Download Name: Date: Block: AP Statistics – Chapter 2 – Density Curves

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AP Statistics – Chapter 2 – Density Curves & Normal Distribution – Study Guide
1.
(2.51) The scores of a reference population on the Wechsler intelligence Score for Children as “gifted” if
their WISC score exceeded 135. They are normally distributed with a mean of 100 and standard deviation
of 15. There are 1300 sixth-graders in the school district. About how many of them are gifted? Show
your work.
2. (2.53) Scores on the ACT test for the 2004 high school graduating class had mean 20.9 and standard
deviation 4.8. In all, 1,171,460 students in this class took the test, and 1,052,490 of them had scores of 27
or lower.
a. If the distribution of scores were normal, what percent of scores would be 27 or lower?
b. What percent of the actual score were 27 or lower?
c. Does the Normal distribution describe the actual data well?
3. (2.54) Joey received a report that he scored in the 97th on the national standardized reading test but in the
72nd percentile on the math portion of the test.
a. Explain to Joey’s grandmother, who knows no statistics, what these numbers mean.
b. Can we determine Joey’s z-score for his reading and math performance? Why or why not?
4. (2.55) The army reports that the distribution of head circumference among soldiers is approximately
Normal with mean 22.8 inches and standard deviation 1.1 inches. Helmets are mass-produced for all
except the smallest 5% and largest 5% of head sizes. Soldiers in the smallest or largest 5% get custommade helmets. What head sizes get custom-made helmets?
5. (2.57) A government report looked at the amount borrowed for college by students who graduated in 2000
and had taken out student loans. The mean amount was $17,776 and the standard deviation was $12,034.
The quartiles were Q1=$9900, M = $15,532, and Q3 = $22,500.
a. Compare the mean and the median M. Also compare the distances of Q1 and Q3 from the median.
What shape is the distribution? Interpret this in the context of student loans.
b. The right-skew increases the standard deviation. So a Normal distribution with the same mean and
standard deviation would have a third quartile larger than the actual third quartile. Find the third
quartile of the Normal Distribution with mean of $17,776 and standard deviation of $12,034 and
compare it with Q3 = $22,500.
6. (2.58) Osteoporosis is a condition in which the bones become brittle due to loss of minerals. To diagnose
osteoporosis, and elaborate apparatus measure bone mineral (BMD). BMD is usually reported in
standardized form. The standardization is based on a population of healthy young adults. The World
Health Organization (WHO) criterion for osteoporosis is a BMD score that is 2.5 standard deviations
below the mean for young adults. BMD measurements in a population of people similar in age and sex
roughly follow a Normal Distribution.
a. What percent of healthy young adults have osteoporosis by the WHO criterion?
b. Women aged 70 to 79 are of course not young adults. The mean BMD in this age group is about 2 on the standard scale for young adults. Suppose that the standard deviation is the same as for
young adults. What percent of this older population has osteoporosis?
7. (2.60) Many companies “grade on a bell curve” to compute the performance of their managers and
professional workers. This forces the use of some low performance ratings, so that not all workers are
listed as “above average”. Ford Motor Company’s “performance management process” for a time
assigned 10% A grades, 80% B grades and 10% C grades to the company’s 18,000 managers. Suppose
that Ford’s performance scores really are Normally distributed. This year, managers with scores less than
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25 received C’s and those with scores above 475 received A’s. What are the mean and standard deviation
of the scores? Show your work.
8. (2.59) The following displays Normal probability plots for four different sets of data. Describe what each
plot tells you about the Normality of the given data set.
Answers:
1. About 13
2. 89.8% of scores would fall below this level.
This agrees well with the actual data.
3a. Joey scored as well as or better than 97% of
alls students who took the reading tests.
Joey scored as well as or better than 72% of
all students who took the math test.
3b. No, unless we knew that the distribution of
scores was Normal, and we do not.
4. Custom-made helmets will be needed for
soldiers with head circumferences less than
21 inches or greater than 24.6 inches
(approximately).
5a. The mean is greater than the median. Q3 is
further from the mean than Q1.
5b. Normal Distribution Q3: $35,889. This is
larger than the actual value of Q3.
6a. About .6%
6b. About 31%
7. Mean = 250, Standard Deviation = 175.8
8a. Reasonably Normal
8b. Data are Normal
8c. Data skewed right with several outliers.
8d. The graph shows three clusters or mounds.
The flat section in the lower left and upper
right illustrate that the data have peaks at the
extremes.