Deck 6: Normal Probability Distributions

Question 1

The heights of adult females are normally distributed. If you were to construct a histogram of 40 randomly selected women, what shape would the histogram of those heights have and what pattern would you expect in
A normal quantile plot of these data?

A) The histogram would by non-symmetric, and the normal quantile plot would have data points that would be non-linear.
B) The histogram would be approximately bell-shaped, and the normal quantile plot would have data points have follow a straight-line pattern.
C) The histogram would by approximately bell-shaped, and the normal quantile plot would have data points that would be bell-shaped.
D) The histogram would by approximately bell-shaped, and the normal quantile plot would have data points that would be non-linear.

Answer 1

B

Question 2

Answer 2

A

Question 3

If a sample size is < ________, the sample size must come from a population having a normal distribution in order to follow normal distribution calculations.

A) 100
B) 50
C) 40
D) 30

Answer 3

D

Question 4

The given values are discrete. Use the continuity correction and describe the region of the normal distribution that corresponds to the indicated probability. The probability of exactly 44 green marbles

A) The area between 44 and 44.5
B) The area between 43.5 and 45.5
C) The area between 43.5 and 44
D) The area between 43.5 and 44.5

Question 5

Estimate the probability of getting exactly 43 boys in 90 births. Estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution.

A) 0.0764
B) 0.0159
C) 0.0729
D) 0.1628

Question 6

For women aged 18-24, systolic blood pressures are normally distributed with a mean of 114.8 mm Hg and a standard deviation of 13.1 mm Hg. If 23 women aged 18-24 are randomly selected, find the probability that
Their mean systolic blood pressure is between 119 and 122 mm Hg.

A) 0.0833
B) 0.9341
C) 0.3343
D) 0.0577

Question 7

If z is a standard normal variable, find the probability that z lies between -2.41 and 0.

A) 0.4910
B) 0.0948
C) 0.4920
D) 0.5080

Question 8

Using the following uniform density curve, answer the question. What is the probability that the random variable has a value greater than 5?

A) 0.250
B) 0.500
C) 0.325
D) 0.375

Question 9

Scores on a test have a mean of 70 and Q3 is 83. The scores have a distribution that is approximately normal. Find P90. (You will need to first find the standard deviation.)

A) 94.8
B) 92.9
C) 93.7
D) 95.6

Question 10

The continuity correction is used to compensate for the fact that a ________ distribution is used to approximate a ________ distribution.

A) discrete; uniform
B) discrete; continuous
C) binomial; uniform
D) continuous; discrete

Question 11

Assume that the weight loss for the first month of a diet program varies between 6 pounds and 12 pounds, and is spread evenly over the range of possibilities, so that there is a uniform distribution. Find the probability that
The given range of pounds lost is between 8 pounds and 11 pounds.

Question 12

Find the probability that in 200 tosses of a fair die, we will obtain at most 30 fives. Use the normal distribution to approximate the desired probability.

A) 0.4936
B) 0.2946
C) 0.1871
D) 0.3229

Question 13

A study of the amount of time it takes a mechanic to rebuild the transmission for a 2010 Chevrolet Colorado shows that the mean is 8.4 hours and the standard deviation is 1.8 hours. If 40 mechanics are randomly selected,
Find the probability that their mean rebuild time is less than 8.9 hours.

A) 0.4276
B) 0.9589
C) 0.9756
D) 0.9608

Question 14

The distribution of certain test scores is a nonstandard normal distribution with a mean of 50 and a standard deviation of 6. What are the values of the mean and standard deviation after all test scores have been
Standardized by converting them to z-scores using ?

A) The mean is 10 and the standard deviation is 100.
B) The mean is 0 and the standard deviation is 1.
C) The mean is 100 and the standard deviation is 10.
D) The mean is 1 and the standard deviation is 0.

Question 15

If a histogram of a sample of men's ages is skewed, what do you expect to see in the normal quantile plot?

A) Points are following a straight-line pattern.
B) Points are not following a straight-line pattern.

Question 16

Estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution. Estimate

A) 0.1015
B) 0.1239
C) 0.8513
D) 0.1958

Question 17

Find the area of the shaded region. The graph depicts the standard normal distribution with mean 0 and standard deviation 1.

A) 0.0344
B) 0.4656
C) -0.0344
D) 0.9656

Question 18

The lengths of human pregnancies are normally distributed with a mean of 268 days and a standard deviation of 15 days. What is the probability that a pregnancy last at least 300 days?

A) 0.0166
B) 0.4834
C) 0.9834
D) 0.0179

Question 19

Question 20

Question 21

Find the area of the shaded region. The graph depicts the standard normal distribution with mean 0 and standard deviation 1.

A) 0.4656
B) 0.0344
C) -0.0344
D) 0.9656

Question 22

An unbiased estimator is a statistic that targets the value of the of the population parameter such that the sampling distribution of the statistic has a ________ equal to the ________ of the corresponding parameter.

A) range; range/4
B) mean; standard deviation
C) mean; mean
D) standard deviation; standard deviation

Question 23

A baseball player has a batting average of 0.346, so the probability of a hit is 0.346. Assume that his hitting attempts are independent of each other. Assume that the batter gets up to bat 4 times in each game. Estimate the
Probability that in 50 consecutive games, there are at least 45 games in which the batter gets at least one hit.
(Hint: first find the probability that in one game the batter gets at least one hit)

A) 0.0918
B) 0.0446
C) 0.8171
D) 0.0643

Question 24

Scores on a test are normally distributed with a mean of 63.2 and a standard deviation of 11.7. Find , which separates the bottom 81% from the top 19%.

A) 0.291
B) 66.6
C) 73.5
D) 0.88

Question 25

In one region, the September energy consumption levels for single -family homes are found to be normally distributed with a mean of 1050 kWh and a standard deviation of 218 kWh. For a randomly selected home, find
The probability that the September energy consumption level is between 1100 kWh and 1225 kWh.

A) 0.2881
B) 0.0910
C) 0.3791
D) 0.1971

Question 26

A coin is tossed 20 times. A person who claims to have extrasensory perception is asked to predict the outcome of each flip in advance. She predicts correctly on 14 tosses. What is the probability of being correct 14 or more
Times by guessing? Does this probability seem to verify her claim? Use the normal distribution to approximate
The desired probability.

A) 0.0582, no
B) 0.4418, no
C) 0.4418, yes
D) 0.0582, yes

Question 27

If z is a standard normal variable, find the probability:

A) 0.4884
B) 0.7557
C) 1.54
D) 0.2211

Question 28

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) 0.0679
B) 0.9331
C) 0.0769
D) 0.0669

Question 29

The weights of college football players are normally distributed with a mean of 200 pounds and a standard deviation of 50 pounds. If a college football player is randomly selected, find the probability that he weighs
Between 170 and 220 pounds.

A) 0.1554
B) 0.3811
C) 0.0703
D) 0.2257

Question 30

A bank's loan officer rates applicants for credit. The ratings are normally distributed with a mean of 200 and a standard deviation of 50. If an applicant is randomly selected, find the probability of a rating that is between 200
And 275.

A) 0.5
B) 0.9332
C) 0.0668
D) 0.4332

Question 31

The given values are discrete. Use the continuity correction and describe the region of the normal distribution that corresponds to the indicated probability.
The probability of no more than 35 defective CDs.

A) The area to the left of 35.5
B) The area to the right of 35.5
C) The area to the left of 35
D) The area to the left of 34.5

Question 32

Estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution. Two percent of hair dryers produced in a certain plant are defective. Estimate the probability that
Of 10,000 randomly selected hair dryers, at least 219 are defective.

A) 0.0934
B) 0.0823
C) 0.0869
D) 0.9066

Question 33

Assume that the red blood cell counts of women are normally distributed with a mean of 4.577 million cells per microliter and a standard deviation of 0.382 million cells per microliter. Approximately what percentage of
Women have red blood cell counts in the normal range from 4.2 to 5.4 million cells per microliter?

A) 4.09%
B) 82.31%
C) 16.11%
D) 17.69%

Question 34

Assume that the red blood cell counts of women are normally distributed with a mean of 4.577 million cells per microliter and a standard deviation of 0.382 million cells per microliter. Find the value closest to the probability
That a randomly selected woman has a red blood cell count above the normal range of 4.2 to 5.4 million cells per
Microliter.

A) 0.1611
B) 0.9842
C) 0.0409
D) 0.0158

Question 35

The Precision Scientific Instrument Company manufactures thermometers that are supposed to give readings of 0°C at the freezing point of water. Tests on a large sample of these thermometers reveal that at the freezing
Point of water, some give readings below 0°C (denoted by negative numbers) and some give readings above
0°C (denoted by positive numbers). Assume that the mean reading is 0°C and the standard deviation of the
Readings is 1.00°C. Also assume that the frequency distribution of errors closely resembles the normal
Distribution. A thermometer is randomly selected and tested. A quality control analyst wants to examine
Thermometers that give readings in the bottom 4%. Find the temperature reading that separates the bottom 4%
From the others.

A) -1.48°
B) -1.75°
C) -1.89°
D) -1.63°

Question 36

If z is a standard normal variable, find P(z > 0.97).

A) 0.1685
B) 0.8340
C) 0.1660
D) 0.1922

Question 37

In a population of 210 women, the heights of the women are normally distributed with a mean of 64.4 inches and a standard deviation of 2.9 inches. If 36 women are selected at random, find the mean and standard
Deviation of the population of sample means. Assume that the sampling is done without replacement and
Use a finite population correction factor.

A) 58.8 inches, 2.65 inches
B) 64.4 inches, 2.07 inches
C) 64.4 inches, 2.9 inches
D) 64.4 inches, 0.44 inches

Question 38

Assume that the red blood cell counts of women are normally distributed with a mean of 4.577 million cells per microliter and a standard deviation of 0.382 million cells per microliter. Find the 80 percentile for the red
Blood cell counts of women.

A) 4.898 million cells per microliter
B) 4.655 million cells per microliter
C) 4.878 million cells per microliter

Question 39

Find the indicated z score. The graph depicts the standard normal distribution with mean 0 and standard deviation 1. Shaded area is 0.4483.

A) 0.3264
B) 0.13
C) 0.6736
D) -0.13

Question 40

Which of the following is a biased estimator?

A) mean
B) variance
C) proportion
D) standard deviation

Question 41

Identify three important criteria to determine if the use of a normal distribution is justified.

Question 42

Lengths of pregnancies are normally distributed with a mean of 268 days and a standard deviation of 15 days.
(a) Find the probability of a pregnancy lasting more than 250 days. (b) Find the probability of a pregnancy
lasting more than 280 days. Draw the diagram for each and discuss the part of the solution that would be
different for finding the requested probabilities.

Question 43

A normal quartile 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 44

You will use a normal distribution to approximate a binomial distribution. Complete the following table for a
distribution in which 16. It might be helpful to make a diagram to help you determine the continuity factor
for each entry.

Question 45

Describe what an unbiased estimator is and give an example of an unbiased estimator and a biased estimator.

Question 46

Three randomly selected households are surveyed as a pilot project for a larger survey to be conducted later.
The numbers of people in the households are 2, 3, and 8. Consider the values of 2, 3, and 8 to be a population.
Assume that samples of size n = 2 are randomly selected with replacement from the population of 2, 3, and 8. The nine
different samples are as follows: (2, 2), (2, 3), (2, 8), (3, 2), (3, 3), (3, 8), (8, 2), (8, 3), and (8, 8).
(i) Find the range of each of the nine samples, then summarize the sampling distribution of the ranges in the format of a
table representing the probability distribution. (ii) Compare the population range to the mean of the sample ranges.
(iii) Do the sample ranges target the value of the population range? In general, do ranges make good estimators
of population ranges? Why or why not?

Question 47

State the central limit theorem. Describe the sampling distribution for a population that is uniform and for a
population that is normal.

Question 48

A normal quartile 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 49

In a recent year, the U.S. Mint in Denver manufactured 270 million quarters. Assume that on each day of
production, a sample of 50 quarters is randomly selected and the mean weight is obtained. Given that the
population of quarters has a mean weight of 5.67 g, what do you know about the mean of the sample means?
What do you know about the shape of the distribution of the sample means?

Question 50

A normal quartile plot is given below for the weekly incomes (in dollars) of a sample of engineers in one town.
Describe what each x value represents and what each y value represents. Use the plot to assess the normality of
the incomes of engineers in this town. Explain your reasoning.

Question 51

The population of current statistics students has ages with mean and standard deviation σ. Samples of
statistics are randomly selected so that there are exactly 40 students in each sample. For each sample, the mean
age is computed. What does the central limit theorem tell us about the distribution of those mean ages?

Question 52

Suppose that you wish to find ) for a continuous uniform distribution having a minimum of -3 and
a maximum of 3. If you incorrectly assume that the distribution is normal instead of uniform, will your answer
be too big, too small, or will you still obtain the correct answer? Explain your thinking.

Question 53

Explain how a nonstandard normal distribution differs from the standard normal distribution. Describe the
process for finding probabilities for nonstandard normal distributions.

Question 54

Heights of adult females are normally distributed. Samples of height of adult females, each of size n = 3, are
randomly collected and the sample means are found. Is it correct to conclude that the sample means cannot be
treated as a normal distribution because the sample size is so small? Explain.

Question 55

SAT verbal scores are normally distributed with a mean of 430 and a standard deviation of 120 (based on data
from the College Board ATP). (a) If a single student is randomly selected, find the probability that the sample
mean is above 500. (b) If a sample of 35 students are selected randomly, find the probability that the sample
mean is above 500. These two problems appear to be very similar. Which problem requires the application of
the central limit theorem, and in what way does the solution process differ between the two problems?

Question 56

The number of books sold over the course of the four-day book fair were 194, 197, 247, and 76. Assume that
samples of size 2 are randomly selected with replacement from this population of four values. List the different
possible samples, and find the mean of each of them.

Question 57

Three randomly selected households are surveyed as a pilot project for a larger survey to be conducted later.
The numbers of people in the households are 5, 7, and 9. Consider the values of 5, 7, and 9 to be a population.
Assume that samples of size n = 2 are randomly selected with replacement from the population of 5, 7, and 9.
The nine different samples are as follows: (5, 5), (5, 7), (5, 9), (7, 5), (7, 7), (7, 9), (9, 5), (9, 7), and (9, 9). (i) Find the
mean of each of the nine samples, then summarize the sampling distribution of the means in the format of a
table representing the probability distribution. (ii) Compare the population mean to the mean of the sample
means. (iii) Do the sample means target the value of the population mean? In general, do means make good
estimators of population means? Why or why not?

Question 58

Define a density curve and describe the two properties that it must satisfy. Show a density curve for a uniform
distribution. Make sure that your graph satisfies both properties.

Question 59

After constructing a new manufacturing machine, five prototype integrated circuit chips are produced and it is
found that two are defective and three are acceptable. Assume that two of the chips are randomly selected with
replacement from this population. After identifying the 25 possible samples, find the proportion of defects in
each of them, using a table to describe the sampling distribution of the proportions of the defects.

Question 60

Define the central limit theorem and its relationship to the sampling distribution of sample means. Define how
you can approximate a normal distribution from an original population that is not normally distributed

Question 61

Find the area of the shaded region. The graph depicts the standard normal distribution with mean 0 and standard
deviation 1.

A) 0.8599
B) 0.1401
C) 0.7198
D) 0.2802

Question 62

Find the indicated z score. The graph depicts the standard normal distribution with mean 0 and standard deviation 1.
Shaded area is 0.4013.

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

Question 63

What is the probability that the random variable has a value less than 6?

A) 0.500
B) 0.750
C) 0.875
D) 0.625

Question 64

Find the area of the shaded region. The graph depicts the standard normal distribution with mean 0 and standard
deviation 1.

A) 0.2420
B) 0.7580
C) 0.8790
D) 0.1210

Question 65

Find the area of the shaded region. The graph depicts the standard normal distribution with mean 0 and standard
deviation 1.

A) 0.4656
B) -0.0344
C) 0.9656
D) 0.0344

Question 66

Find the area of the shaded region. The graph depicts the standard normal distribution with mean 0 and standard
deviation 1.

A) 0.2190
B) 0.2224
C) 0.2776
D) 0.7224

Question 67

Find the area of the shaded region. The graph depicts the standard normal distribution with mean 0 and standard
deviation 1.

A) 0.8907
B) 0.8708
C) 0.1292
D) 0.8485

Question 68

Find the area of the shaded region. The graph depicts the standard normal distribution with mean 0 and standard
deviation 1.

A) 0.8812
B) 0.9406
C) 0.1188
D) 0.0594

Question 69

Find the area of the shaded region. The graph depicts the standard normal distribution with mean 0 and standard
deviation 1.

A) 0.9398
B) 0.0301
C) 0.0602
D) 0.9699

Question 70

What is the probability that the random variable has a value between 5.3 and 5.7?

A) 0.3000
B) 0.0500
C) 0.1750
D) 0.0750

Question 71

Find the area of the shaded region. The graph depicts the standard normal distribution with mean 0 and standard
deviation 1.

A) 0.6424
B) 0.3576
C) 0.8212
D) 0.1788

Question 72

Assume that the weight loss for the first month of a diet program varies between 6 pounds and 12 pounds, and is spread
evenly over the range of possibilities, so that there is a uniform distribution. Find the probability of the given range of
pounds lost.
Less than 11 pounds

Question 73

Assume that the weight loss for the first month of a diet program varies between 6 pounds and 12 pounds, and is spread
evenly over the range of possibilities, so that there is a uniform distribution. Find the probability of the given range of
pounds lost.
Between 8.5 pounds and 10 pounds

Question 74

What is the probability that the random variable has a value greater than 1.3?

A) 0.7875
B) 0.7125
C) 0.8375
D) 0.9625

Question 75

Find the indicated z score. The graph depicts the standard normal distribution with mean 0 and standard deviation 1.
Shaded area is 0.9599.

A) 1.82
B) 1.75
C) 1.03
D) -1.38

Question 76

What is the probability that the random variable has a value between 0.4 and 0.8?

A) 0.3
B) 0.175
C) 0.075
D) 0.05

Question 77

Assume that the weight loss for the first month of a diet program varies between 6 pounds and 12 pounds, and is spread
evenly over the range of possibilities, so that there is a uniform distribution. Find the probability of the given range of
pounds lost.
Between 8 pounds and 11 pounds

Question 78

What is the probability that the random variable has a value less than 2.7?

A) 0.4625
B) 0.2125
C) 0.3375
D) 0.0875

Question 79

Assume that the weight loss for the first month of a diet program varies between 6 pounds and 12 pounds, and is spread
evenly over the range of possibilities, so that there is a uniform distribution. Find the probability of the given range of
pounds lost.
More than 10 pounds

Question 80

What is the probability that the random variable has a value greater than 5?

A) 0.500
B) 0.250
C) 0.325
D) 0.375