Problem 11
A town's January high temperatures average \(36^{\circ} \mathrm{F}\) with a standard deviation of \(10^{\circ}\), while in July the mean high temperature is \(74^{\circ}\) and the standard deviation is \(8^{\circ} .\) In which month is it more unusual to have a day with a high temperature of \(55^{\circ} ?\) Explain.
Problem 12
An incoming freshman took her college's placement exams in French and mathematics. In French, she scored 82 and in math 86 . The overall results on the French exam had a mean of 72 and a standard deviation of 8 , while the mean math score was 68 , with a standard deviation of \(12 .\) On which exam did she do better compared with the other freshmen?
Problem 13
The first Stats exam had a mean of 65 and a standard deviation of 10 points; the second had a mean of 80 and a standard deviation of 5 points. Derrick scored an 80 on both tests. Julie scored a 70 on the first test and a 90 on the second. They both totaled 160 points on the two exams, but Julie claims that her total is better. Explain.
Problem 14
The first Stat exam had a mean of 80 and a standard deviation of 4 points; the second had a mean of 70 and a standard deviation of 15 points. Reginald scored an 80 on the first test and an 85 on the second. Sara scored an 88 on the first but only a 65 on the second. Although Reginald's total score is higher, Sara feels she should get the higher grade. Explain her point of view.
Problem 15
Anna, a language major, took final exams in both French and Spanish and scored 83 on each. Her roommate Megan, also taking both courses, scored 77 on the French exam and 95 on the Spanish exam. Overall, student scores on the French exam had a mean of 81 and a standard deviation of 5, and the Spanish scores had a mean of 74 and a standard deviation of 15 . a) To qualify for language honors, a major must maintain at least an 85 average for all language courses taken. So far, which student qualifies? b) Which student's overall performance was better?
Problem 16
Two companies market new batteries targeted at owners of personal music players. DuraTunes claims a mean battery life of 11 hours, while RockReady advertises 12 hours. a) Explain why you would also like to know the standard deviations of the battery lifespans before deciding which brand to buy. b) Suppose those standard deviations are 2 hours for DuraTunes and \(1.5\) hours for RockReady. You are headed for 8 hours at the beach. Which battery is most likely to last all day? Explain. c) If your beach trip is all weekend, and you probably will have the music on for 16 hours, which battery is most likely to last? Explain.
Problem 17
The Virginia Cooperative Extension reports that the mean weight of yearling Angus steers is 1152 pounds. Suppose that weights of all such animals can be described by a Normal model with a standard deviation of 84 pounds. a) How many standard deviations from the mean would a steer weighing 1000 pounds be? b) Which would be more unusual, a steer weighing 1000 pounds or one weighing 1250 pounds?
Problem 18
John Beale of Stanford, \(\mathrm{CA}\), recorded the speeds of cars driving past his house, where the speed limit read \(20 \mathrm{mph}\). The mean of 100 readings was \(23.84 \mathrm{mph}\), with a standard deviation of \(3.56 \mathrm{mph}\). (He actually recorded every car for a two-month period. These are 100 representative readings.) a) How many standard deviations from the mean would a car going under the speed limit be? b) Which would be more unusual, a car traveling \(34 \mathrm{mph}\) or one going \(10 \mathrm{mph}\) ?
Problem 19
Recall that the beef cattle described in Exercise 17 had a mean weight of 1152 pounds, with a standard deviation of 84 pounds. a) Cattle buyers hope that yearling Angus steers will weigh at least 1000 pounds. To see how much over (or under) that goal the cattle are, we could subtract 1000 pounds from all the weights. What would the new mean and standard deviation be? b) Suppose such cattle sell at auction for 40 cents a pound. Find the mean and standard deviation of the sale prices for all the steers.
Problem 24
A popular band on tour played a series of concerts in large venues. They always drew a large crowd, averaging 21,359 fans. While the band did not announce (and probably never calculated) the standard deviation, which of these values do you think is most likely to be correct: \(20,200,2000\), or 20,000 fans? Explain your choice.
