Download Distributions Sample Test

Sample Test # 3
Multiple Choice
Identify the letter of the choice that best completes the statement or answers the question.
____
____
1. The heights of men, in centimetres, are normally distributed with a mean of 175 and a standard deviation of
20. If Mario is 180 cm tall, what percent of men is he taller than?
a. 0.62%
c. 59.87%
b. 40.13%
d. 99.38%
2. The masses of bolts made in a plant are normally distributed. Bolts will be rejected if their z-scores are
____
3.
____
4.
____
5.
____
6.
____
7.
____
greater than 2.15 or less than 2.10. What percent of bolts will be rejected?
a. 0%
c. 1.79%
b. 1.58%
d. 3.37%
The heights, in centimetres, of the 700 female students in a high school are normally distributed with a mean
of 158 and a standard deviation of 6. How many of these students have a height between 151 cm and 165 cm?
a. 527
c. 530
b. 528
d. 531
The men that shop for clothes at the Hard-to-Fit Shoppe are typically unusually short or unusually tall. The
distribution of their heights is likely to be
a. U-shaped
c. mound-shaped
b. uniform
d. skewed
The distributions of men’s masses and women’s masses are both mound-shaped. But since men tend to have
greater masses than women, the distribution of the masses of all adults will likely be
a. bimodal
c. right-skewed
b. assymetric
d. left-skewed
How many of the numbers in the set below are within two standard deviations of the mean?
0, 2, 4, 6, 7, 8
a. 3
c. 6
b. 4
d. 8
Which of the following measures of spread could be negative?
a. the range
c. the standard deviation
b. the IQR
d. none of the above
8. If X~N(12.4, ) and 95% of the data lie in the interval 11.8–13.0 the equals
a. 1.2
c. 0.09
b. 0.3
d. 0.06
____ 9. The variable has a normal distribution with 99.7% of the area under its curve falling symmetrically between x
= 50 and x = 170. Its mean and standard deviation are respectively
a. 110 and 20
c. 120 and 20
b. 110 and 202
d. 120 and 202
____ 10. A university accepts only applicants scoring in the top 16% on an entrance test. Each year the test scores are
normally distributed with a standard deviation of 30. What is the highest value that the mean can have for
Fred to be accepted with a score of 520?
a. 460
c. 520
b. 490
d. 550
Short Answer
11. Calculate the z-score, to one decimal place, of x = 7.2 if
= 8.1 and
= 3.
12. Suppose X~N(50, 42 ). What value of x would have a z-score of 2.10?
13. The maximum attention spans, in seconds, of 40 two-year-olds are normally distributed with a mean of 75
and a standard deviation of 10. How many of these toddlers will have an attention span of less than 71 s?
14. How many peaks are there in a perfectly U-shaped distribution?
15. What type of distribution has an assymetrical histogram?
16. “Some of you did well, some of you did not do so well, but most of you have a mark near the middle.” What
shape of distribution would these marks likely have?
17. Find the range of the following data set: 2, 7, 0, 1, 5, 9
18. What is the interquartile range for the following average January temperatures (in /C): 2.0, 1.5, 0.2, 1.7,
2.6, 4.1, 4.1, 5.0, 7.2, 8.1, 8.1
19. The variance of a set of data is 13.2. Find the standard deviation to one decimal place.
20. Find the standard deviation, to one decimal place, of the IQ scores listed below:
IQ
Frequency
80
3
90
4
100
8
110
3
120
2
21. Use the standard deviation to determine which bowling scores were more consistent and state the standard
deviations of those scores to the nearest decimal place.
Onorio
Marianne
22. X~N(12.2,
170
161
165
172
181
148
174
163
149
178
) and 99.7% of data fall within the interval 11.6–12.8. What is the standard deviation?
23. An internet site has an average of 2500 hits per day. The number of daily hits is approximately normally
distributed with a standard deviation of 220. For how many of the next 50 days would you expect the site to
receive fewer than 2280 hits?
24. A traffic study showed that vehicle speeds on a particular highway were normally distributed with a mean of
102.5 km/h and a standard deviation of 5.1 km/h. What percent of vehicles had a speed between 92.3 km/h and
107.6 km/h?
Problem
25. What is the approximate range of a set of data that has been displayed in a histogram with 10 bars each with
width 0.5?
26. The distribution of the ages of people at a Punctured Lung concert was mound-shaped with eighteen-year-olds
representing the largest age group in attendance. Fred attended the concert with his ten year old son. Why did
he not see many people near either of their ages at the Lung concert?
27. Why is the mode, but not the mean or median an appropriate measure of central tendency for qualitative data?
28. A set of seven numbers is arranged from smallest to largest. The IQR is zero and the third number is 14. Is it
possible to determine the median? Explain.
29. The masses of largemouth bass, in kilograms, are normally distributed with a mean of 1.1 and a standard
deviation of 0.3. Any bass caught with a mass between 0.8 kg and 1.7 kg must be released. What percent of the
bass does this represent?
30. Suppose X~N(2, 32 ). Which set of data contains a greater percent of the data?
a) x
1 together with x
31. For X~N( ,
5 or b) 2
x
5?
), approximately 44% of the data are less than 2.9. Find the value of , to one decimal place.
32. Why do normal z-score tables typically not include z values greater than 2.99?
33. What are the mean and standard deviation of the standard normal distribution?
34. How many of the 250 perch netted by a fisherman would you expect to have a mass more than 400 g, if the
distribution of their masses in approximately normal with a mean of 300 g and a standard deviation of 80 g?
35. The masses of Burgerville’s half-pounders are normally distributed. Of these burgers, 33% have masses greater
then 253.52 g and 40.9% of them have masses less than 248.16 g. Find the mean and standard deviation of the
half-pounder masses.
36. Outliers are data values much larger or smaller than most of the other data values. Where would any intervals
containing outliers occur for each of these distributions: U-shaped, uniform, mound-shaped, right-skewed, leftskewed?
37. The noon-day temperature (in /C) in Chillsville last winter can be represented by the normal distribution
X~N( 6, 32 ). How often was it above 0/C at noon in Chillsville last winter?
38. Construct a suitable histogram for the following temperatures (in /C) on the TI-83+.
1, 0, 1, 1, 2, 3, 4, 4, 4, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 10, 11
39. Explain how measures of spread are used to measure consistency.
40. If each number in a data set is multiplied by a constant m, then the standard deviation should also be multiplied
by m. Verify this statement for the data sets below:
x: 0, 2, 4, 10
y: 0, 6, 12, 30
41. State the mean, standard deviation, and variance if X~N(4, 32 ).
Sample Test # 3
Answer Section
MULTIPLE CHOICE
1. ANS:
OBJ:
TOP:
2. ANS:
OBJ:
TOP:
3. ANS:
LOC:
4. ANS:
OBJ:
TOP:
5. ANS:
OBJ:
TOP:
6. ANS:
LOC:
7. ANS:
LOC:
8. ANS:
LOC:
9. ANS:
LOC:
10. ANS:
LOC:
C
REF: Knowledge and Understanding
3.5 Applying the Normal Distribution: Z-Scores
LOC: ST3.03
Tools for Analyzing Data
D
REF: Knowledge and Understanding
3.5 Applying the Normal Distribution: Z-Scores
LOC: ST3.03
Tools for Analyzing Data
D
REF: Application OBJ: 3.5 Applying the Normal Distribution: Z-Scores
ST3.03
TOP: Tools for Analyzing Data
A
REF: Knowledge and Understanding
3.1 Graphical Displays of Information
LOC: ST3.01
Tools for Analyzing Data
A
REF: Knowledge and Understanding
3.1 Graphical Displays of Information
LOC: ST3.01
Tools for Analyzing Data
C
REF: Knowledge and Understanding
OBJ: 3.3 Measures of Spread
ST2.01
TOP: Tools for Analyzing Data
D
REF: Application OBJ: 3.3 Measures of Spread
ST2.02
TOP: Tools for Analyzing Data
B
REF: Knowledge and Understanding
OBJ: 3.4 Normal Distribution
ST3.02
TOP: Tools for Analyzing Data
A
REF: Application OBJ: 3.4 Normal Distribution
ST3.02
TOP: Tools for Analyzing Data
B
REF: Application OBJ: 3.4 Normal Distribution
ST3.02
TOP: Tools for Analyzing Data
SHORT ANSWER
11. ANS:
The z-score is 0.3.
REF: Knowledge and Understanding
OBJ: 3.5 Applying the Normal Distribution: Z-Scores
LOC: ST2.03
TOP: Tools for Analyzing Data
12. ANS:
The value of x would be 58.4.
REF: Knowledge and Understanding
OBJ: 3.5 Applying the Normal Distribution: Z-Scores
LOC: ST3.03
TOP: Tools for Analyzing Data
13. ANS:
Of these toddlers, 14 will have an attention span of less than 71 s.
REF:
Application
OBJ: 3.5 Applying the Normal Distribution: Z-Scores
LOC: ST3.03
TOP: Tools for Analyzing Data
14. ANS:
The number of peaks in a perfectly U-shaped distribution is 2.
REF: Knowledge and Understanding
OBJ: 3.1 Graphical Displays of Information
LOC: ST3.01
TOP: Tools for Analyzing Data
15. ANS:
A skewed distribution has an assymetrical histogram.
REF: Knowledge and Understanding
OBJ: 3.1 Graphical Displays of Information
LOC: ST3.01
TOP: Tools for Analyzing Data
16. ANS:
These marks would likely have a distribution that has a mound-shape.
REF: Knowledge and Understanding
OBJ: 3.1 Graphical Displays of Information
LOC: ST3.01
TOP: Tools for Analyzing Data
17. ANS:
The range is 10.
REF: Knowledge and Understanding
OBJ: 3.3 Measures of Spread
LOC: ST2.01
TOP: Tools for Analyzing Data
18. ANS:
The average January temperature is 7.0ºC.
REF: Knowledge and Understanding
OBJ: 3.3 Measures of Spread
LOC: ST2.01
TOP: Tools for Analyzing Data
19. ANS:
The standard deviation is 3.6.
REF: Knowledge and Understanding
OBJ: 3.3 Measures of Spread
LOC: ST2.01
TOP: Tools for Analyzing Data
20. ANS:
The standard deviation is 11.5.
REF: Knowledge and Understanding
OBJ: 3.3 Measures of Spread
LOC: ST2.01
TOP: Tools for Analyzing Data
21. ANS:
Marianne’s bowling scores were more consistent. The standard deviation for her scores is 10.2.
REF: Knowledge and Understanding
OBJ: 3.3 Measures of Spread
LOC: ST2.02
TOP: Tools for Analyzing Data
22. ANS:
The standard deviation is 0.2.
REF: Knowledge and Understanding
OBJ: 3.4 Normal Distribution
LOC: ST3.02
TOP: Tools for Analyzing Data
23. ANS:
The site would receive fewer than 2280 hits on 8 of the next 50 days.
REF: Application OBJ: 3.4 Normal Distribution
LOC: ST3.02
TOP: Tools for Analyzing Data
24. ANS:
The percent of vehicles with a speed between 91.3 km/h and 107.6 km/h is 81.5%.
REF:
TOP:
Application OBJ: 3.4 Normal Distribution
Tools for Analyzing Data
LOC: ST3.02
PROBLEM
25. ANS:
Bin width = range
0.5 = range
number intervals
10
Therefore, the range is 5.
REF: Knowledge and Understanding
OBJ: Chapter 3 Prob
LOC: ST3.01
TOP: Tools for Analyzing Data
26. ANS:
Mound-shaped distributions have decreased frequencies on either side of the interval with greatest frequency.
So there would be few people much younger or older than 18 at the concert.
REF: Communication
OBJ: Chapter 3 Prob
LOC: ST3.01
TOP: Tools for Analyzing Data
27. ANS:
The mean and median need the data to have properties of numbers to be evaluated, for example, the capacity to
be added and divided, and order. The mode can be evaluated with any type of data since only frequency is
required.
REF: Communication
OBJ: Chapter 3 Prob
LOC: ST2.01
TOP: Tools for Analyzing Data
28. ANS:
Yes, it is possible to determine the median.
IQR = Q3
=0
Q1
Q1 = Q3
Since the numbers are arranged in increasing order, this means that all of the numbers between Q1 and Q3,
inclusive, are equal. But Q1 is the second number and Q3 is the sixth number, so this includes the third number
and the fourth number which is the median. Therefore, the median equals 14.
REF: Thinking/Inquiry/PS
OBJ: Chapter 3 Prob
LOC: ST2.01
TOP: Tools for Analyzing Data
29. ANS:
0.8 is one standard deviation below the mean and 1.7 is two standard deviations above the mean. Therefore, the
percent released is 68% + 13.5% = 81.5%.
REF:
Application
OBJ: Chapter 3 Prob
LOC: ST3.02
TOP: Tools for Analyzing Data
30. ANS:
The first set consists of data not within one standard deviation of the mean, so it contains 32% of the data. The
second set consists of the data within one standard deviation above the mean, and it contains 34% of the data.
Therefore, the set in b) contains a greater percent of the data.
REF: Knowledge and Understanding
OBJ: Chapter 3 Prob
LOC: ST3.02
TOP: Tools for Analyzing Data
31. ANS:
If P(Z < z)
0.44, then z = 0.15
REF: Knowledge and Understanding
OBJ: Chapter 3 Prob
LOC: ST3.03
TOP: Tools for Analyzing Data
32. ANS:
Almost all of the data in a normal distribution are within three standard deviations of the mean. Thus, data with
z-scores of three or greater represent a negligible proportion of the total.
REF: Communication
OBJ: Chapter 3 Prob
LOC: ST3.03
TOP: Tools for Analyzing Data
33. ANS:
The mean = 0.
The standard deviation = 1.
REF: Knowledge and Understanding
OBJ: 3.5 Applying the Normal Distribution: Z-Scores
LOC: ST3.02
TOP: Tools for Analyzing Data
34. ANS:
Therefore, 26 fish should have a mass more than 400 g.
REF: Application
LOC: ST3.03
35. ANS:
For Z~N(0, 1):
OBJ: 3.5 Applying the Normal Distribution: Z-Scores
TOP: Tools for Analyzing Data
If
, then
And If
so
, then
.
.
So
and
Solving this linear system gives
g and
g.
REF: Thinking/Inquiry/PS
OBJ: 3.5 Applying the Normal Distribution: Z-Scores
LOC: ST3.03
TOP: Tools for Analyzing Data
36. ANS:
The intervals would appear on either the extreme right or extreme left side and have a small frequency, assuming
that the bin width is not too great. Therefore, for U-shaped and uniform distributions there would not be outlier
intervals. For mound-shaped distributions, the intervals containing outliers would occur on both the left and right
sides of the distribution. For right-skewed distributions, the intervals containing outliers would be on the right
side and for left-skewed distributions they would be on the left side.
REF: Thinking/Inquiry/PS
OBJ: 3.1 Graphical Displays of Information
LOC: ST3.01
TOP: Tools for Analyzing Data
37. ANS:
x > 0 = –6 + 2(3), so a positive temperature is more than 2 standard deviations above the mean. Therefore, this
would have occured on 2.5% of the days.
REF: Application OBJ: 3.4 Normal Distribution
TOP: Tools for Analyzing Data
38. ANS:
LOC: ST3.02
REF: Application OBJ: 3.1 Graphical Displays of Information
LOC: STV.02
TOP: Tools for Analyzing Data
39. ANS:
If a set of data has a small measure of spread, then the data are relatively close to each other and therefore
consistent. Conversely, if the measure of spread is large, then the data are more dispersed and therefore not as
consistent.
REF: Communication
OBJ: 3.3 Measures of Spread
LOC: ST2.02
TOP: Tools for Analyzing Data
40. ANS:
m=3
and
and
REF: Knowledge and Understanding
OBJ: 3.3 Measures of Spread
LOC: ST2.01
TOP: Tools for Analyzing Data
41. ANS:
The mean = 4.
The standard deviation = 3.
The variance = 9.
REF: Knowledge and Understanding
OBJ: 3.4 Normal Distribution
LOC: ST3.02
TOP: Tools for Analyzing Data