Deck 6: The Normal Probability Distribution

Question 1

Which of the following values is the z-score representing the first quartile of the standard normal distribution?

A) 0.67
B) -0.67
C) 1.28
D) -1.28

Answer 1

B

Question 2

Which of the following statements best describes continuous random variables?

A) They can assume values at all points on an interval with no breaks between possible values.
B) We can gauge the likely occurrence of specific values of such variables with the help of one or another of certain probability distributions, including the binomial, Poisson, or hypergeometric distributions.
C) They model random variables such as waiting times or lifetimes.
D) They calculate probabilities for the binomial random variable .

Answer 2

A

Question 3

If z is a standard normal random variable, what is the value of P(-1.25 z -0.75)?

A) 0.6678
B) 0.2266
C) 0.1210
D) 0.1056

Answer 3

C

Question 4

What proportion of the data from a normal distribution is within two standard deviations from the mean?

A) 0.3413
B) 0.4772
C) 0.6826
D) 0.9544

Question 5

If the random variable x is normally distributed with a mean of 88 and a standard deviation of 12, what is the value of P(x 96)?

A) 0.1243
B) 0.2486
C) 0.2514
D) 0.4972

Question 6

If x is a normal random variable with a mean of 1228 and a standard deviation of 120, how many standard deviations are there from 1228 to 1380?

A) 11.50
B) 10.233
C) 3.1989
D) 1.267

Question 7

If z is a standard normal random variable, the area between z = 0.0 and z = 1.20 is 0.3849, while the area between z = 0.0 and z = 1.40 is 0.4192. What is the area between z = -1.20 and z = 1.40?

A) 0.0343
B) 0.0808
C) 0.1151
D) 0.8041

Question 8

Which of the following is always true for all probability density functions of continuous random variables?

A) They are symmetrical.
B) They are skewed to the right.
C) They are skewed to the left.
D) The area under the curve is 1.0.

Question 9

Let be a z-score that is unknown but identifiable by position and area. If the symmetrical area between a negative and a positive is 0.8132, what would be the value of ?

A) z = 2.64
B) z = 1.78
C) z = 1.32
D) z = 0.89

Question 10

Given that z is a standard normal variable, which of the following is the value for which P(z ) = 0.242?

A) 0.70
B) 0.65
C) -0.65
D) -0.70

Question 11

If z is a standard normal random variable, how does the area between z = 0.0 and z = 1.25 compare to the area between z = 1.25 and z = 2.5?

A) The latter area will be larger than the former.
B) The latter area will be smaller than the former.
C) The two areas are the same.
D) The two areas are not the same.

Question 12

Which of the following is a property of the standard normal distribution?

A) It has a mean of 0 and a standard deviation of 1.
B) It has a mean of 0 and a standard deviation of 0.
C) It has a mean of 1 and a standard deviation of 0.
D) It has a mean of 1 and a standard deviation of 1.

Question 13

Let x be a continuous random variable and let c be a constant. Which of the following statements does NOT apply to the probability of x?

A) The probability that x assumes a value in the intervaltois the area under the probability density function betweenand.
B) P(x = c) = c
C) P(x < c) = P(x < c) and P(x > c) = P(x > c)
D) P(x = c) = c and P(x > c) = P(x > c)

Question 14

Given that Z is a standard normal random variable, what is P(-1.2 Z 1.5)?

A) 0.8181
B) 0.5228
C) 0.4772
D) 0.3849

Question 15

Using the standard normal table, what is the total probability to the right of z = 2.0 and to the left of z = -2.0?

A) 0.0228
B) 0.0456
C) 0.4772
D) 0.9544

Question 16

What is the z-score representing the third quartile of the standard normal distribution?

A) 0.67
B) -0.67
C) 1.28
D) -1.28

Question 17

Which of these statements is valid with regard to probabilities of continuous random variables?

A) The height of the curve shows the probability of an event.
B) The probability of exactly an event A occurring is always equal to one.
C) Probabilities of events are determined from areas under the curve.
D) The probability distribution is always mound-shaped.

Question 18

Which of the following incorrectly describes both of the discrete probability distributions and the continuous probability density functions?

A) They both reflect the distribution of a random variable.
B) They both have outcomes, which occur only in integers.
C) They both assist in determining the probability of a given outcome.
D) They are both relative frequency distributions.

Question 19

Many different types of continuous random variables give rise to a large variety of probability density functions. Which of those listed below are generated by such variables?

A) the binomial probability distributions
B) the hypergeometric probability distributions
C) the normal probability distributions
D) the Poisson probability distributions

Question 20

Let be a z-score that is unknown but identifiable by position and area. If the area to the right of is 0.7088, then what is the value of ?

A) 1.06
B) 0.55
C) -0.55
D) -1.06

Question 21

Suppose you are given that x is a normally distributed random variable with a mean of and a standard deviation of 0.15. If P(x < 2.10) = 0.025, what is the value of ?

A) 2.394
B) 2.104
C) 2.096
D) 1.806

Question 22

Which of the following is a valid statement about the mean, median, and mode?

A) They are equal to one another in any Poisson probability distribution.
B) They are equal to one another in any normal probability distribution.
C) They are different measures of centre and therefore cannot possibly be equal to one another.
D) They are different measures of centre, but can equal each other only if the probability distribution is negatively or positively skewed.

Question 23

Continuous random variables that can assume values at all points on an interval of values, with no breaks between possible values, are quite common. Which of the following items could NOT be classified as a continuous random variable?

A) profit per dollar of sales
B) cost per credit taken by graduate students
C) the average time it takes to assemble a car, or write a test
D) heights of human beings rounded to the nearest inch

Question 24

The z-score of a random variable value of x = 2 is z = -2, while the standard deviation of the random variable x equals 2. What is the mean of x?

A) 6
B) 4
C) 2
D) 0

Question 25

The random variable x is normally distributed, with a mean equal to 0.45 and a standard deviation equal to 0.40. What is the value of P(x 0.75)?

A) 0.7734
B) 0.7500
C) 0.2734
D) 0.2266

Question 26

Given that Z is a standard normal variable, and P(Z b) = 0.2580, which of these values is b?

A) -0.65
B) 0.242
C) 0.70
D) 0.758

Question 27

If a normal curve appears more peaked, what may be said of the population parameters?

A) The smaller is the value of the standard deviation.
B) The larger is the value of the standard deviation.
C) The smaller is the value of the mean.
D) The larger is the value of the mean.

Question 28

Which of the following statements best describes a standard normal distribution?

A) It is a normal distribution with a mean of 0 and a standard deviation of 1.
B) It is a normal distribution with a mean of 1 and a standard deviation of 0.
C) It is a normal distribution with a mean usually larger than the standard deviation.
D) It is a normal distribution with a mean always larger than the standard deviation.

Question 29

x is a normally distributed random variable with a mean of 8.20 and variance of 4.41, and P(x > b) = 0.08. What is the value of b?

A) 3.409
B) 3.448
C) 3.452
D) 11.914

Question 30

The time it takes Jessica to bicycle to school is normally distributed, with a mean time of 15 minutes and a variance of 4 minutes. Jessica has to be at school at 8:00
A) Twenty-three minutes to school is not an unusually long commute time.

A)m. Suppose it took her 23 minutes to get to school. What can you reasonably infer or conclude?
B) A commuting time of 23 minutes is highly unusual or atypical.
C) The distribution of commute times must not be normal with mean 15 minutes and standard deviation 2 minutes.
D) Both b and c

Question 31

If the random variable X is normally distributed, with a mean of 75 and a standard deviation of 8, which of the following values is the probability that X is greater than or equal to 75?

A) 0.125
B) 0.500
C) 0.625
D) 0.975

Question 32

If Z is a standard normal random variable, then which of these values is P(-1.75 Z -1.25)?

A) 0.8543
B) 0.1056
C) 0.0655
D) 0.0401

Question 33

Members of the normal probability distribution family differ from one another only by which of the following measures?

A) mean and standard deviation
B) mean and median
C) mean and mode
D) mode and standard deviation

Question 34

Which of the following is NOT a characteristic of the normal distribution?

A) It is symmetric.
B) It is not bell-shaped.
C) The mean equals the median.
D) The mean equals the mode.

Question 35

Which of the following correctly describes the normal probability distribution?

A) It is single-peaked above the random variable's mean, median, and mode, all of which are equal to one another.
B) It features tails extending indefinitely in both directions from the centre, approaching (but never touching) the horizontal axis, which implies a positive probability for finding values of the random variable anywhere between minus infinity and plus infinity.
C) Both a and b

Question 36

Which of the following correctly pertains to a continuous random variable?

A) We cannot list all the probabilities for each one of the infinite number of conceivable values of the variable.
B) We commonly associate probabilities with ranges of values along the continuum of possible values that the random variable might take on.
C) Both a and b

Question 37

Given that Z is a standard normal random variable, which of the following is the value of P(-1.0 Z 1.5)?

A) 0.0919
B) 0.7745
C) 0.8413
D) 0.9332

Question 38

Which of the following probability distributions can be used to describe the distribution for a continuous random variable?

A) binomial distribution
B) normal distribution
C) Poisson distribution
D) hypergeometric distribution

Question 39

If Z is a standard normal random variable, then which of these values is the value of z for which P(-z Z z) equals 0.8764?

A) 3.08
B) 1.54
C) 1.16
D) 0.3764

Question 40

Given a normal distribution with a mean of 80 and a standard deviation of 20, which z-score would correspond to an observation of x = 50?

A) z = +3.0
B) z = +2.0
C) z = +1.5
D) z = -1.5

Question 41

If Z is a standard normal random variable, how would one describe the area between z = 0.0 and z =1.50 compared to the area between z = 1.5 and z = 3.0?

A) the same
B) larger
C) smaller

Question 42

If Z is a standard normal random variable, then the area between z = 0.0 and z =1.30 is 0.4032, while the area between z = 0.0 and z = 1.50 is 0.4332. What is the area between z = -1.30 and z = 1.50?

A) 0.0300
B) 0.0668
C) 0.0968
D) 0.8364

Question 43

What proportion of the data from a normal distribution is within two standard deviations from the mean?

A) 0.3413
B) 0.4772
C) 0.6826
D) 0.9544

Question 44

Given that Z is a standard normal random variable, which of the following is the value for P(Z > -1.58)?

A) -0.4429
B) 0.0571
C) 0.5571
D) 0.9429

Question 45

Given that the random variable X is normally distributed with a mean of 80 and a standard deviation of 10, which of these is the value of P(85 X 90)?

A) 0.5328
B) 0.3413
C) 0.1915
D) 0.1498

Question 46

Which of the following distributions is considered to be the cornerstone distribution of statistical inference?

A) binomial distribution
B) normal distribution
C) Poisson distribution
D) uniform distribution

Question 47

Suppose that the z-value for a given value x of the random variable X is z = 1.96, and that the distribution of X is normally distributed with a mean of 60 and a standard deviation of 6. To what x-value does this z-value correspond?

A) 71.76
B) 67.96
C) 61.96
D) 48.24

Question 48

For some positive value of z, the probability that a standard normal variable is between 0 and z is 0.3770. What is the value of that z?

A) 0.18
B) 0.81
C) 1.16
D) 1.47

Question 49

For some value of z, the probability that a standard normal variable is below z is 0.2090. What is the value of that z?

A) -0.81
B) -0.31
C) 0.31
D) 1.96

Question 50

Given that Z is a standard normal random variable, which of the following best describes the area to the left of a value z?

A) P(Zz)
B) P(Zz)
C) P(0Zz)
D) P(Z-z)

Question 51

As a general rule, the normal distribution is used to approximate the sampling distribution of the sample proportion only if which of the following conditions holds?

A) The sample size n is greater than 30.
B) The population proportion p is close to 0.50.
C) The underlying population is normal.
D) np and n(1 - p) are both greater than 5.

Question 52

Which of the following characteristics is NOT a property of a normal distribution?

A) It is unimodal.
B) It is symmetrical.
C) It is discrete.
D) It has a bell shape.

Question 53

Given that Z is a standard normal random variable, which of these is the value of z if the area to the right of z is 0.1949?

A) 0.51
B) -0.51
C) 0.86
D) -0.86

Question 54

Which of the following must be specified in order to completely determine the probability density function f(x) of a random variable X that is normally distributed?

A) the mean and median of X
B) the median and mode of X
C) the mean and mode of X
D) the mean and standard deviation of X

Question 55

For some positive value of x, the probability that a standard normal variable is between 0 and +1.5x is 0.4332. What is the value of x?

A) 0.10
B) 0.50
C) 1.00
D) 1.50

Question 56

Given that X is a binomial random variable, the binomial probability P(X x) is approximated by the area under a normal curve to the right of which of the following values?

A) x - 0.5
B) x + 0.5
C) x - 1
D) x + 1

Question 57

Which of the following distributions are always symmetrical?

A) exponential
B) normal
C) binomial
D) all continuous distributions

Question 58

Given that Z is a standard normal random variable, which of these is the value of z if the area to the right of z is 0.9066?

A) 1.32
B) -1.32
C) 0.66
D) -0.66

Question 59

Even though the normal probability distribution deals with continuous variables, under certain circumstances we can use it to approximate various discrete probability distributions. Under which of these conditions is the use of the normal probability distribution permissible?

A) in the case of binomial probabilities, when np > 5 and also nq > 5
B) in the case of Poisson probabilities, when> 100
C) in the case of hypergeometric probabilities, when np > 1 and also nq > 1

Question 60

For some positive value of x, the probability that a standard normal variable is between 0 and +2x is 0.1255. Which of the following is the value of that x?

A) 0.99
B) 0.40
C) 0.32
D) 0.16

Question 61

The relative frequency associated with a particular class in the population is the fraction of measurements in the population falling in that class.

Question 62

Given a normal random variable x with mean and standard deviation , the standard normal random variable associated with x is .

Question 63

The relative frequency associated with a particular class in the population is the probability of drawing a measurement from that class.

Question 64

The shape of the normal probability distribution is determined by the population standard deviation $\sigma$ . Small values of $\sigma$ reduce the height of the curve and increase the spread; large values of $\sigma$ increase the height of the curve and reduce the spread.

Question 65

The mean, median, and standard deviation of a normally distributed random variable are all at the same position on the horizontal axis since the distribution is symmetric.

Question 66

Given that X is a binomial random variable, the binomial probability P(X x) is approximated by the area under a normal curve to the left of which of the following values?

A) x
B) -x
C) x + 0.5
D) x - 0.5

Question 67

The proportion of the total area under the normal curve that lies within one standard deviation of the mean is approximately 0.75.

Question 68

For any continuous random variable x, P(x = 5) = 1 is always true.

Question 69

The normal probability distribution is important because a large number of random variables observed in nature possess a frequency distribution that is approximately a normal probability distribution.

Question 70

The total area under the curve f(x) of any continuous random variable x is equal to one.

Question 71

Given that X is a binomial random variable, then between which of the following values would the binomial probability P(X = x) be approximated by the area under a normal curve?

A) x - 0.5 and 0.0
B) 0.0 and x + 0.5
C) 1 - x and 1 + x
D) x - 0.5 and x + 0.5

Question 72

For a continuous random variable x, P(6 < x < 8) is represented by the area under the probability function over the interval from 6 to 8, including both end points.

Question 73

If x denotes a continuous random variable, then P(x = c) = 0 for every number
c.

Question 74

Suppose that the probability p of success on any trial of a binomial distribution equals 0.90. For which of the following number of trials, n, would the normal distribution provide a good approximation to the binomial distribution?

A) 25
B) 35
C) 45
D) 55

Question 75

Continuous random variables can assume infinitely many values corresponding to points on a line interval.

Question 76

For a continuous random variable x, the probabilities P(0 < x < 1) and P(0 < x < 1) are always equal.

Question 77

The left half of the normal curve is slightly smaller than the right half.

Question 78

Using the standard normal curve, the area between z = 0 and z = 2.2 is 0.4868.

Question 79

A normal random variable x is standardized by expressing its value as the number of standard deviations it lies to the right or left of its mean .

Question 80

Any continuous random variable cannot take on negative values.