5.11 Scores on the Wechsler Adult Intelligence Scale (WAIS) approximate a normal curve with a mean of 100 and a standard deviation of 15. What proportion of IQ scores are
(a) above Kristen’s 125?
(b) below 82?
(c) within 9 points of the mean? (
d) more than 40 points from the mean?
5.13 IQ scores on the WAIS test approximate a normal curve with a mean of 100 and a standard deviation of 15. What IQ score is identiﬁed with
(a) the upper 2 percent, that is, 2 percent to the right (and 98 percent to the left)?
(b) the lower 10 percent?
(c) the upper 60 percent?
(d) the middle 95 percent? [Remember, the middle 95 percent straddles the line perpendicular to the mean (or the 50th percentile), with half of 95 percent, or 47.5 percent, above this line and the remaining 47.5 percent below this line.]
(e) the middle 99 percent?
Finding Proportions and Scores
IMPORTANT NOTE : When doing Questions 5. 15 and 5.16, remember to decide ﬁrst whether a proportion or a score is to be found.
*5. 15 An investigator polls common cold sufferers, asking them to estimate the number of hours of physical discomfort caused by their most recent colds. Assume that their estimates approximate a normal curve with a mean of 83 hours and a standard deviation of 20 hours.
(a) What is the estimated number of hours for the shortest suffering 5 percent?
(b) What proportion of sufferers estimate that their colds lasted longer than 48 hours?
(c) What proportion suffered for fewer than 61 hours?
(d) What is the estimated number of hours suffered by the extreme 1 percent either above or below the mean?
(e) What proportion suffered for between 1 and 3 days, that is, between 24 and 72 hours?
(f) What is the estimated number of hours suffered by the middle 95 percent? [See the comment about “middle 95 percent” in Question 5.13(d) .]
(g) What proportion suffered for between 2 and 4 days?
(h) A medical researcher wishes to concentrate on the 20 percent who suffered the most. She will work only with those who estimate that they suffered for more than hours.
(i) Another researcher wishes to compare those who suffered least with those who suffered most. If each group is to consist of only the extreme 3 percent, the mild group will consist of those who suffered for fewer than hours, and the severe group will consist of those who suffered for more than hours.
(j) Another survey found that people with colds who took daily doses of vitamin C suffered, on the average, for 61 hours. What proportion of the original survey (with a mean of 83 hours and a standard deviation of 20 hours) suffered for more than 61 hours?
(k) What proportion of the original survey suffered for exactly 61 hours? (Be careful!)
*5.18 The body mass index (BMI) measures body size in people by dividing weight (in pounds) by the square of height (in inches) and then multiplying by a factor of 703. A BMI less than 18.5 is deﬁned as underweight; between 18.5 to 24.9 is normal; between 25 and 29.9 is overweight; and 30 or more is obese. It is well-established that Americans have become heavier during the last half century. Assume that the positively skewed distribution of BMIs for adult American males has a mean of 28 with a standard deviation of 4.
(a) Would the median BMI score exceed, equal, or be exceeded by the mean BMI score of 28?
(b) What z score deﬁnes overweight?
(c) What z score deﬁnes obese?
8.10 Television stations sometimes solicit feedback volunteered by viewers about a televised event. Following a televised debate between Barack Obama and Mitt Romney in the 2012 U.S. presidential election campaign, a TV station conducted a telephone poll to determine the “winner.” Callers were given two phone numbers, one for Obama and the other for Romney, to register their opinions automatically.
(a) Comment on whether or not this was a random sample.
(b) How might this poll have been improved?
*8.14 The probability of a boy being born equals .50, or 1 / 2 , as does the probability of a girl being born. For a randomly selected family with two children, what’s the probability of
(a) two boys, that is, a boy and a boy? (Reminder: Before using either the addition or multiplication rule, satisfy yourself that the various events are either mutually exclusive or independent, respectively.)
(b) two girls?
(c) either two boys or two girls?
POPULATIONS, SAMPLES, AND PROBABILITY
8.16 A traditional test for extra-sensory perception (ESP) involves a set of playing cards, each of which shows a different symbol (circle, square, cross, star, or wavy lines). If C represents a correct guess and I an incorrect guess, what is the probability of
(b) CI (in that order) for two guesses?
(c) CCC for three guesses?
(d) III for three guesses?
8.19 A sensor is used to monitor the performance of a nuclear reactor. The sensor accurately reﬂects the state of the reactor with a probability of .97. But with a probability of .02, it gives a false alarm (by reporting excessive radiation even though the reactor is performing normally), and with a probability of .01, it misses excessive radiation (by failing to report excessive radiation even though the reactor is performing abnormally).
(a) What is the probability that a sensor will give an incorrect report, that is, either a false alarm or a miss?
(b) To reduce costly shutdowns caused by false alarms, management introduces a second completely independent sensor, and the reactor is shut down only when both sensors report excessive radiation. (According to this perspective, solitary reports of excessive radiation should be viewed as false alarms and ignored, since both sensors provide accurate information much of the time.) What is the new probability that the reactor will be shut down because of simultaneous false alarms by both the ﬁrst and second sensors?
(c) Being more concerned about failures to detect excessive radiation, someone who lives near the nuclear reactor proposes an entirely different strategy: Shut down the reactor whenever either sensor reports excessive radiation. (According to this point of view, even a solitary report of excessive radiation should trigger a shutdown, since a failure to detect excessive radiation is potentially catastrophic.) If this policy were adopted, what is the new probability that excessive radiation will be missed simultaneously by both the ﬁrst and second sensors?
*8.21 Assume that the probability of breast cancer equals .01 for women in the 50–59 age group. Furthermore, if a woman does have breast cancer, the probability of a true positive mammogram (correct detection of breast cancer) equals .80 and the probability of a false negative mammogram (a miss) equals .20. On the other hand, if a woman does not have breast cancer, the probability of a true negative mammogram (correct nondetection) equals .90 and the probability of a false positive mammogram (a false alarm) equals .10. (Hint: Use a frequency analysis to answer questions. To facilitate checking your answers with those in the book, begin with a total of 1,000 women, then branch into the number of women who do or do not have breast cancer, and ﬁnally, under each of these numbers, branch into the number of women with positive and negative mammograms.)
(a) What is the probability that a randomly selected woman will have a positive mammogram?
(b) What is the probability of having breast cancer, given a positive mammogram?
(c) What is the probability of not having breast cancer, given a negative mammogram?