Deck 6: The Normal Distribution

Question 1

Provide an appropriate response.
A normal probability plot is given below for a sample of scores on an aptitude test. Use the
plot to assess the normality of scores on this test. Explain your reasoning.

Answer 1

Since the normal probability plot is roughly linear, it appears that scores on this test
are approximately normally distributed.

Question 2

Provide an appropriate response.
On the same axes sketch normal distributions with

Answer 2

Responses can vary, but a correct response should resemble the following: where
Distribution a is represented by the solid-line curve,
Distribution b is represented by the dashed-line curve on the right, and
Distribution c is represented by the dashed-line curve on the left.

Question 3

Provide an appropriate response.
Which is larger, the area under the standard normal curve between -1 and 1, or the area
under the standard normal curve between 0 and 2? Explain your reasoning.

Answer 3

The area under the standard normal curve
between -1 and 1 is larger. The area under the curve between 0 and 1 is common to
both areas. The area under the curve between -1 and 0 is larger than the area under
the curve between 1 and 2 because it is closer to the mean.

Question 4

Construct a normal probability plot of the given data.
The resting heart rates from a group of 9 men before starting a workout program are given
below.

Question 5

Construct a normal probability plot of the given data.
The prices per gallon (in dollars)of regular unleaded gasoline at twelve service stations are
given below.

Question 6

Provide an appropriate response.
Scores on an aptitude test are normally distributed with a mean of 400 and a standard
deviation of 60. Explain how you would find any given percentile.

Question 7

Provide an appropriate response.
A variable is normally distributed with a mean of 100 and a standard deviation of 10.
Which is larger, the percentage of observations between 80 and 90 or the percentage of
observations between 120 and 130? Explain your reasoning.

Question 8

Provide an appropriate response.
Suppose that you know the area under the standard normal curve between 1 and 3 and the
area under the standard normal curve to the left of 3. Without further consulting a table of
areas, how could you find the area under the standard normal curve to the left of 1?
Explain your reasoning by using a sketch of the standard normal curve.

Question 9

Provide an appropriate response.
A student wished to use a table of areas for the standard normal curve to find the z-score
having an area to its right of 0.52. The student started by looking for the closest area to 0.52
in the body of the table and reading off the corresponding z-score which was 0.05. She
then subtracted this z-score from 1 to get 0.95. Was her reasoning correct? If not, where did
she go wrong and how would you have solved the problem?

Question 10

Provide an appropriate response.
Suppose that scores on a test are normally distributed with a mean of 80 and a standard
deviation of 8. Read the two questions below.
A. What is the 90th percentile?
B. What percentage of students score less than 90?
Explain the difference between the two questions. Describe how the method for solving A
would differ from the the method for solving B. Be sure to include in your explanation a
description of how the table of areas would be used in each case.

Question 11

Construct a normal probability plot of the given data.
The weekly incomes (in dollars)of a sample of 12 nurses working at a Los Angeles hospital
are given below.

Question 12

Provide an appropriate response.
According to data from the American Medical Association, 10% of us are left-handed.
Suppose a group of 500 people is randomly selected. You wish to find the probability that
at least 80 are left-handed. Describe the characteristics of this problem which help you to
recognize that the problem is about a binomial distribution that you are to solve by
estimating with the normal distribution. (Assume that you would not use a computer, a
table, or the binomial probability formula.)

Question 13

Provide an appropriate response.
A normal probability plot is given below for a sample of scores on an aptitude test. Use the
plot to identify outliers, if any. Explain your reasoning.

Question 14

Provide an appropriate response.
In assessing the normality of data, why is a normal probability plot especially
advantageous for small samples?

Question 15

Provide an appropriate response.
Fill in the blanks.

Question 16

Provide an appropriate response.
Fill in the blanks.

Question 17

Provide an appropriate response.
How does the standard normal distribution differ from a nonstandard normal distribution?
Why is it necessary to standardize in order to find percentages for nonstandard normal
variables?

Question 18

Provide an appropriate response.
Sketch a standard normal curve and shade the area to the right of the z-score 1.6.

Question 19

Provide an appropriate response.
Generally, the normal probability plot for a data set must be roughly linear in order to
assume that the variable is approximately normally distributed. Should this rule be
interpreted more strictly for small data sets or for large data sets? Explain.

Question 20

Provide an appropriate response.
Two random variables are normally distributed with the same mean. One has a standard
deviation of 10 while the other has a standard deviation of 15. How will the graphs of the
two variables differ and how will they be alike?

Question 21

Provide an appropriate response.
A normal probability plot is given below for the lifetimes (in hours)of a sample of batteries
of a particular brand. Use the plot to assess the normality of the lifetimes of these batteries.
Explain your reasoning.

Question 22

Provide an appropriate response.
Suppose that you know the area under the standard normal curve to the right of -2. How
could you use this to find the area under the standard normal curve to the left of 2? Explain
your reasoning.

Question 23

Provide an appropriate response.
A variable is normally distributed. 42% of the possible observations of the variable lie
between 20 and 28. What information does this give you about the graph of the normal
curve for this variable?

Question 24

Provide an appropriate response.
In assessing the normality of a data set, why is it easier to interpret a normal probability
plot than it is to interpret a histogram?

Question 25

Provide an appropriate response.
A normal probability plot is given below for the weekly incomes (in dollars)of a sample of
engineers in one town. Use the plot to identify outliers, if any. Explain your reasoning.

Question 26

Provide an appropriate response.
Suppose that you know the area under the standard normal curve to the right of 2 and the
area under the standard normal curve to the right of 1. Without further consulting a table
of areas, how could you find the area under the standard normal curve between 1 and 2?
Explain your reasoning.

Question 27

Provide an appropriate response.
A variable is normally distributed with a mean of 100 and a standard deviation of 10. A
student wanted to find the percentage of observations of the variable lying between 106
and 110. What is wrong with his solution?

Question 28

Provide an appropriate response.
A curve has area 0.296 to the left of 5 and area 0.642 to the right of 5. Could this curve be a
density curve for some variable? Explain your answer.

Question 29

Provide an appropriate response.
Under what conditions are you allowed to use the normal distribution to approximate the
binomial distribution? Under what conditions might you want to use the normal
distribution to approximate the binomial as opposed to using the binomial probability
formula, a table of binomial probabilities, or a computer?

Question 30

Provide an appropriate response.
Sketch a standard normal curve and shade the area between the z-scores -2.5 and -1.

Question 31

Provide an appropriate response.
Suppose that you know the area under the standard normal curve to the right of -1.7. How
could you use this to find the area under the standard normal curve to the right of 1.7?
Explain your reasoning.

Question 32

Provide an appropriate response.
Explain why a continuity correction factor is necessary when approximating the binomial
distribution by the normal distribution. Refer to the terms "discrete" and "continuous", and
draw a diagram to support your answer.

Question 33

Construct a normal probability plot of the given data.
The systolic blood pressure (in mmHg)is given below for a sample of 12 men aged
between 60 and 65.

Question 34

Construct a normal probability plot of the given data.
The final exam scores for 15 students in a statistics course are given below.

Question 35

Provide an appropriate response.
A normal probability plot is given below for a sample of scores on an aptitude test. Use the
plot to assess the normality of scores on this test. Explain your reasoning.

Question 36

Provide an appropriate response.
When a normal probability plot is constructed, which axis is used for the normal scores
(horizontal or vertical)?

Question 37

Provide an appropriate response.
Fill in the blanks.

Question 38

Provide an appropriate response.
Fill in the blanks.

Question 39

Provide an appropriate response.
A normal probability plot is given below for the weekly incomes (in dollars)of a sample of
engineers in one town. Use the plot to assess the normality of the incomes of engineers in
this town. Explain your reasoning.

Question 40

Provide an appropriate response.
The area under the standard normal curve to the right of a z-score is 0.56. Explain how
you could use a table of areas to find the z-score.

Question 41

Use the empirical rule to solve the problem.
The annual precipitation for one city is normally distributed with a mean of 35.6 inches and a standard deviation of 2.9 inches. Fill in the blanks.
In 99.74% of the years, the precipitation in this city is between ___ and ___ inches.

A)29.8, 35.6
B)35.6, 44.3
C)29.8, 41.4
D)26.9, 44.3

Question 42

Provide an appropriate response.
A normal probability plot is given below for the lifetimes (in hours)of batteries of a
particular type. Use the plot to identify outliers, if any. Explain your reasoning.

Question 43

Provide an appropriate response.
Suppose that scores on a test are normally distributed with a mean of 80 and a standard
deviation of 8. You have been asked to find the 70th percentile. After sketching a standard
normal curve and shading the area of interest, the next step in solving this problem is to
use the table of areas. Would you look for 0.7 in the body of the table or in the left-hand
column? Explain your reasoning.

Question 44

Provide an appropriate response. Assume that the variable under consideration has a density curve.
The area under the density curve that lies to the right of 12 is 0.405. What percentage of all possible observations of the variable are at most 12?

A)65.5%
B)59.5%
C)40.5%
D)34.5%

Question 45

Use a table of areas for the standard normal curve to find the required z-score.
Find the z-score having area 0.09 to its left under the standard normal curve.

A)-1.26
B)-1.34
C)-1.45
D)-1.39

Question 46

Provide an appropriate response.
A normal probability plot is given below for a sample of scores on an aptitude test. Use the
plot to identify outliers, if any. Explain your reasoning.

Question 47

Use a table of areas to find the specified area under the standard normal curve.
The area that lies between 0 and 3.01

A)0.4987
B)0.1217
C)0.9987
D)0.5013

Question 48

Solve the problem.
Frequency data were reported for the ages of full-time employees at a company. The age distribution is given in the table. Obtain a relative-frequency histogram of these data and
Determine whether the ages are approximately normally distributed.

Question 49

Fill in the blanks by standardizing the normally distributed variable.
Dave drives to work each morning at about the same time. His commute time is normally distributed with a mean of 35 minutes and a standard deviation of 5 minutes. The percentage of
Time that his commute time is less than 44 minutes is equal to the area under the standard normal
Curve that lies to the ___ of __.

A)right, 1.1
B)right, 1.8
C)left, 1.8
D)left, -1.8

Question 50

Estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution.
The probability that a radish seed will germinate is 0.7. Estimate the probability that of 140 randomly selected seeds, exactly 100 will germinate.

A)0.0769
B)0.9331
C)0.0679
D)0.0669

Question 51

Choose the one alternative that best completes the statement or answers the question.Use a table of areas for the standard normal curve to find the required z-score.
Find the z-score for which the area under the standard normal curve to its left is 0.40

A)0.25
B)-0.25
C)0.57
D)-0.57

Question 52

Solve the problem.

Question 53

Provide an appropriate response.
For a variable with a density curve, what is the relationship between the percentage of all
possible observations of the variable that lie within any specified range and the
corresponding area under its density curve?

Question 54

Provide an appropriate response.
True or false, the standard deviation of a normally distributed variable can be any real number.

Question 55

Find the indicated probability or percentage for the normally distributed variable.
The diameters of bolts produced by a certain machine are normally distributed with a mean of 0.30 inches and a standard deviation of 0.01 inches. What percentage of bolts will have a diameter
Greater than 0.32 inches?

A)97.72%
B)2.28%
C)47.72%
D)37.45%

Question 56

Provide an appropriate response.
Does the presence of an outlier in your data set necessarily mean that you cannot use the
normal model to interpret the data? Explain.

Question 57

Provide an appropriate response.
Use a sketch of the standard normal curve to explain the difference between z-scores and
areas under the standard normal curve. What are the possible values for an area and what
are the possible values for a z-score?

Question 58

Estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution.
A multiple choice test consists of 60 questions. Each question has 4 possible answers of which one is correct. If all answers are random guesses, estimate the probability of getting at least 20% correct.

A)0.3508
B)0.8508
C)0.1492
D)0.0901

Question 59

Provide an appropriate response. Assume that the variable under consideration has a density curve.
The area under the density curve that lies between 26 and 48 is 0.409. What percentage of all possible observations of the variable are either less than 26 or greater than 48?

A)65.9%
B)40.9%
C)59.1%
D)34.1%

Question 60

Solve the problem.
Frequency data were reported for the ages of women who became mothers during one year in a selected U.S. city. The age distribution is given in the table. Obtain a relative-frequency histogram
Of these data and determine whether the ages are approximately normally distributed.

A)No. The distribution is right-skewed.
B)Yes. The distribution is bell-shaped.
C)No. The distribution is J-shaped.
D)No. The distribution is left-skewed.

Question 61

Use a table of areas to obtain the shaded area under the standard normal curve.

Question 62

Use the empirical rule to solve the problem.
The amount of Jen's monthly phone bill is normally distributed with a mean of $73 and a standard deviation of $9. What percentage of her phone bills are between $46 and $100?

A)95.44%
B)99.74%
C)99.99%
D)68.26%

Question 63

Estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution.
With n = 20 and p = 0.60, estimate P(less than or equal to 8).

A)0.4953
B)0.4332
C)0.0548
D)0.0668

Question 64

Fill in the blanks by standardizing the normally distributed variable.
The amount of time that customers wait in line during peak hours at one bank is normally distributed with a mean of 14 minutes and a standard deviation of 3 minutes. The percentage of
Time that the waiting time exceeds 12 minutes is equal to the area under the standard normal curve
That lies to the ___ of ___.

A)left, -0.67
B)right, -0.67
C)left, 0.64
D)right, 0.67

Question 65

Find the indicated probability or percentage for the normally distributed variable.
The volumes of soda in quart soda bottles are normally distributed with a mean of 32.3 oz and a standard deviation of 1.2 oz. What is the probability that the volume of soda in a randomly selected
Bottle will be less than 32 oz?

A)0.5987
B)0.0987
C)0.4013
D)0.3821

Question 66

Use a table of areas to obtain the shaded area under the standard normal curve.

Question 67

Use a table of areas for the standard normal curve to find the required z-score.
Find the z-score for having area 0.09 to its right under the standard normal curve, that is, find

A)1.34
B)1.45
C)1.26
D)1.39

Question 68

Use the empirical rule to solve the problem.
The lifetimes of lightbulbs of a particular type are normally distributed with a mean of 400 hours and a standard deviation of 12 hours. What percentage of the bulbs have lifetimes that lie within 2
Standard deviations to either side of the mean?

A)97.72%
B)99.74%
C)95.44%
D)68.26%

Question 69

Use the empirical rule to solve the problem.
The systolic blood pressure of 18-year-old women is normally distributed with a mean of 120 mmHg and a standard deviation of 12 mmHg. What percentage of 18-year-old women have a
Systolic blood pressure that lies within 3 standard deviations to either side of the mean?

A)68.26%
B)99.99%
C)95.44%
D)99.74%

Question 70

Find the specified percentile, quartile, or decile.
The lifetimes of lightbulbs of a particular type are normally distributed with a mean of 263 hours and a standard deviation of 8 hours. Find the first quartile, Q1 .

A)257.64
B)268.36
C)265
D)261

Question 71

Use a table of areas for the standard normal curve to find the required z-score.
Find the z-score for having area 0.07 to its right under the standard normal curve, that is, find

A)1.39
B)1.45
C)1.48
D)1.26

Question 72

Solve the problem.
Frequency data were reported for the ages of women who became mothers during one year in a selected U.S. city. The age distribution is given in the table. Obtain a relative-frequency histogram
Of these data and determine whether the ages are approximately normally distributed.

A)No. The distribution is left-skewed.
B)No. The distribution is uniform.
C)Yes. The distribution is bell-shaped.
D)No. The distribution is right-skewed.

Question 73

Find the specified percentile, quartile, or decile.
At one college, GPAs are normally distributed with a mean of 2.8 and a standard deviation of 0.6. Find the third quartile,

A)3.25
B)2.398
C)3.1
D)3.202

Question 74

Provide an appropriate response.
The area under the standard normal curve between 1 and 2 is equal to 0.1359. Scores on a particular aptitude test are normally distributed with a mean of 100 and a standard deviation of 10. Which of
The following are equal to 13.59%?

Question 75

Solve the problem.

-Data were reported for household size (number of people in household)in a small community. The size distribution is given in the table. Obtain a relative-frequency histogram for these data and Determine whether the household sizes are approximately normally distributed. $\begin{array}{c|c}\begin{array}{c}\text { Size } \\\text { (\# of people) }\end{array} & \text { Frequency } \\\hline \text { 1-under 2 } & 4 \\\text { 2-under 3 } & 33 \\\text { 3-under 4 } & 30 \\\text { 4-under 5 } & 12 \\\text { 5-under 6 } & 6 \\\text { 6-under 7 } & 3\end{array}$

A)No. The distribution is right-skewed.
B)No. The distribution has outliers.
C)Yes. The distribution is bell-shaped.
D)No. The distribution is left-skewed.

Question 76

Solve the problem.
Frequency data were reported for the ages of women who became mothers during one year in a selected U.S. city. The age distribution is given in the table. Obtain a relative-frequency histogram
Of these data and determine whether the ages are approximately normally distributed.

A)No. The distribution is J-shaped.
B)No. The distribution is left-skewed.
C)Yes. The distribution is bell-shaped.
D)No. The distribution is right-skewed.

Question 77

Estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution.
A certain question on a test is answered correctly by 22% of the respondents. Estimate the probability that among the next 150 responses there will be at most 40 correct answers.

A)0.1003
B)0.0694
C)0.9306
D)0.8997

Question 78

Find the specified percentile, quartile, or decile.
Suppose that replacement times for washing machines are normally distributed with a mean of 11 years and a standard deviation of 1 years. Find the 82nd percentile.

A)10.1 years
B)11.2 years
C)11.1 years
D)11.9 years

Question 79

Solve the problem.
A southeastern college has an enrollment of 2951 female students. Records show that the mean height of these students is 64.7 inches and that the standard deviation is 2.3 inches. The table shows
Frequency and relative-frequency data for these heights. If you assume that the distribution of
Heights is approximately normal, then you can use the table to estimate areas under the associated
Normal curve (that is, under the normal curve that has parameters µ = 64.7 and Ϭ = 2.3). Making
This assumption, estimate the area under the associated normal curve to the left of 65.

A)0.4591
B)0.5407
C)0.3415
D)0.1782

Question 80

Solve the problem.
Frequency data were reported for the ages of full-time employees at a company. The age distribution is given in the table. Obtain a relative-frequency histogram of these data and
Determine whether the ages are approximately normally distributed.

A)No. The distribution is left-skewed.
B)No. The distribution is right-skewed.
C)Yes. The distribution is bell-shaped.
D)No. The distribution is uniform.