Solution: IQ Scores. In Exercises 1320, assume that adults have IQ scores that are

Pulse Rates Pulse rates of women are normally distributed with a mean of 77.5 beats per minute and a standard deviation of 11.6 beats per minute (based on Data Set 1 in Appendix B). a. What are the values of the mean and standard deviation after converting all pulse rates of women to z scores using z = ( x ) / ? b. The original pulse rates are measured with units of beats per minute. What are the units of the corresponding z scores?

IQ Scores The Wechsler Adult Intelligence Scale is an IQ score obtained through a test, and the scores are normally distributed with a mean of 100 and a standard deviation of 15. A bellshaped graph is drawn to represent this distribution. a. For the bellshaped graph, what is the area under the curve? b. What is the value of the median? c. What is the value of the mode? d. What is the value of the variance?

Normal Distributions What is the difference between a standard normal distribution and a nonstandard normal distribution?

Random Digits Computers are commonly used to randomly generate digits of telephone numbers to be called when conducting a survey. Can the methods of this section be used to find the probability that when one digit is randomly generated, it is less than 3? Why or why not? What is the probability of getting a digit less than 3?

IQ Scores. In Exercises 58, find the area of the shaded region. The graphs depict IQ scores of adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15 (as on the Wechsler test).

IQ Scores. In Exercises 58, find the area of the shaded region. The graphs depict IQ scores of adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15 (as on the Wechsler test).

IQ Scores. In Exercises 58, find the area of the shaded region. The graphs depict IQ scores of adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15 (as on the Wechsler test).

IQ Scores. In Exercises 58, find the area of the shaded region. The graphs depict IQ scores of adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15 (as on the Wechsler test).

IQ Scores. In Exercises 912, find the indicated IQ score, and round to the nearest whole number. The graphs depict IQ scores of adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15 (as on the Wechsler test).

IQ Scores. In Exercises 912, find the indicated IQ score, and round to the nearest whole number. The graphs depict IQ scores of adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15 (as on the Wechsler test).

IQ Scores. In Exercises 912, find the indicated IQ score, and round to the nearest whole number. The graphs depict IQ scores of adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15 (as on the Wechsler test).

IQ Scores. In Exercises 912, find the indicated IQ score, and round to the nearest whole number. The graphs depict IQ scores of adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15 (as on the Wechsler test).

IQ Scores. In Exercises 1320, assume that adults have IQ scores that are normally distributed with a mean of 100 and a standard deviation of 15 (as on the Wechsler test). For a randomly selected adult, find the indicated probability or IQ score. Round IQ scores to the nearest whole number. (Hint: Draw a graph in each case.) Find the probability of an IQ less than 85

IQ Scores. In Exercises 1320, assume that adults have IQ scores that are normally distributed with a mean of 100 and a standard deviation of 15 (as on the Wechsler test). For a randomly selected adult, find the indicated probability or IQ score. Round IQ scores to the nearest whole number. (Hint: Draw a graph in each case.)Find the probability of an IQ greater than 70 (the requirement for being a statistics textbook author)

IQ Scores. In Exercises 1320, assume that adults have IQ scores that are normally distributed with a mean of 100 and a standard deviation of 15 (as on the Wechsler test). For a randomly selected adult, find the indicated probability or IQ score. Round IQ scores to the nearest whole number. (Hint: Draw a graph in each case.)Find the probability that a randomly selected adult has an IQ between 90 and 110 (referred to as the normal range).

IQ Scores. In Exercises 1320, assume that adults have IQ scores that are normally distributed with a mean of 100 and a standard deviation of 15 (as on the Wechsler test). For a randomly selected adult, find the indicated probability or IQ score. Round IQ scores to the nearest whole number. (Hint: Draw a graph in each case.)Find the probability that a randomly selected adult has an IQ between 110 and 120 (referred to as bright normal).

IQ Scores. In Exercises 1320, assume that adults have IQ scores that are normally distributed with a mean of 100 and a standard deviation of 15 (as on the Wechsler test). For a randomly selected adult, find the indicated probability or IQ score. Round IQ scores to the nearest whole number. (Hint: Draw a graph in each case.)Find P 90 , which is the IQ score separating the bottom 90% from the top 10%.

IQ Scores. In Exercises 1320, assume that adults have IQ scores that are normally distributed with a mean of 100 and a standard deviation of 15 (as on the Wechsler test). For a randomly selected adult, find the indicated probability or IQ score. Round IQ scores to the nearest whole number. (Hint: Draw a graph in each case.)Find the first quartile Q 1 , which is the IQ score separating the bottom 25% from the top 75%

IQ Scores. In Exercises 1320, assume that adults have IQ scores that are normally distributed with a mean of 100 and a standard deviation of 15 (as on the Wechsler test). For a randomly selected adult, find the indicated probability or IQ score. Round IQ scores to the nearest whole number. (Hint: Draw a graph in each case.)Find the third quartile Q 3 , which is the IQ score separating the top 25% from the others

IQ Scores. In Exercises 1320, assume that adults have IQ scores that are normally distributed with a mean of 100 and a standard deviation of 15 (as on the Wechsler test). For a randomly selected adult, find the indicated probability or IQ score. Round IQ scores to the nearest whole number. (Hint: Draw a graph in each case.)Mensa Mensa International calls itself the international high IQ society, and it has more than 100,000 members. Mensa states that candidates for membership of Mensa must achieve a score at or above the 98th percentile on a standard test of intelligence (a score that is greater than that achieved by 98 percent of the general population taking the test). Find the 98th percentile for the population of Wechsler IQ scores. This is the lowest score meeting the requirement for Mensa membership

In Exercises 2124, use these parameters (based on Data Set 1 in Appendix B):Navy Pilots The U.S. Navy requires that fighter pilots have heights between 62 in. and 78 in. a. Find the percentage of women meeting the height requirement. Are many women not qualified because they are too short or too tall? b. Find the percentage of men meeting the height requirement. Are many men not qualified because they are too short or too tall? c. If the Navy changes the height requirements so that all women are eligible except the shortest 2% and the tallest 2%, what are the new height requirements for women? d. If the Navy changes the height requirements so that all men are eligible except the shortest 1% and the tallest 1%, what are the new height requirements for men?

In Exercises 2124, use these parameters (based on Data Set 1 in Appendix B):Air Force Pilots The U.S. Air Force requires that pilots have heights between 64 in. and 77 in. a. Find the percentage of women meeting the height requirement. b. Find the percentage of men meeting the height requirement. c. If the Air Force height requirements are changed to exclude only the tallest 3% of men and the shortest 3% of women, what are the new height requirements?

In Exercises 2124, use these parameters (based on Data Set 1 in Appendix B): Disney Characters Most of the live characters at Disney World have height requirements with a minimum of 4 ft 8 in. and a maximum of 6 ft 3 in. a. Find the percentage of women meeting the height requirement. b. Find the percentage of men meeting the height requirement. c. If the height requirements are changed to exclude only the tallest 5% of men and the shortest 5% of women, what are the new height requirements?

In Exercises 2124, use these parameters (based on Data Set 1 in Appendix B):Executive Jet Doorway The Gulfstream 100 is an executive jet that seats six, and it has a doorway height of 51.6 in. a. What percentage of adult men can fit through the door without bending? b. What percentage of adult women can fit through the door without bending? c. Does the door design with a height of 51.6 in. appear to be adequate? Why didn't the engineers design a larger door? d. What doorway height would allow 60% of men to fit without bending?

Water Taxi Safety When a water taxi sank in Baltimore's Inner Harbor, an investigation revealed that the safe passenger load for the water taxi was 3500 lb. It was also noted that the mean weight of a passenger was assumed to be 140 lb. Assume a worstcase scenario in which all of the passengers are adult men. (This could easily occur in a city that hosts conventions in which people of the same gender often travel in groups.) Assume that weights of men are normally distributed with a mean of 182.9 lb and a standard deviation of 40.8 lb (based on Data Set 1 in Appendix B). a. If one man is randomly selected, find the probability that he weighs less than 174 lb (the new value suggested by the National Transportation and Safety Board). b. With a load limit of 3500 lb, how many male passengers are allowed if we assume a mean weight of 140 lb? c. With a load limit of 3500 lb, how many male passengers are allowed if we use a new mean weight of 182.9 lb? d. Why is it necessary to periodically review and revise the number of passengers that are allowed to board?

Designing a Work Station A common design requirement is that an environment must fit the range of people who fall between the 5th percentile for women and the 95th percentile for men. In designing an assembly work table, we must consider sitting knee height, which is the distance from the bottom of the feet to the top of the knee. Males have sitting knee heights that are normallly distributed with a mean of 21.4 in. and a standard deviation of 1.2 in. females have sitting knee heights that are normally distributed with a mean of 19.6 in. and a standard deviation of 1.1 in. (based on data from the Department of Transportation). a. What is the minimum table clearance required to satisfy the requirement of fitting 95% of men? Why is the 5th percentile for women ignored in this case? b. The author is writing this exercise at a table with a clearance of 23.5 in. above the floor. What percentage of men fit this table, and what percentage of women fit this table? Does the table appear to be made to fit almost everyone?

Lengths of Pregnancies The lengths of pregnancies are normally distributed with a mean of 268 days and a standard deviation of 15 days. a. One classical use of the normal distribution is inspired by a letter to Dear Abby in which a wife claimed to have given birth 308 days after a brief visit from her husband, who was serving in the Navy. Given this information, find the probability of a pregnancy lasting 308 days or longer. What does the result suggest? b. If we stipulate that a baby is premature if the length of pregnancy is in the lowest 3%, find the length that separates premature babies from those who are not premature. Premature babies often require special care, and this result could be helpful to hospital administrators in planning for that care.

Body Temperatures Based on the sample results in Data Set 3 in Appendix B, assume that human body temperatures are normally distributed with a mean of 98.20 F and a standard deviation of 0.62 F . a. Bellevue Hospital in New York City uses 100.6 F as the lowest temperature considered to be a fever. What percentage of normal and healthy persons would be considered to have a fever? Does this percentage suggest that a cutoff of 100.6 F is appropriate? b. Physicians want to select a minimum temperature for requiring further medical tests. What should that temperature be, if we want only 5.0% of healthy people to exceed it? (Such a result is a false positive, meaning that the test result is positive, but the subject is not really sick.)

Earthquakes Based on Data Set 16 in Appendix B, assume that Richter scale magnitudes of earthquakes are normally distributed with a mean of 1.184 and a standard deviation of 0.587. a. Earthquakes with magnitues less than 2.000 are considered microearthquakes that are not felt. What percentage of earthquakes fall into this category? b. Earthquakes above 4.0 will cause indoor items to shake. What percentage of earhquakes fall into this category? c. Find the 95th percentile. Will all earthquakes above the 95th percentile cause indoor items to shake?

Aircraft Seat Width Engineers want to design seats in commercial aircraft so that they are wide enough to fit 99% of all males. (Accommodating 100% of males would require very wide seats that would be much too expensive.) Men have hip breadths that are normally distributed with a mean of 14.4 in. and a standard deviation of 1.0 in. (based on anthropometric survey data from Gordon, Clauser, et al.). Find P 99 . That is, find the hip breadth for men that separates the smallest 99% from the largest 1%

Chocolate Chip Cookies The Chapter Problem for Chapter 3 includes Table 31, which lists the numbers of chocolate chips in Chips Ahoy regular cookies. Those numbers have a distribution that is approximately normal with a mean of 24.0 chocolate chips and a standard deviation of 2.6 chocolate chips. Find P 1 and P 99 . How might those values be helpful to the producer of Chips Ahoy regular cookies?

Quarters After 1964, quarters were manufactured so that the weights had a mean of 5.67 g and a standard deviation of 0.06 g. Some vending machines are designed so that you can adjust the weights of quarters that are accepted. If many counterfeit coins are found, you can narrow the range of acceptable weights with the effect that most counterfeit coins are rejected along with some legitimate quarters. a. If you adjust vending machines to accept weights between 5.64 g and 5.70 g, what percentage of legal quarters are rejected? Is that percentage too high? b. If you adjust vending machines to accept all legal quarters except those with weights in the top 2.5% and the bottom 2.5%, what are the limits of the weights that are accepted?

Large Data Sets. In Exercises 33 and 34, refer to the data sets in Appendix B and use computer software or a calculator.Appendix B Data Set: Pulse Rates of Males Refer to Data Set 1 in Appendix B and use the pulse rates of males. a. Find the mean and standard deviation, and verify that the pulse rates have a distribution that is roughly normal. b. Treating the unrounded values of the mean and standard deviation as parameters, and assuming that male pulse rates are normally distributed, find the pulse rate separating the lowest 2.5% and the pulse rate separating the highest 2.5%. These values could be helpful when physicians try to determine whether pulse rates are unusually low or unusually high.

Large Data Sets. In Exercises 33 and 34, refer to the data sets in Appendix B and use computer software or a calculator.Appendix B Data Set: Weights of Diet Pepsi Refer to Data Set 19 in Appendix B and use the weights (pounds) of diet Pepsi.a. Find the mean and standard deviation, and verify that the data have a distribution that is roughly normal. b. Treating the unrounded values of the mean and standard deviation as parameters, and assuming that the weights are normally distributed, find the weight separating the lowest 0.5% and the weight separating the highest 0.5%. These values could be helpful when quality control specialists try to control the manufacturing process so that underweight or overweight cans are rejected.

Curving Test Scores A statistics professor gives a test and finds that the scores are normally distributed with a mean of 40 and a standard deviation of 10. She plans to curve the scores. a. If she curves by adding 35 to each grade, what is the new mean? What is the new standard deviation? b. Is it fair to curve by adding 35 to each grade? Why or why not? c. If the grades are curved so that grades of B are given to scores above the bottom 70% and below the top 10%, find the numerical limits for a grade of B. d. Which method of curving the grades is fairer: Adding 35 to each original score or using a scheme like the one given in part (c)? Explain.

Using Continuity Correction There are many situations in which a normal distribution can be used as a good approximation to a random variable that has only discrete values. In such cases, we can use this continuity correction: Represent each whole number by the interval extending from 0.5 below the number to 0.5 above it. The Chapter Problem for Chapter 3 includes Table 31, which lists the numbers of chocolate chips in Chips Ahoy regular cookies. Those numbers have a distribution that is approximately normal with a mean of 24.0 chocolate chips and a standard deviation of 2.6 chocolate chips. Find the percentage of such cookies having between 20 and 30 chocolate chips inclusive. a. Find the result without using the continuity correction. b. Find the result using the continuity correction. c. Does the use of the continuity correction make much of a difference in this case?

Outliers For the purposes of constructing modified boxplots as described in Section 34, outliers defined as data values that are above Q 3 by an amount greater than 1.5 IQR or below Q 1 by an amount greater than 1.5 IQR , where IQR is the interquartile range. Using this definition of outliers, find the probability that when a value is randomly selected from a normal distribution, it is an outlier

SAT and ACT Tests Based on recent results, scores on the SAT test are normally distributed with a mean of 1511 and a standard deviation of 312. Scores on the ACT test are normally distributed with a mean of 21.1 and a standard deviation of 5.1. Assume that the two tests use different scales to measure the same aptitude. a. If someone gets an SAT score that is in the 95th percentile, find the actual SAT score and the equivalent ACT score. b. If someone gets an SAT score of 2100, find the equivalent ACT score.