Deck 6: The Normal Distribution

Question 1

Which of the following characteristics does not apply to a theoretical normal distribution?

A) It is bell-shaped.
B) The mean, median, and mode are equal.
C) It is bimodal.
D) It is never negative.

Answer 1

C

Question 2

In applied statistics, it is the area under the normal distribution curve which is most important, not the value of single points on the curve.

Answer 2

True

Question 3

For a normal distribution curve with a mean of 7 and a standard deviation of 5, which of the following ranges of the variable will define an area under the curve corresponding to a probability of approximately 34%?

A) from 4.5 to 9.5
B) from 2 to 12
C) from 7 to 12
D) from -3 to 17

Answer 3

C

Question 4

Which of the following properties distinguishes the standard normal distribution from other normal distributions?

A) The mean is 0 and the standard deviation is 1.
B) The total area under the curve is equal to 1.00.
C) The curve is continuous.
D) The mean is located at the center of the distribution.

Question 5

Find the area under the standard normal distribution curve between z = 0 and z = 2.16.

A) 2.1600
B) 0.9846
C) 0.4846
D) 0.3708

Question 6

For a normal distribution curve with a mean of 15 and a standard deviation of 3, which range of the variable defines an area under the curve corresponding to a probability of approximately 68%?

A) from 13.5 to 16.5
B) from 12 to 18
C) from 15 to 21
D) from 9 to 21

Question 7

The number of standard deviations a particular X value is from the mean is commonly referred to as ________.

A) z
B) P(X)
C)
D)

Question 8

The normal distribution curve can be used as a probability distribution curve for normally distributed variables.

Question 9

Identify the type of distribution pattern that occurs when the majority of the data values fall to the left of the mean?

A) symmetrical
B) positively skewed
C) negatively skewed
D) left skewed

Question 10

The area under a normal distribution curve is always positive even if the z value is negative.

Question 11

Stating that the area under the standard normal distribution curve between z = 0 and z = 1.00 is 0.3413, is the same as stating that the _____________ of randomly selecting a standard normally distributed variable z with a value between 0 and 1.00 is 0.3413.

Question 12

Which choice is another term that can be used to describe a normal distribution:

A) independent variable
B) discrete distribution
C) negatively or positively skewed
D) bell curve

Question 13

To find the area under the standard normal distribution curve between two z values, one first finds the difference between the two z values, then locates the value corresponding to that difference in the Standard Normal Distribution table.

Question 14

The area under the normal distribution curve that lies within three standard deviations of the mean is approximately 95%.

Question 15

When the data values are evenly distributed about the mean, the distribution is said to be ______________.

Question 16

When the majority of the data values fall to the right of the mean, the distribution is said to be left-skewed.

Question 17

If a normal distribution has a mean of 20 and a standard deviation of 10, then

A) the median is 20 and the mode is 20.
B) the median is 10 and the mode is 30.
C) the median is 30 and the mode is 10.
D) the median is 20 and the mode is 30.

Question 18

One normal curve has a mean of 20 and a standard deviation of 5. A second normal curve has a mean of 5 and a standard deviation of 20. The curve that is more dispersed, or spread out, is

A) the first normal curve.
B) the second normal curve.
C) both; they are equally spread out.
D) neither; by definition,a normal curve shows no spread.

Question 19

On an easy test, the mean score was 98 out of a possible 100 points. The distribution of the test scores is likely to be

A) symmetric.
B) negatively skewed.
C) positively skewed.
D) diagonally skewed.

Question 20

The figure below is an example of a negatively skewed distribution.

Question 21

The probability is 0.3340.

Question 22

Find the area under the standard normal distribution curve between z = -2.05 and z = 2.05.

A) 0.4938
B) 0.4798
C) 0.9596
D) 0.9876

Question 23

Find the probability P(-0.77 < z < -0.16) using the standard normal distribution.

A) 0.0400
B) 0.7842
C) 0.3458
D) 0.2158

Question 24

Find the probability P(-1.09 < z < 1.06) using the standard normal distribution.

A) 0.8410
B) 0.2824
C) 0.7176
D) 0.6744

Question 25

Find the probability , using the standard normal distribution.

A) 45.25%
B) 45.54%
C) 42.07%
D) 35.54%

Question 26

Find the z value to the right of the mean so that 62.93% of the area under the distribution curve lies to the left of it.

Question 27

The area under the standard normal distribution curve to the right of z = 1.83 is 0.4664.

Question 28

Find the probability using the standard normal distribution.

A) 0.6591
B) 0.3409
C) 0.8409
D) 0.5900

Question 29

Find the z value that corresponds to the given area.

A) 1.33
B) -0.51
C) 0.41
D) -0.22

Question 30

Find the probability using the standard normal distribution.

A) 0.2517
B) 0.7517
C) 0.7483
D) 0.2483

Question 31

Find the area under the standard normal distribution curve between and .

A) -2.1600
B) 0.9846
C) 0.4846
D) 0.3708

Question 32

Find the probability using the standard normal distribution.

A) 0.7800
B) 0.4129
C) 0.5871
D) 0.0871

Question 33

Find the probability P(0.21 < z < 1.28) using the standard normal distribution.

A) 0.3166
B) 0.6834
C) 0.8166
D) 0.4100

Question 34

Find the z value to the right of the mean such that 85% of the total area under the standard normal distribution curve lies to the left of it?

Question 35

Find the probability using the standard normal distribution.

A) 0.2946
B) 0.7054
C) 0.7946
D) 0.2054

Question 36

Find the z value that corresponds to the given area.

A) -0.79
B) 0.57
C) 0.44
D) 0.27

Question 37

In a standard normal distribution, what z value corresponds to 17% of the data between the mean and the z value?

A) 1.25
B) 0.44
C) 0.52
D) 2.10

Question 38

Find the z value that corresponds to the given area.

A) -1.46
B) 1.23
C) 1.46
D) 0.07

Question 39

What is the area under the standard normal distribution curve between z = 1.50 and z = 2.50?

A) 0.0802
B) 0.0606
C) 0.0764
D) 1.00

Question 40

Find the area under the standard normal distribution curve to the left of z = 1.69.

A) 0.4545
B) 0.4452
C) 0.9545
D) 0.9452

Question 41

What is the z value such that 50% of the total area under the standard normal distribution curved lies to the right of it?

Question 42

The data shown represent quiz scores from Ms. Mayer's 5th grade class. Use the Pearson coefficient to check for skewness.
21 25 9 21 23 22 22 15 23 22 16 11 20 20 17 22 21

A) so data is skewed
B) so data is skewed
C) so data is skewed
D) so data is not skewed

Question 43

The average gas mileage of a certain model car is 28.0 miles per gallon. If the gas mileages are normally distributed with a standard deviation of 0.45 miles per gallon, find the probability that a car has a gas mileage of between 27.8 and 28.2 miles per gallon.

A) 0.343
B) 0.157
C) 0.403
D) 0.172

Question 44

The average hourly wage of workers at a fast food restaurant is $6.50/hr. Assume the wages are normally distributed with a standard deviation of $0.45. If a worker at this fast food restaurant is selected at random, what is the probability that the worker earns more than $6.75?

A) 28.8%
B) 27.6%
C) 42.1%
D) 5.2%

Question 45

The average length of crocodiles in a swamp is 12 feet. If the lengths are normally distributed with a standard deviation of 1.9 feet, find the probability that a crocodile is more than 11.5 feet long.

A) 0.90
B) 0.40
C) 0.10
D) 0.60

Question 46

X is a normally distributed random variable with a mean of 10.00. If the probability that X is less than 11.54 is 0.67 (as shown below), then what is the standard deviation of X? (Note: the diagram is not necessarily to scale.)

A) 1.75
B) 3.50
C) 4.20
D) 12.25

Question 47

X is a normally distributed random variable with a standard deviation of 2.00. Find the mean of X if 12.71% of the area under the distribution curve lies to the right of 8.28. (Note: the diagram is not necessarily to scale.)

A) 7.6
B) 6.6
C) 6.0
D) 5.5

Question 48

Mrs. Smith's reading class can read an average of 175 words per minute with a standard deviation of 20 words per minute. The top 3% of the class is to receive a special award. What is the minimum number of words per minute a student would need to read in order to get the award? Assume the data is normally distributed.

Question 49

For a normal distribution with a mean of 4 and a standard deviation of 2, the value 5 has a z value of

A) -0.5
B) 0.5
C) 1.5
D) 2.5

Question 50

At a large department store, the average number of years of employment for a cashier is 5.7 with a standard deviation of 1.8 years, and the distribution is approximately normal. If an employee is picked at random, what is the probability that the employee has worked at the store for over 10 years?

A) 99.2%
B) 49.2%
C) 1.7%
D) 0.8%

Question 51

The following formula can be used to find the specific value of the variable in a data set that corresponds to a particular value of the standard normal variable:

Question 52

The average height of flowering cherry trees in a certain nursery is 10.5 feet. If the heights are normally distributed with a standard deviation of 1.5 feet, find the probability that a tree is less than 12.5 feet tall.

A) 0.73
B) 0.91
C) 0.96
D) 0.82

Question 53

X is a normally distributed random variable with a mean of 5.0. Find the standard deviation of the distribution if 59.10% of the data lies to the right of 4.54. (Note: the diagram is not necessarily to scale.)

A) 2.0
B) 1.3
C) 1.0
D) 0.5

Question 54

For a normal distribution with a mean of 8 and a standard deviation of 4, the value 6 has a z value of

A) 1.5
B) -2.5
C) -1.5
D) -0.5

Question 55

In order to be accepted into a certain top university, applicants must score within the top 5% on the SAT exam. Given that the exam has a mean of 1000 and a standard deviation of 200, what is the lowest possible score a student needs to qualify for acceptance into the university?

A) 1400
B) 1330
C) 1250
D) 1100

Question 56

X is a normally distributed random variable with a mean of 7 and a standard deviation of 2.5. Find the value of X for which 70.54% of the area under the distribution curve lies to the right of it. (Note: the diagram is not necessarily to scale.)

A) 6.10
B) 6.32
C) 5.65
D) 8.43

Question 57

A recent study found that the life expectancy of a people living in Africa is normally distributed with an average of 53 years with a standard deviation of 7.5 years. If a person in Africa is selected at random, what is the probability that the person will die before the age of 65?

A) 94.5%
B) 44.5%
C) 61.0%
D) 89.0%

Question 58

X is a normally distributed random variable with a standard deviation of 3.50. Find the mean of X when 64.8% of the area lies to the left of 10.33. (Note: the diagram is not necessarily to scale.)

A) 7.1
B) 8.7
C) 9.0
D) 10.2

Question 59

If a normally distributed group of test scores have a mean of 70 and a standard deviation of 12, find the percentage of scores that will fall below 50.

A) 4.75%
B) 45.25%
C) 6.75%
D) 35.54%

Question 60

X is a normally distributed random variable with a mean of 9.0 and a standard deviation of 3.00. Find the value x such that P(X < x) is equal to 0.86. (Note: the diagram is not necessarily to scale.)

A) 12.24
B) 10.47
C) 13.95
D) 9.00

Question 61

The difference between a sample mean and the population mean may be referred to as ______________________.

A) the standard deviation
B) the variance
C) skewness
D) sampling error

Question 62

A(n) _______________________ is employed when a continuous distribution is used to approximate a discrete distribution.

Question 63

Of the members of a Boy Scout troop, 15% have received the first aid merit badge. If 40 boy scouts are selected at random, find the probability that four or more will have the first aid merit badge?

A) 86.6%
B) 81.3%
C) 36.6%
D) 31.3%

Question 64

The average age of doctors in a certain hospital is 45.0 years old. Suppose the distribution of ages is normal and has a standard deviation of 4.0 years. If 16 doctors are chosen at random for a committee, find the probability that the average age of those doctors is less than 45.5 years. Assume that the variable is normally distributed.

A) 19.1%
B) 30.9%
C) 61.2%
D) 69.1%

Question 65

As the sample size n increases, the shape of the distribution of the sample means taken with replacement from a population with mean $\mu$ and standard deviation $\sigma$ , will approach a normal distribution. This distribution will have a mean of $\mu$ and a standard deviation of . This is a statement of the _______________________.

Question 66

A(n) __________________________________ is a distribution of means obtained from samples of a specific size taken from a population.

Question 67

The average number of mosquitos in a stagnant pond is 60 per square meter with a standard deviation of 16. If 25 square meters are chosen at random for a mosquito count, find the probability that the average of those counts is more than 62.9 mosquitos per square meter. Assume that the variable is normally distributed.

A) 0.3%
B) 18.4%
C) 31.6%
D) 81.6%

Question 68

The average diameter of sand dollars on a certain island is 3.50 centimeters with a standard deviation of 1.00 centimeters. If 25 sand dollars are chosen at random for a collection, find the probability that the average diameter of those sand dollars is more than 3.28 centimeters. Assume that the variable is normally distributed.

A) 0.636
B) 0.800
C) 0.364
D) 0.864

Question 69

The mean weight of loads of rock is 47.0 tons with a standard deviation of 6.0 tons. If 16 loads are chosen at random for a weight check, find the probability that the mean weight of those loads is less than 46.2 tons. Assume that the variable is normally distributed.

A) 21.23
B) 28.77
C) 71.23
D) 78.77

Question 70

A survey of 250 lobster fishermen found that they catch an average of 32.0 pounds of lobster per day with a standard deviation of 4.0 pounds. If a random sample of 36 lobster fishermen is selected, what is the probability that their average catch is less than 31.5 pounds? Assume the distribution of the weights of lobster is normal.

Question 71

The average age of vehicles registered in the United States is 96 months. Assume the population is normally distributed with a standard deviation of 15 months. Find the probability that the mean age of a sample of 36 vehicles is between 98 and 100 months?

A) 44.5%
B) 28.8%
C) 15.7%
D) 6.4%

Question 72

If a baseball player's batting average is 0.340 , find the probability that the player will have a bad season and get at most 60 hits in 200 times at bat?

A) 38.3%
B) 11.7%
C) 36.9%
D) 13.1%

Question 73

If the standard deviation of a normally distributed population is 60.0 and we take a sample of size 25, then the standard error of the mean is

A) 12.0
B) 5.0
C) 60.0
D) 2.4

Question 74

In order to have the standard error of the mean be 11, one would need to take _______ samples from a normally distributed population with a standard deviation of 55.

A) 5
B) 25
C) 125
D) 625

Question 75

A biologist estimates that 40% of the deer in a region carry a certain type of tick. For a sample of 300 deer selected at random, what is the chance that 124 or fewer deer have this tick?

A) 0.864
B) 0.702
C) 0.404
D) 0.298

Question 76

The length of country and western songs is normally distributed and has a mean of 200 seconds and a standard deviation of 40 seconds. Find the probability that a random selection of 16 songs will have mean length of 186.30 seconds or less. Assume the distribution of the lengths of the songs is normal.

A) 0.91
B) 0.41
C) 0.09
D) 0.59

Question 77

The _________________ correction factor is important if a relatively large number of samples are taken without replacement from a small population.

Question 78

Use the normal approximation to the binomial to find that probability for the specific value of X.
N = 20, p = 0.3, X = 7

A) 0.17
B) 0.35
C) 0.11
D) 0.3

Question 79

The standard deviation of a normal distribution is 20. What is the standard error of the mean obtained by averaging 225 samples?

Question 80

The standard deviation of sample means will be larger than the standard deviation of the population.