Chapter 2 practice
Other
1. An internet reaction time test asks subjects to click their mouse button as soon as a light flashes on
the screen. The light is programmed to go on at a random time from 2 to 5 seconds after the subject
clicks “start.” Below is a density curve for the time at which the light goes on.
(a) Find the height of the curve (that is, find the value of “A”).
(b) Find the proportion of times that the light goes on between 2.5 and 4 seconds after the subject
clicks “start.”
2. A Normal probability plot for the weights of 40 squirrels trapped and released on a college campus
is shown below. Is the distribution of squirrel weights approximately Normal? Justify your answer.
3. A real estate company compiled data on the prices at which 35 homes sold during a one month in a
county in New Jersey. A histogram and some summary statistics from Minitab for the home prices
are given below. (Note that home prices are in thousands of dollars.)
Descriptive Statistics: home prices
Variable
N Mean SE Mean StDev Minimum Q1 Median Q3 Maximum
home prices 35 260.8 24.6
145.6 80.0
165.0 220.0 307.0 626.0
(a) One of the houses in this data set sold for 350 thousand dollars. Six houses sold for more than
that. Calculate and interpret the percentile and z-score for this house’s price.
(b) What was the “typical” price for a home in this county during the month in which these data
were collected? Justify your answer.
4. (a) Find the proportion of observations from a standard Normal distribution that satisfies 0.51 < Z <
2.84. Sketch the Normal curve and shade the area under the curve that is the answer to the question.
(b) What z-score in a Normal distribution has 22% of all scores above it?
5. The Program for International Student Assessment (PISA) conducts tests in mathematics in countries
throughout the world. For 15-year-olds in all developed countries, the scores in 2012 were Normally
distributed with a mean of 494 and standard deviation of 92.
(a) Make an accurate sketch of the distribution of PISA math scores. Be sure to provide a scale on
the horizontal axis.
(b) Use the 68–95–99.7 rule to find the proportion of scores between 402 and 678.
(c) The mean score for U.S. students was 481. What proportion of all scores were below the U.S.
mean?
(d) What proportion of all PISA scores were above 700?
(e) Calculate and interpret the 34th percentile of the distribution of PISA scores.
6. Twenty students were asked to guess the age of a man in a photograph. Here are their guesses:
44
43
48
37
44
40
33
42
43
41
50
49
43
46
46
45
43
38
39
41
Are these guesses approximately Normally distributed? Provide evidence to support your answer.
7.The scores of a reference population on the Wechsler Intelligence Scale for Children (WISC) are
approximately Normally distributed with µ = 100 and s = 15.
(a) What score would represent the 50th percentile? Explain.
(b) A score in what range would represent the top 1% of the scores?
(c) What proportion of the reference population has WISC scores below 110?
(d) What proportion of the reference population has WISC scores between 80 and 110?
(e) What is the interquartile range of WISC scores for the reference population?
.
8. A researcher wishes to calculate the average height of patients suffering from a particular disease.
From patient records, the mean was computed to be 156 cm, with a standard deviation of 5 cm.
Further investigation reveals that the scale was misaligned, and that all readings are 2 cm too large,
for example, a patient whose height is really 180 cm was measured as 182 cm. Furthermore, the
researcher would like to work with statistics based on meters (1 meter = 100 centimeters). What
would be the revised values for the mean and standard deviation of the patients’ heights?
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
1. For the density curve shown above, which statement is true?
a. The area under the curve between 0 and 1 is 1.
b. The density curve is symmetric.
c. The density curve is skewed right.
d. The density curve is Normal.
e. None of the above is correct.
____
2. For the density curve shown above, which statement is true?
a. The mean and median are equal.
b. The mean is greater than the median.
c. The mean is less than the median.
d. The mean could be either greater than or less than the median.
e. None of the above is correct.
____
3. The distribution of the time it takes for different people to solve a certain crossword puzzle is
strongly skewed to the right, with a mean of 30 minutes and a standard deviation of 15 minutes. The
distribution of z-scores for those times is
a. Normally distributed, with mean 30 and standard deviation 15.
b. Skewed to the right, with mean 30 and standard deviation 15.
c. Normally distributed, with mean 0 and standard deviation 1.
d. Skewed to the right, with mean 0 and standard deviation 1.
e. Skewed to the right, but the mean and standard deviation cannot be determined
without more information.
____
4. The cumulative relative frequency graph at right shows the distribution of lengths (in centimeters) of
fingerlings at a fish hatchery. The interquartile range for this distribution is approximately:
a.
b.
c.
d.
e.
____
0.18 to 0.85 centimeters
5 to 7 centimeters
5.5 to 6.7 centimeters
1.2 centimeters
2 centimeters
5. A fire department in a rural county reports that its response time to fires is approximately Normally
distributed with a mean of 22 minutes and a standard deviation of 11.9 minutes. Approximately
what proportion of their response times is over 30 minutes?
a. 0.03
b. 0.21
c. 0.25
d. 0.75
e. 0.79
Suppose that 16-ounce bags of chocolate chip cookies are produced with weights that follow a
Normal distribution with mean weight 16.1 ounces and standard deviation 0.1 ounce.
____
6. For the distribution of cookie bags described above, approximately what percent of the bags will
likely be underweight (that is, less than 16 ounces)?
a.
b.
c.
d.
e.
10
16
32
64
None of the above
____
7. The plot shown at the right is a Normal probability plot for a set of test scores. Which statement is
true for these data?
a.
b.
c.
d.
e.
____
The data are clearly Normally distributed.
The data are approximately Normally distributed
The data are clearly skewed to the left.
The data are clearly skewed to the right.
There is insufficient information to determine the shape of the distribution.
8. Which of the following statements are true?
I.
II.
III.
a.
b.
c.
d.
e.
____
The area under a Normal curve is always 1, regardless of the mean and
standard deviation.
The mean is always equal to the median for any Normal distribution.
The interquartile range for any Normal curve extends from
to
.
I and II
I and III
II and III
I, II, and III
None of the above
9. The proportion of scores in a standard Normal distribution that are greater than 1.25 is closest to:
a. .1056
b. .1151
c. .1600
d. .8849
e. .8944
____ 10. At right is a cumulative relative frequency graph for the 48 racers who finished the grueling 50km
cross-country ski race at the 2010 Vancouver Olympics. Approximately what proportion of the
racers finished the race in more than 2.15 hours?
a.
b.
c.
d.
e.
0.17
0.40
0.45
0.50
0.55
____ 11. Referring to the above graph, the mean finish time is 2.164 hours and the standard deviation is 0.85
hours. The distribution is skewed right. What are the mean, standard deviation, and shape of the
distribution of z-scores of the same data?
a. Mean = 2.164, Standard deviation = 0.85, skewed right
b. Mean = 2.164, Standard deviation = 0.85, skewed left
c. Mean = 2.164, Standard deviation = 0.85, approximately normal
d. Mean = 0, Standard deviation = 1, skewed right
e. Mean = 0, Standard deviation = 1, approximately normal
____ 12. Which of these variables is least likely to have a Normal distribution?
a. Annual income for all 150 employees at a local high school
b. Lengths of 50 newly hatched pythons
c. Heights of 100 white pine trees in a forest
d. Amount of soda in 60 cups filled by an automated machine at a fast-food
restaurant
e. Weights of 200 of the same candy bar in a shipment to a local supermarket
____ 13. The proportion of observations from a standard Normal distribution that take values larger than is
about
a.
b.
c.
d.
e.
0.2266
0.7704
0.7734
0.7764
0.8023
____ 14. The density curve shown to the right takes the value 0.5 on the interval 0 + x + 2 and takes the value
0 everywhere else. What percent of the observations lie between 0.5 and 1.2?
a.
b.
c.
d.
e.
25%
35%
50%
68%
70%
____ 15. If a store runs out of advertised material during a sale, customers become upset, and the store loses
not only the sale but also goodwill. From past experience, a music store finds that the mean number
of CDs sold in a sale is 845, the standard deviation is 15, and a histogram of the demand is
approximately Normal. The manager is willing to accept a 2.5% chance that a CD will be sold out.
About how many CDs should the manager order for an upcoming sale?
a.
b.
c.
d.
e.
1295
1070
935
875
860
____ 16. If your score on a test is at the 60th percentile, you know that your score lies
a. below the first quartile.
b. between the first quartile and the median.
c. between the median and the third quartile.
d. above the third quartile.
e. There is not enough information to say where it lies relative to the quartiles.
____ 17. In some courses (but certainly not in an intro stats course!), students are graded on a “Normal
curve.” For example, students between 0 and 0.5 standard deviations above the mean receive a C+;
between 0.5 and 1.0 standard deviations above the mean receive a B –; between 1.0 and 1.5 standard
deviations above the mean receive a B; between 1.5 and 2.0 standard deviations above the mean
receive a B+, etc. The class average on an exam was 60 with a standard deviation of 10.Which of the
following is bounds for a B and the percent of students who will receive a B if the marks are actually
Normally distributed?
a. (65, 75), 24.17%
b. (65, 75), 12.08%
c. (70, 75), 18.38%
d. (70, 75), 9.19%
e. (70, 75), 6.68%
____ 18. The mean age (at inauguration) of all U.S. Presidents is approximately Normally distributed with a
mean of 54.6. Barack Obama was 47 when he was inaugurated, which is the 11th percentile of the
distribution. Which of the following is closest to the standard deviation of presidents’ ages?
a.
b.
c.
d.
e.
-9.20
-6.18
6.18
7.60
9.20
____ 19. Which of the following is true about all Normal distributions?
a. There is no area under the curve for z-scores greater than 3.49.
b. The area underneath the curve is equal to 1.
c. The points at which the curvature changes from “up” to “down” (the points of
inflection) are at the first and third quartiles of the distribution.
d. About 68% of the values of the variable are more than one standard deviation
away from the mean.
e. At the mean, the height of the curve is 0.5.
____ 20. The 16th percentile of a Normally distributed variable has a value of 25 and the 97.5th percentile has
a value of 40. Which of the following is the best estimate of the mean and standard deviation of the
variable?
a. Mean 32.5; Standard deviation 2.5
b. Mean 32.5; Standard deviation 5
c. Mean 32.5; Standard deviation 10
d. Mean 30; Standard deviation 2.5
e. Mean 30; Standard deviation 5
Chapter 2 practice
Answer Section
OTHER
1. ANS:
(a) Since the area under the curve must be 1, the height of the curve must be 1/3. (b) Area under
curve between 2.5 and 4 is (1/3)(4-2.5) = 0.5.
PTS: 1
2. ANS:
The normal probability plot is roughly linear, so the distribution of squirrel weights is approximately
Normal. (Some students may argue that there is slight skew to the right, based on the lowest two or
three values for squirrel weight.)
PTS: 1
3. ANS:
(a) The house that sold for $350 thousand has 28 out of 35 house prices below it, so it is at the 80th
percentile. The z-score for $350 thousand is (350 – 260.8)/145.6 = 0.61, which is 0.61 standard
deviations above the mean house price. (b) $220 thousand—the median is the best measure of the
center of a skewed distribution.
PTS: 1
4. ANS:
(a) Proportion = 0.3027. See graph below, right.
(b) z = 0.77.
PTS: 1
5. ANS:
(a) See graph below, left.
(b) 402 is 1 standard deviation below 494, so 68/2 or 34% of the scores are between 402 and 494.
678 is 2 standard deviation above the mean, so 95/2 or 47.5% of the scores are between 494 and 678.
Thus 34 + 47.5 = 81.5% of the scores are between 402 and 678. (c) z-score for 481 is (481-494)/92
= -0.14, which (by Table A) has a percentile of 0.4443, so about 44% of the scores are below 481.
(d) z-score for 700 is (700-494)/92 = 2.24, which (by Table A) has 1 – .9875 = 0.0125 or 1.25% of
the scores above it. (e) The 34th percentile corresponds to z = – 0.41. This corresponds to a score of
– 0.41(92) + 494 = 456. So about 34% of the PISA scores are below 456.
PTS: 1
6. ANS:
Answers may vary. Students may choose to provide a dotplot, boxplot, histogram, or normal
probability plot. See examples below. Since the Normal probability plot is roughly linear, the data
is approximately normal. The boxplot shows only slight skew to the left, and there is no strong skew
in the dotplot. None of the plots suggest strong departure from Normality.
PTS: 1
7. ANS:
(a) 50th percentile = median, which is equal to the mean in a symmetric distribution, so it’s 100. (b)
Top 1% is equivalent to 99th percentile, which corresponds to z = 2.33. This would be a score of
2.33(15) + 100 135 or higher. (c) . By Table A, this has 0.7486 or 74.86% of the scores below it.
(d) , which by Table A has 0.0918 or 9.18% or the scores below it. Using the answer to part (c), the
percentage of z-scores between – 1.33 and 0.67 is .7486 – .0918 = .6568 or 65.68%. (e) Q1 (the 25th
percentile) for a Normal distribution (from Table A) is at approximately z = –0.67. By symmetry, Q3
is at z = 0.67. These correspond to WISC scores of . Thus the interquartile range is approximately
20 points.
PTS: 1
8. ANS:
Mean = 0.01(156 – 2) = 1.54 meters; standard deviation = 0.01(5) = 0.05 meters.
PTS: 1
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A
C
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E
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