and electricity. 4

5 26) On a given day, gasoline prices in the state of Colorado had a mean price of $2.20/gallon with a standard deviation of $0.09. A particular Colorado gas station had gasoline for $2.11/gallon. Interpret the z-score for this gas station. 26) A) The gas price of this Colorado station falls 9 standard deviations above the mean gas price of all Colorado stations. B) The gas price of this station falls 1 standard deviation above the mean gas price of all Colorado stations. C)The gas price of this station falls 1 standard deviation below the mean gas price of all Colorado stations. D) The gas price of this Colorado station falls 9 standard deviations below the mean gas price of all Colorado stations. SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 27) Test scores for a history class had a mean of 79 with a standard deviation of 4.5. Test scores for a physics class had a mean of 69 with a standard deviation of 3.7. One student earned a 82 on the history test and a 84 on the physics test. Calculate the z-score for each test. On which test did the student perform better? 27) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 28) The speeds of the fastballs thrown by major league baseball pitchers were measured by radar gun. The mean speed was 88 miles per hour. The standard deviation of the speeds was 6 mph. Which of the following speeds would be classified as an outlier? 28) A) 82 mph B) 107 mph C)97 mph D) 76 mph 29) At the U.S. Open Tennis Championship a statistician keeps track of every serve that a player hits during the tournament. The statistician reported that the mean serve speed of a particular player was 98 miles per hour (mph) and the standard deviation of the serve speeds was 9 mph. Using the z-score approach for detecting outliers, which of the following serve speeds would represent outliers in the distribution of the player's serve speeds? 29) Speeds: 67 mph, 107 mph, and 116 mph A) 67 and 107 are both outliers, but 116 is not. B) 67 is the only outlier. C)67, 107, and 116 are all outliers. D) None of the three speeds is an outlier. 30) The mean x of a data set is 36.71, and the sample standard deviation s is Find the interval representing measurements within one standard deviation of the mean. 30) A) (33.49, 39.93) B) (27.05, 46.37) C)(35.71, 37.71) D) (30.27, 43.15) 5

6 31) The following is a list of 25 measurements: 31) How many of the measurements fall within one standard deviation of the mean? A) 18 B) 13 C)25 D) 16 32) A standardized test has a mean score of 500 points with a standard deviation of 100 points. Five students' scores are shown below. 32) Adam: 575 Beth: 690 Carlos: 750 Doug: 280 Ella: 440 Which of the students have scores within two standard deviations of the mean? A) Adam, Beth, Carlos, Ella B) Carlos, Doug C) Adam, Beth D) Adam, Beth, Ella 33) The amount of television viewed by today's youth is of primary concern to Parents Against Watching Television (PAWT). 300 parents of elementary school-aged children were asked to estimate the number of hours per week that their child watches television. The mean and the standard deviation for their responses were 16 and 2, respectively. PAWT constructed a stem-and-leaf display for the data that showed that the distribution of times was a symmetric, mound-shaped distribution. Give an interval where you believe approximately 95% of the television viewing times fell in the distribution. 33) A) between 12 and 20 hours per week B) less than 20 C)less than 14 and more than 18 hours per week D) between 10 and 22 hours per week 34) A study was designed to investigate the effects of two variables (1) a student's level of mathematical anxiety and (2) teaching method on a student's achievement in a mathematics course. Students who had a low level of mathematical anxiety were taught using the traditional expository method. These students obtained a mean score of 380 with a standard deviation of 30 on a standardized test. Assuming a mound-shaped and symmetric distribution, in what range would approximately 68% of the students score? 34) A) below 410 B) between 350 and 410 C)below 350 and above 410 D) above ) The distribution of scores on a test is mound-shaped and symmetric with a mean score of 78. If 68% of the scores fall between 72 and 84, which of the following is most likely to be the standard deviation of the distribution? 35) A) 12 B) 2 C)6 D) 3 6

7 Answer Key Testname: CH5REVIEW NORMAL DIST 1) C 2) D 3) ) ) ) ) 8.6 ounces 8) B 9) A 10) A 11) A 12) C 13) B 14) D 15) C 16) C 17) C 18) B 19) D 20) C 21) Let x be the internal rate of return. Then x is a normal random variable with μ = 20% and σ = 3%. To determine the probability that x is at least 15.5%, we need to find the z-value for x = 15.5%. z = x - μ σ = = -1.5 P(x 15.5%) = P(-1.5 z) = P(-1.5 z 0) = = ) B 23) D 24) D 25) A 26) C 27) history z-score = 0.67; physics z-score = 4.05; The student performed better on the physics test. 28) B 29) B 30) A 31) D 32) D 33) A 34) B 35) C 7