Deck 7: Continuous Random Variables

Question 1

The normal probability distribution is a discrete probability distribution.

Answer 1

False

Question 2

If the random variable, x, is normally distributed, 68.26 percent of all possible observed values of x will be within two standard deviations of the mean.

Answer 2

False

Question 3

A continuous random variable may assume only integer values in a given interval.

Answer 3

False

Question 4

The mean and median are the same for a normal distribution.

Question 5

The z value tells us the number of standard deviations that a value, x, is from the mean.

Question 6

The mean of a standard normal distribution is always equal to 1.

Question 7

A uniform distribution f(x) is a continuous probability distribution, which says the probability that x is in any two intervals of equal length is the same.

Question 8

The number of defective pencils in a total lot of 1000 is an example of a continuous random variable.

Question 9

The exponential probability distribution is based on a continuous random variable.

Question 10

The standard deviation of a standard normal distribution is always equal to 1.

Question 11

If the distances between the ordered data values are proportional to the distances between the actual values, then the normal probability plot will be a straight line.

Question 12

A continuous probability distribution that has a rectangular shape, where the probability is evenly distributed over an interval of numbers, is called a uniform probability distribution.

Question 13

The actual weight of hamburger patties is an example of a continuous random variable.

Question 14

For a binomial probability experiment, with n = 150 and p = .2, it is appropriate to use the normal approximation to the binomial distribution.

Question 15

The mean and the median are the same for an exponential distribution.

Question 16

In a statistical study, the random variable X = 1 if the house is colonial, and X = 0 if the house is not colonial. The random variable X is continuous.

Question 17

For a continuous distribution, the exact probability of a particular value is zero.

Question 18

All continuous random variables are normally distributed.

Question 19

For a continuous random variable x, the height of the probability curve f(x) at a particular point indicates the value of the probability for that value.

Question 20

For a continuous distribution, P(X ≤ 100) = P(X < 100).

Question 21

The standardized normal quantile value O_i is the z value that gives an area of i/(n − 1) to its right under the normal curve.

Question 22

If the random variable, X, is normally distributed and has a mean of µ and a standard deviation σ, then the mean and standard deviation, respectively, of (X − µ)/σ are

A) μ and σ.
B) and s.
C) 1 and 0.
D) 0 and 1.

Question 23

The first step to constructing a normal probability plot is to

A) calculate O_i.
B) compute area i/(n + 1).
C) construct a box-and-whiskers plot.
D) order the values in the data.

Question 24

The random variable x has a uniform distribution when x lies between the values of 4 and 10. When x = 10, what is the value of f(x)?

A) 0
B) 1
C) 1/6
D) 1/14

Question 25

The grade a student received on an examination was transformed to a z value, which was negative. Therefore, we know that the student scored

A) higher than 16 percent of the class.
B) higher than 45 percent of the class.
C) above the first quartile.
D) below the mean.
E) lower than 16 percent of the class.

Question 26

The price-to-earnings ratio for firms in a given industry is distributed according to the normal distribution. In this industry, a firm with a standard normal variable value of z = 1

A) has an above average price-to-earnings ratio.
B) has a below average price-to-earnings ratio.
C) has an average price-to-earnings ratio.
D) may have an above average or below average price-to-earnings ratio depending on the value of the mean.

Question 27

If the wages of workers for a given company are normally distributed with a mean of $15 per hour, then the proportion of the workers earning more than $13 per hour

A) is greater than the proportion earning less than $13 per hour.
B) is greater than the proportion earning less than $18 per hour.
C) is less than 50 percent.
D) is less than the proportion earning more than the mean wage.

Question 28

The normal approximation of the binomial distribution is appropriate when

A) np ≥ 5.
B) n(1 − p) ≥ 5.
C) np ≤ 5.
D) n(1 − p) ≤ 5 and np ≤ 5.
E) np ≥ 5 and n(1 − p) ≥ 5.

Question 29

Which of the following is a continuous distribution?

A) Poisson
B) hypergeometric
C) exponential
D) binomial

Question 30

Values of the standard normal random variable are measured

A) with reference to specific units.
B) in the units in which the mean is measured.
C) in the number of standard deviations from the mean.
D) in squared units in which the mean is measured.
E) None of these answers are correct.

Question 31

The area under the normal curve between z = 0 and z = 1 is ________ the area under the normal curve between z = 1 and z = 2.

A) less than
B) greater than
C) equal to
D) less than, greater than, or equal to, depending on the value of the mean,
E) less than, greater than, or equal to, depending on the value of the standard deviation,

Question 32

A study shows that employees who begin their workday at 9:00 a.m. vary their times of arrival uniformly from 8:40 a.m. to 9:30 a.m. The probability that a randomly chosen employee reports to work between 9:00 and 9:10 is

A) 40 percent.
B) 20 percent.
C) 10 percent.
D) 30 percent.
E) 16.7 percent.

Question 33

A property of continuous distributions is that

A) as with discrete random variables, the probability distribution can be approximated by a smooth curve.
B) probabilities for continuous variables can be approximated using discrete random variables.
C) unlike discrete random variables, probabilities can be found using tables.
D) unlike discrete random variables, the probability that a continuous random variable equals a specific value is zero [P(X = x) = 0].

Question 34

It is appropriate to use the uniform distribution to describe a continuous random variable x when

A) the area under the probability curve = 1.
B) the probability curve f(x) > 0.
C) the shape of the histogram of all possible values of x is nonsymmetrical.
D) relative frequencies of all possible values of x are about the same.

Question 35

In order to approximate the binomial distribution using the normal distribution, the following condition(s) must be met if p is near 1.

A) np > 5 only
B) n must be larger than just meeting the condition of np > 5.
C) n can be as small as np > 5.
D) n > 5

Question 36

The relationship between the standard normal random variable, z, and normal random variable, X, is that

A) only the normal random variable X is continuous.
B) only the standard normal variable z is continuous.
C) the standard normal variable z counts the number of standard deviations that the value of the normal random variable X is away from its mean.
D) the values of the standard normal random variable z cannot be negative.
E) the values of the normal random variable X cannot be negative.

Question 37

The ________ is a graphic that is used to visually check whether data come from a normal population.

A) exponential plot
B) normal probability plot
C) box-and-whiskers plot
D) normal distribution graph

Question 38

The mean life of a pair of shoes is 40 months with a standard deviation of 8 months. If the life of the shoes is normally distributed, how many pairs of shoes out of one million will need replacement before 36 months?

A) 500,073
B) 808,573
C) 191,537
D) 308,537

Question 39

A standard normal distribution has a mean of ________ and standard deviation of ________.

A) zero, zero
B) zero, one
C) one, one
D) one, zero

Question 40

The ________ distribution would most likely be used to describe the distribution of time between arrivals of customers at the grocery store.

A) normal
B) exponential
C) Poisson
D) binomial
E) uniform

Question 41

If the random variable x is normally distributed, ________ percent of all possible observed values of x will be within three standard deviations of the mean.

A) 68.26
B) 95.44
C) 99.73
D) 100
E) None of these answers are correct.

Question 42

The number of standard deviations that a value, x, is from the mean is a(n) ________.

A) variance
B) exponential value
C) z score
D) area

Question 43

The area under the curve of a valid continuous probability distribution must ________.

A) equal 1
B) be between 0 and 1
C) be infinite
D) be less than 1

Question 44

A probability distribution that describes the time or space between successive occurrences of an event is a(n) ________ probability distribution.

A) binomial
B) uniform
C) exponential
D) normal

Question 45

A normal distribution with mean equal to zero and a standard deviation equal to one is called the ________ normal distribution.

A) continuous
B) uniform
C) exponential
D) standard

Question 46

The specific shape of each normal distribution is determined by its ________ and ________.

A) mean, standard deviation
B) median, standard deviation
C) mean, variance
D) median, variance

Question 47

The standard deviation of a standard normal distribution is always equal to ________.

A) 1
B) .5
C) 0
D) −1

Question 48

Given that X is a normal random variable, the probability that a given value of X is below its mean is ________.

A) 1
B) equal to .5
C) less than .5
D) greater than .5

Question 49

A continuous probability distribution that is useful in describing the time or space between successive occurrences of an event is a(n)

A) uniform probability distribution.
B) normal probability distribution.
C) Poisson probability distribution.
D) exponential probability distribution.

Question 50

When a ________ probability distribution is used to approximate a ________ probability distribution, we must make a ________ correction.

A) continuous, discrete, continuity
B) discrete, continuous, continuity
C) continuous, discrete, finite population
D) None of these answers are correct.

Question 51

The thickness of a randomly selected metal piece is a ________ random variable.

A) Poisson
B) continuous
C) discrete
D) binomial

Question 52

________ values of the standard deviation result in a normal curve that is wider and flatter.

A) Smaller
B) Equal
C) Larger
D) Uniform

Question 53

The z value tells us the number of ________ that a value of x is from the mean.

A) variances
B) standard deviations
C) medians
D) standard means

Question 54

________ values of the standard deviation result in a normal curve that is narrower and more peaked.

A) Smaller
B) Equal
C) Larger
D) Uniform

Question 55

A continuous probability distribution having a rectangular shape, where the probability is evenly distributed over an interval of numbers is a(n) ________ distribution.

A) exponential
B) binomial
C) normal
D) uniform

Question 56

If the random variable of x is normally distributed, ________ percent of all possible observed values of x will be within two standard deviations of the mean.

A) 99.73
B) 68.26
C) 95.00
D) 95.44

Question 57

The fill weight of a certain brand of adult cereal is normally distributed with a mean of 910 grams and a standard deviation of 5 grams. We calculated the value of z for a specific box of this brand of cereal, and the z value was negative. This negative z value indicates that

A) we made a mistake in our calculations, as z must always be a nonnegative number.
B) the fill weight is more than 910 grams.
C) the fill weight is less than 910 grams.
D) the fill weight is equal to 910 grams.
E) the fill weight may be more or less than 910 grams depending on the value of the standard deviation.

Question 58

________ theory is a methodology that attempts to determine the number of servers that strikes an optimum balance between the time customers wait for service and the cost of providing service.

A) Exponential
B) Binomial
C) Uniform distribution
D) Queuing

Question 59

Which of the following statements is not a property of the normal probability distribution?

A) The normal distribution is symmetric.
B) 95.44 percent of all possible observed values of the random variable x are within plus or minus three standard deviations of the population mean.
C) The mean, median, and mode are equal.
D) The area under the normal curve to the right of the mean is equal to the area under the normal curve to the left of the mean.
E) All of these answers are properties of the normal distribution.

Question 60

The mean of a standard normal distribution is always equal to ________.

A) 1
B) .5
C) 0
D) −1

Question 61

The time it takes a woman to travel from her apartment to the bus station follows a uniform distribution over the interval from 20 to 30 minutes. If she leaves home at 9:05 a.m., what is the probability that she will get to the station between 9:25 and 9:30 a.m.?

A) 1.0
B) 0
C) .5
D) .25

Question 62

Consider a normal population with a mean of 10 and a variance of 4. Find P(X < 6).

A) .0228
B) .1587
C) .8413
D) .9772

Question 63

The lifetime of a stereo component is exponentially distributed with mean 1,000 days. What is the probability that the lifetime of the stereo component exceeds 1,000 days?

A) .6321
B) .5000
C) .3679
D) 1.000

Question 64

The flying time of a drone airplane has a normal distribution with mean 4.76 hours and standard deviation of .04 hours. What is the probability that the drone will fly between 4.70 and 4.82 hours?

A) .1336
B) .8664
C) .9332
D) .4332

Question 65

Suppose the daily change in price of a stock is normally distributed with mean = .20 and standard deviation = .30. What price change is associated with the 25th percentile?

A) .1925
B) .2075
C) .401
D) −.002

Question 66

An aptitude test has a mean score of 80 and a standard deviation of 5. The population of scores is normally distributed. What proportion of tests has scores over 90?

A) .9772
B) .0228
C) .9544
D) .0456

Question 67

The flying time of a drone airplane has a normal distribution with mean 4.76 hours and standard deviation of .04 hours. What is the probability that the drone will fly more than 4.80 hours?

A) .1587
B) .8413
C) .6587
D) .3413

Question 68

The probability that an appliance is currently being repaired is .5. If an apartment complex has 100 such appliances, what is the probability that at least 60 are currently being repaired? Use the normal approximation to the binomial.

A) .5000
B) .0287
C) .6000
D) .9713

Question 69

If the mileage per gallon for a car is normally distributed, 32 mpg has a z score of 1.2, and 24 mpg has a z score of −.4, what is the mean mpg of the distribution?

A) 28
B) 26
C) 30
D) 38

Question 70

If x is a binomial random variable where n = 100 and p = .1, find the probability that x is less than or equal to 10, using the normal approximation to the binomial.

A) .9544
B) .0446
C) .5662
D) .4325

Question 71

The life of a light bulb is exponentially distributed with a mean of 1,000 hours. What is the probability that the bulb will last more than 1,200 hours?

A) .3012
B) .3679
C) .4345
D) .6988

Question 72

The flying time of a drone airplane has a normal distribution with mean 4.76 hours and standard deviation of .04 hours. What is the probability that the drone will fly less than 4.66 hours?

A) −.0062
B) .5062
C) .0062
D) .9938

Question 73

The time between breakdowns of an alarm system is exponentially distributed with mean 10 days. What is the probability that there is less than 1 breakdown on a given day?

A) .9048
B) .3679
C) .0952
D) .000

Question 74

Given that the length an athlete throws a hammer is a normal random variable with mean 50 feet and standard deviation 5 feet, what is the probability he throws it no less than 55 feet?

A) .8413
B) .1587
C) .6826
D) .3174

Question 75

The lifetime of a stereo component is exponentially distributed with mean 1,000 days. What is the probability that the lifetime of the stereo component is greater than or equal to 700 days?

A) .7603
B) .5034
C) .2397
D) .4966

Question 76

An aptitude test has a mean score of 80 and a standard deviation of 5. The population of scores is normally distributed. What raw score corresponds to the 70th percentile?

A) 77.4
B) 83.5
C) 82.6
D) 76.5

Question 77

The life of a light bulb is exponentially distributed with a mean of 1,000 hours. What is the probability that the bulb will last less than 800 hours?

A) .6321
B) .5507
C) .7135
D) .4493

Question 78

What is the probability that a standard normal random variable will be between −2 and 2?

A) .4772
B) .0228
C) .9772
D) .9544

Question 79

Given that the length an athlete throws a hammer is a normal random variable with mean 50 feet and standard deviation 5 feet, what is the probability he throws it between 50 feet and 60 feet?

A) .9972
B) .5000
C) .9544
D) .4772

Question 80

What is the probability that a standard normal random variable will be between .3 and 3.2?

A) .6179
B) .3814
C) .3808
D) .9993