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LESSON 2.2 • Transforming Data 97

25. (a) About the 60th percentile.
(a) The median household income in North Dakota in (a) Judy has not taken a statistics class in a few years.

2018 was $66,505. Estimate North Dakota’s percentile. Explain in simple language what the standardized (b) About $57,500.
(b) Maine is at the 30th percentile of the distribution. score tells her about her bone density.

Estimate its median household income in 2018. (b) Use the information provided to calculate the stan- 26. (a) About the 40th percentile.

dard deviation of bone density in the population of (b) About 1000 hours.
26. Light life The cumulative relative frequency graph 25-year-old women.

describes the lifetimes (in hours) of 200 lamps. 8 27. (a) Judy’s hip bone density is 1.45 Lesson 2.2
100 Recycle and Review
standard deviations below the mean
80 28. Birthrates in Africa (1.6, 1.7, 1.8) One of the import- hip bone density (956 g/cm ) of all
2
Percentile 60 is the birthrate per 1000 individuals in a population. 25-year-old women. This means that
ant factors in determining population growth rates
Here are a dotplot and five-number summary for the
Judy’s hip bone density is lower
40
20 birthrates per 1000 individuals in 54 African nations. than that of most women her age.
d
0 d d d 948 −956
d d
(C) 2021 BFW Publishers -- for review purposes only.
500 700 900 1100 1300 1500 d d d d d d d (b) −1.45 = gives SD = 5.52.
d
d
d
dd
d d d d
Lifetimes (h) d d d d d dd dd dd dd d dd ddd d d d d d dd d d d SD
d dd
18 24 30 36 42 48 54
(a) Estimate the percentile for a lamp that lasted 900 hours. Birthrate (per 1000 population) The standard deviation of hip bone

(b) Estimate the 60th percentile of this distribution. density in the population of 25-year-old

Min Q 1 Med Q 3 Max

2

27. Medical exam results People with low bone density women is 5.52g/cm .
have a high risk of broken bones. Currently, the most 14 29 37.5 41 53
=
Q = 29,Med 37.5,
common method for testing bone density is dual- (a) Construct a boxplot for these data. 28. (a) Min14, 1 =

energy X-ray absorptiometry (DEXA). A patient Q = 41,Max 53, no outliers.
=
who undergoes a DEXA test usually gets bone den- (b) Suppose the maximum value of 53 was in error and 3
2
sity results in grams per square centimeter (g/cm) should have been 45. For each statistic, indicate whether
and in standardized units. this correction would result in an increase, a decrease,
Judy, who is 25 years old, has her bone density or no change. Justify your answer in each case.
measured using DEXA. Her results indicate a bone • Mean
2

density in the hip of 948 g/cm and a standardized • Median

score of =−z 1.45 In the population of 25-year-old
.

women like Judy, the mean bone density in the hip • Standard deviation 10 15 20 25 30 35 40 45 50 55

2 9

is 956 g/cm . • Interquartile range Birth rate (per 1000 population)

(b) The mean would decrease; when
adding all of the values, we will be adding
a smaller value of 45 rather than 53.
The median will not change; changing
Lesson 2.2 53 to 45 will still leave the value in the
upper 50% of the data and will leave the
Transforming Data middle number unchanged. The standard
deviation will decrease; changing 53 to
45 is bringing an extreme value closer to
the mean, which decreases the typical
L E AR N I N G TAR G E T S distance from the mean. The IQR will stay
the same; the values of Q 1 and Q 3 will not

• Describe the effect of adding or subtracting a constant on a distribution
of quantitative data. change when the data value changes
• Describe the effect of multiplying or dividing by a constant on a distribution from 53 to 45.

of quantitative data.
• Analyze the effect of adding or subtracting a constant and multiplying or

dividing by a constant on measures of center, location, and variability. LEARNING T AR GET KEY
The problems in the test bank are
keyed to the learning targets using
these numbers:
03_StarnesSPA4e_24432_ch02_088_153.indd 97 07/09/20 1:54 PM • 2.2.1
TEACHING TIP BELL RINGER
• 2.2.2
Lesson 2.2 can be safely skipped if you are Write your height (in inches) on the • 2.2.3
pressed for time. It does provide some of board, along with the rest of the class. Use
the theoretical justification for standardizing technology to create a dotplot and calculate
normal distributions in Lesson 2.5, but it is the mean and standard deviation. Now
not necessary. In the 90-day pacing guide, we compute and interpret the percentile and
omit Lesson 2.2. z-score for your height. Share your answers
with a partner.
LESSON 2.2 • Transforming Data 97
03_TysonTEspa4e_25177_ch02_088_153_4pp.indd 97 10/11/20 7:43 PM