AP Statistics Chapter 2

m = 41.43 minutes and s = 17.86 minutes.

An airline flies the same route at the same time each day. The flight time varies according to a Normal distribution with unknown mean and stan- dard deviation. On 15% of days, the flight takes more than an hour. On 3% of days, the flight lasts 75 minutes or more. Use this information to determine the mean and standard deviation of the flight time distribution.

0.84 = x − 266/16 gives x = 279.44. The longest 20% of pregnancies last longer than 279.47 days.

The length of human pregnancies from conception to birth varies according to a distribution that is approximately Normal with mean 266 days and standard deviation 16 days. How long do the longest 20% of pregnancies last?

$2.90, $0.30

When Sam goes to a restaurant, he always tips the server $2 plus 10% of the cost of the meal. If Sam's distribution of meal costs has a mean of $9 and a stan- dard deviation of $3, what are the mean and standard deviation of the distribution of his tips?

4.6

A different species of cockroach has weights that follow a Normal distribution with a mean of 50 grams. After measuring the weights of many of these cockroaches, a lab assistant reports that 14% of the cockroaches weigh more than 55 grams. Based on this report, what is the approximate standard deviation of weights for this species of cockroaches?

0.50 - 0.37/0.04 = 3.25 P(x > 0.5) = P(z > 3.25) = 1 - 0.9994 = 0.0006 = 0.06%

An important measure of the performance of a locomotive is its "adhesion," which is the locomotive's pulling force as a multiple of its weight. The adhesion of one 4400-horsepower diesel locomotive varies in actual use according to a Normal distribution with mean m = 0.37 and standard deviation s = 0.04. An adhesion greater than 0.50 for the locomotive will result in a problem because the train will arrive too early at a switch point along the route. On what proportion of days will this happen?

0.30-0.37/0.04 = -1.75 P(x>= 0.30) = P(z > -1.75) = 1 - {(z < -1.75) = 1 -0.0401 = 0.9599 = 95.99%

An important measure of the performance of a locomotive is its "adhesion," which is the locomotive's pulling force as a multiple of its weight. The adhesion of one 4400-horsepower diesel locomotive varies in actual use according to a Normal distribution with mean m = 0.37 and standard deviation s = 0.04. For a certain small train's daily route, the locomotive needs to have an adhesion of at least 0.30 for the train to arrive at its destination on time. On what proportion of days will this happen?

Mean = 9 (25) + 32 = 77°F and standard deviation = 95 5 (2) = 3.6°F

Coach Ferguson uses a thermometer to measure the temperature (in degrees Celsius) at 20 different locations in the school swimming pool. An analysis of the data yields a mean of 25°C and a standard deviation of 2°C. Find the mean and stan- dard deviation of the temperature readings in degrees Fahrenheit (recall that °F = (9/5)°C + 32).

z = 240 − 266/16 = −1.63 and z= 270 − 266/1616 = 0.25. From Table A, the proportion of z-scores between −1.63 and 0.25 is 0.5987 − 0.0516 = 0.5471. About 55% of pregnancies last between 240 and 270 days.

The length of human pregnancies from conception to birth varies according to a distribution that is approximately Normal with mean 266 days and standard deviation 16 days. What percent of pregnancies last between 240 and 270 days (roughly between 8 months and 9 months)?

0.30 + (-2.05)(0.04) = 0.30 + 0.082 = 0.382

The locomotive's manufacturer is considering two changes that could reduce the percent of times that the train arrives late. One option is to increase the mean adhesion of the locomotive. The other possibility is to decrease the variability in adhesion from trip to trip, that is, to reduce the standard deviation. If the standard deviation remains at s = 0.04, to what value must the manufacturer change the mean adhesion of the locomotive to reduce its proportion of late arrivals to only 2% of days?

Bill

George has an average bowling score of 180 and bowls in a league where the average for all bowlers is 150 and the standard deviation is 20. Bill has an average bowling score of 190 and bowls in a league where the average is 160 and the standard deviation is 15. Who ranks higher in his own league, George or Bill?

between median and third quartile

Jorge's score on Exam 1 in his statistics class was at the 64th percentile of the scores for all students. His score falls

Subtract 68 from everybody's score, then divide these values by 15

Mr. Olsen uses an unusual grading system in his class. After each test, he transforms the scores to have a mean of 0 and a standard deviation of 1. Mr. Olsen then assigns a grade to each student based on the transformed score. On his most recent test, the class's scores had a mean of 68 and a standard deviation of 15. What transformations should he apply to each test score? Explain.

Cumulative Frequency

Number of individuals in all classes up to this point

Relative Frequency

Percent of data the class contains

95th percentile

Scores on the Wechsler Adult Intelligence Scale (a standard IQ test) for the 20 to 34 age group are approximately Normally distributed with m = 110 and s = 25. At what percentile is an IQ score of 150?

P(125 < x < 250) = P(0.60 < z < 1.60) = P(z<160() - P(z < 0.60) = 0.9452 - 0.7257 = 0.2195 = 21.95%

Scores on the Wechsler Adult Intelligence Scale (a standard IQ test) for the 20 to 34 age group are approximately Normally distributed with m = 110 and s = 25. What percent of people aged 20 to 34 have IQs between 125 and 150?

z = 240 − 266/16 = − 1.63, the proportion of z scores less than -1.63 is 0.0516. About 5% of pregnancies last less than 240 days, so 240 days is at the 5th percentile of pregnancy lengths.

The length of human pregnancies from conception to birth varies according to a distribution that is approximately Normal with mean 266 days and standard deviation 16 days. At what percentile is a pregnancy that lasts 240 days (that's about 8 months)?

Multiply each score by 4 and add 27

The scores on Ms. Martin's statistics quiz had a mean of 12 and a standard deviation of 3. Ms. Martin wants to transform the scores to have a mean of 75 and a standard deviation of 12. What transformations should she apply to each test score? Explain.

z = 4.05 − 3.98 = 3.5. From Table A, 0.02 the proportion of z-scores above 3.50 is approximately 0. Approximately, 0% large lids are too big to fit.

The supplier is considering two changes to reduce the percent of its large-cup lids that are too small to 1%. One strategy is to adjust the mean diameter of its lids. Another option is to alter the production process, thereby decreasing the standard deviation of the lid diameters. If the mean diameter stays at m=3.98 inches,what value of the standard deviation will result in only 1% of lids that are too small to fit?

z = 3.95 − 3.98/0.02 = −1.5 From Table A, the proportion of z-scores below −1.5 is 0.0668. About 7% of the large lids are too small to fit.

The supplier is considering two changes to reduce the percent of its large-cup lids that are too small to 1%. One strategy is to adjust the mean diameter of its lids. Another option is to alter the production process, thereby decreasing the standard deviation of the lid diameters. If the standard deviation remains at s = 0.02 inches, at what value should the supplier set the mean diameter of its large-cup lids so that only 1% are too small to fit?

No. If it was Normal, then the minimum value should be around 2 or 3 standard deviations below the mean. However, the actual minimum has a z-score of just z = −1.09. Also, if the distri- bution was Normal, the minimum and maximum should be about the same distance from the mean. However, the maximum is much farther from the mean (20,209) than the minimum (8741).

We collected data on the tuition charged by colleges and universities in Michigan. Here are some numerical summaries for the data: Mean Std. Dev. Min Max 10614 8049 1873 30823 Based on the relationship between the mean, stan- dard deviation, minimum, and maximum, is it rea- sonable to believe that the distribution of Michigan tuitions is approximately Normal? Explain.

Frequency

number of individuals in each class

Percentile

numerical measures that give the relative statistical position (Location) of a data value relative to the entire data set

Relative Cumulative Frequency

percent of the data in all classes up to this point