Population Parameters Use the same population of {4, 5, 9} from Example 1 in Section 64

Standard Error of the Mean The population of current statistics students has ages with mean and standard deviation . Samples of statistics students are randomly selected so that there are exactly 40 students in each sample. For each sample, the mean age is computed. What does the central limit theorem tell us about the distribution of those mean ages?

Small Sample Heights of adult females are normally distributed. Samples of heights of adult females, each of size n = 3 , are randomly collected and the sample means are found. Is it correct to conclude that the sample means cannot be treated as a normal distribution because the sample size is too small? Explain.

Notation The population of distances that adult females can reach forward is normally distributed with a mean of 60.5 cm and a standard deviation of 6.6 cm (from the Federal Aviation Administration). If samples of 36 adult females are randomly selected, what do x and x represent, and what are their values?

Lottery Numbers In each drawing for the Texas Pick 3 lottery, three digits between 0 and 9 inclusive are randomly selected. What is the distribution of the selected digits? If the mean is calculated for each drawing, can the distribution of the sample means be treated as a normal distribution?

Using the Central Limit Theorem. In Exercises 510, use this information about the overhead reach distances of adult females: = 205.5 cm , = 8.6 cm , and overhead reach distances are normally distributed (based on data from the Federal Aviation Administration). The overhead reach distances are used in planning assembly work stations a. If 1 adult female is randomly selected, find the probability that her overhead reach is less than 222.7 cm. b. If 49 adult females are randomly selected, find the probability that they have a mean overhead reach less than 207.0 cm

Using the Central Limit Theorem. In Exercises 510, use this information about the overhead reach distances of adult females: = 205.5 cm , = 8.6 cm , and overhead reach distances are normally distributed (based on data from the Federal Aviation Administration). The overhead reach distances are used in planning assembly work stations a. If 1 adult female is randomly selected, find the probability that her overhead reach is less than 196.9 cm. b. If 36 adult females are randomly selected, find the probability that they have a mean overhead reach less than 205.0 cm.

Using the Central Limit Theorem. In Exercises 510, use this information about the overhead reach distances of adult females: = 205.5 cm , = 8.6 cm , and overhead reach distances are normally distributed (based on data from the Federal Aviation Administration). The overhead reach distances are used in planning assembly work stationsa. If 1 adult female is randomly selected, find the probability that her overhead reach is greater than 218.4 cm. b. If 9 adult females are randomly selected, find the probability that they have a mean overhead reach greater than 204.0 cm. c. Why can the normal distribution be used in part (b), even though the sample size does not exceed 30?

Using the Central Limit Theorem. In Exercises 510, use this information about the overhead reach distances of adult females: = 205.5 cm , = 8.6 cm , and overhead reach distances are normally distributed (based on data from the Federal Aviation Administration). The overhead reach distances are used in planning assembly work stationsa. If 1 adult female is randomly selected, find the probability that her overhead reach is greater than 195.0 cm. b. If 25 adult females are randomly selected, find the probability that they have a mean overhead reach greater than 203.0 cm. c. Why can the normal distribution be used in part (b), even though the sample size does not exceed 30?

Using the Central Limit Theorem. In Exercises 510, use this information about the overhead reach distances of adult females: = 205.5 cm , = 8.6 cm , and overhead reach distances are normally distributed (based on data from the Federal Aviation Administration). The overhead reach distances are used in planning assembly work stationsa. If 1 adult female is randomly selected, find the probability that her overhead reach is between 179.7 cm and 231.3 cm. b. If 40 adult females are randomly selected, find the probability that they have a mean overhead reach between 204.0 cm and 206.0 cm.

Using the Central Limit Theorem. In Exercises 510, use this information about the overhead reach distances of adult females: = 205.5 cm , = 8.6 cm , and overhead reach distances are normally distributed (based on data from the Federal Aviation Administration). The overhead reach distances are used in planning assembly work stationsa. If 1 adult female is randomly selected, find the probability that her overhead reach is between 180.0 cm and 200.0 cm. b. If 50 adult females are randomly selected, find the probability that they have a mean overhead reach between 198.0 cm and 206.0 cm

Elevator Safety Example 2 referred to an Ohio elevator with a maximum capacity of 2500 lb. When rating elevators, it is common to use a 25% safety factor, so the elevator should actually be able to carry a load that is 25% greater than the stated limit. The maximum capacity of 2500 lb becomes 3125 lb after it is increased by 25%, so 16 male passengers can have a mean weight of up to 195.3 lb. If the elevator is loaded with 16 male passengers, find the probability that it is overloaded because they have a mean weight greater than 195.3 lb. (As in Example 2, assume that weights of males are normally distributed with a mean of 182.9 lb and a standard deviation of 40.8 lb.) Does this elevator appear to be safe?

Elevator Safety Exercise 11 uses 182.9 lb , which is based on Data Set 1 in Appendix B. Repeat Exercise 11 using 174 lb (instead of 182.9 lb), which is the assumed mean weight that was commonly used just a few years ago. Assuming that the mean weight of males is now 182.9 lb, not the value of 174 lb that was used just a few years ago, what do you conclude about the effect of using an outdated mean that is substantially lower than it should be?

Designing Hats Women have head circumferences that are normally distributed with a mean of 22.65 in. and a standard deviation of 0.80 in. (based on data from the National Health and Nutrition Examination Survey). a. If the Hats by Leko company produces women's hats so that they fit head circumferences between 21.00 in. and 25.00 in., what percentage of women can fit into these hats? b. If the company wants to produce hats to fit all women except for those with the smallest 2.5% and the largest 2.5% head circumferences, what head circumferences should be accommodated? continued c. If 64 women are randomly selected, what is the probability that their mean head circumference is between 22.00 in. and 23.00 in.? If this probability is high, does it suggest that an order for 64 hats will very likely fit each of 64 randomly selected women? Why or why not?

Designing Manholes According to the web site www.torchmate.com, manhole covers must be a minimum of 22 in. in diameter, but can be as much as 60 in. in diameter. Assume that a manhole is constructed to have a circular opening with a diameter of 22 in. Men have shoulder breadths that are normally distributed with a mean of 18.2 in. and a standard deviation of 1.0 in. (based on data from the National Health and Nutrition Examination Survey). a. What percentage of men will fit into the manhole? b. Assume that the Connecticut Light and Power company employs 36 men who work in manholes. If 36 men are randomly selected, what is the probability that their mean shoulder breadth is less than 18.5 in.? Does this result suggest that money can be saved by making smaller manholes with a diameter of 18.5 in.? Why or why not?

Water Taxi Safety Passengers died when a water taxi sank in Baltimore's Inner Harbor. Men are typically heavier than women and children, so when loading a water taxi, assume a worstcase scenario in which all passengers are men. 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). The water taxi that sank had a stated capacity of 25 passengers, and the boat was rated for a load limit of 3500 lb. a. Given that the water taxi that sank was rated for a load limit of 3500 lb, what is the mean weight of the passengers if the boat is filled to the stated capacity of 25 passengers? b. If the water taxi is filled with 25 randomly selected men, what is the probability that their mean weight exceeds the value from part (a)? c. After the water taxi sank, the weight assumptions were revised so that the new capacity became 20 passengers. If the water taxi is filled with 20 randomly selected men, what is the probability that their mean weight exceeds 175 lb, which is the maximum mean weight that does not cause the total load to exceed 3500 lb? d. Is the new capacity of 20 passengers safe?

Loading M&M Packages M&M plain candies have a mean weight of 0.8565 g and a standard deviation of 0.0518 g (based on Data Set 20 in Appendix B). The M&M candies used in Data Set 20 came from a package containing 465 candies, and the package label stated that the net weight is 396.9 g. (If every package has 465 candies, the mean weight of the candies must exceed 396.9 / 465 = 0.8535 g for the net contents to weigh at least 396.9 g.) a. If 1 M&M plain candy is randomly selected, find the probability that it weighs more than 0.8535 g. b. If 465 M&M plain candies are randomly selected, find the probability that their mean weight is at least 0.8535 g. c. Given these results, does it seem that the Mars Company is providing M&M consumers with the amount claimed on the label?

Gondola Safety A ski gondola in Vail, Colorado, carries skiers to the top of a mountain. It bears a plaque stating that the maximum capacity is 12 people or 2004 lb. That capacity will be exceeded if 12 people have weights with a mean greater than 2004 / 12 = 167 lb . Because men tend to weigh more than women, a worstcase scenario involves 12 passengers who are all men. 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. Find the probability that if an individual man is randomly selected, his weight will be greater than 167 lb b. Find the probability that 12 randomly selected men will have a mean weight that is greater than 167 lb (so that their total weight is greater than the gondola maximum capacity of 2004 lb). c. Does the gondola appear to have the correct weight limit? Why or why not?

Pulse Rates of Women Women have pulse rates that 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. Find the percentiles P 1 and P 99 . b. Dr. Puretz sees exactly 25 female patients each day. Find the probability that 25 randomly selected women have a mean pulse rate between 70 beats per minute and 85 beats per minute. c. If Dr. Puretz wants to select pulse rates to be used as cutoff values for determining when further tests should be required, which pulse rates are better to use: the results from part (a) or the pulse rates of 70 beats per minute and 85 beats per minute from part (b)? Why?

Redesign of Ejection Seats When women were allowed to become pilots of fighter jets, engineers needed to redesign the ejection seats because they had been originally designed for men only. The ACESII ejection seats were designed for men weighing between 140 lb and 211 lb. Weights of women are now normally distributed with a mean of 165.0 lb and a standard deviation of 45.6 lb (based on Data Set 1 in Appendix B). a. If 1 woman is randomly selected, find the probability that her weight is between 140 lb and 211 lb. b. If 36 different women are randomly selected, find the probability that their mean weight is between 140 lb and 211 lb. c. When redesigning the fighter jet ejection seats to better accommodate women, which probability is more relevant: the result from part (a) or the result from part (b)? Why?

Loading a Tour Boat The Ethan Allen tour boat capsized and sank in Lake George, New York, and 20 of the 47 passengers drowned. Based on a 1960 assumption of a mean weight of 140 lb for passengers, the boat was rated to carry 50 passengers. After the boat sank, New York State changed the assumed mean weight from 140 lb to 174 lb. a. Given that the boat was rated for 50 passengers with an assumed mean of 140 lb, the boat had a passenger load limit of 7000 lb. Assume that the boat is loaded with 50 male passengers, and 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). Find the probability that the boat is overloaded because the 50 male passengers have a mean weight greater than 140 lb. b. The boat was later rated to carry only 14 passengers, and the load limit was changed to 2436 lb. If 14 passengers are all males, find the probability that the boat is overloaded because their mean weight is greater than 174 lb (so that their total weight is greater than the maximum capacity of 2436 lb). Do the new ratings appear to be safe when the boat is loaded with 14 male passengers?

Doorway Height The Boeing 757200 ER airliner carries 200 passengers and has doors with a height of 72 in. Heights of men are normally distributed with a mean of 69.5 in. and a standard deviation of 2.4 in. (based on Data Set 1 in Appendix B). a. If a male passenger is randomly selected, find the probability that he can fit through the doorway without bending. b. If half of the 200 passengers are men, find the probability that the mean height of the 100 men is less than 72 in. c. When considering the comfort and safety of passengers, which result is more relevant: the probability from part (a) or the probability from part (b)? Why? d. When considering the comfort and safety of passengers, why are women ignored in this case?

Loading Aircraft Before every flight, the pilot must verify that the total weight of the load is less than the maximum allowable load for the aircraft. The Bombardier Dash 8 aircraft can carry 37 passengers, and a flight has fuel and baggage that allows for a total passenger load of 6200 lb. The pilot sees that the plane is full and all passengers are men. The aircraft will be overloaded if the mean weight of the passengers is greater than 6200 lb / 37 = 167.6 lb . What is the probability that the aircraft is overloaded? Should the pilot take any action to correct for an overloaded aircraft? 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).

Correcting for a Finite Population In a study of babies born with very low birth weights, 275 children were given IQ tests at age 8, and their scores approximated a normal distribution with = 95.5 and = 16.0 (based on data from Neurobehavioral Outcomes of Schoolage Children Born Extremely Low Birth Weight or Very Preterm, by Anderson et al., Journal of the American Medical Association, Vol. 289, No. 24). Fifty of those children are to be randomly selected for a followup study. a. When considering the distribution of the mean IQ scores for samples of 50 children, should x be corrected by using the finite population correction factor? Why or why not? What is the value of x ? b. Find the probability that the mean IQ score of the followup sample is between 95 and 105.

Correcting for a Finite Population The Orange County Spa began with 300 members. Those members had weights with a distribution that is approximately normal with a mean of 177 lb and a standard deviation of 40 lb. The facility includes an elevator that can hold up to 16 passengers. a. When considering the distribution of sample means from weights of samples of 16 passengers, should x be corrected by using the finite population correction factor? Why or why not? What is the value of x ? b. If the elevator is designed to safely carry a load of up to 3000 lb, what is the maximum safe mean weight of passengers when the elevator is loaded with 16 passengers? c. If the elevator is filled with 16 randomly selected club members, what is the probability that the total load exceeds the safe limit of 3000 lb? Is this probability low enough? d. What is the maximum number of passengers that should be allowed if we want at least a 0.999 probability that the elevator will not be overloaded when it is filled with randomly selected club members?

Population Parameters Use the same population of {4, 5, 9} from Example 1 in Section 64. Assume that samples of size n = 2 are randomly selected without replacement. a. Find and s for the population. b. After finding all samples of size n = 2 that can be obtained without replacement, find the population of all values of x by finding the mean of each sample of size n = 2 . c. Find the mean x and standard deviation x for the population of sample means found in part (b). d. Verify that x = and x = n N n N 1