Deck 7: Continuous Probability Distributions

Question 1

In any normal distribution,the mean,median,mode,and standard deviation are all at the same position on the horizontal axis.

Answer 1

False

Question 2

A continuous probability distribution represents a random variable:

A) having outcomes,which occur in counting numbers.
B) having an infinite number of outcomes which may assume any number of values within an interval.
C) which is best described in a histogram.
D) which has a definite probability for the occurrence of a given integer.
E) None of these is correct.

Answer 2

having an infinite number of outcomes which may assume any number of values within an interval.

Question 3

The standard normal distribution has a mean of 0.0 and standard deviation of 1.0.

Answer 3

True

Question 4

If Excel and Minitab packages are both used to generate its own sample of 10,000 observations from a theoretical continuous distribution,the two sets of observations are expected to be identical.

Question 5

In a continuous probability distribution,the probability that x will take on an exact value:

A) is equal to the height of the curve at that value.
B) is calculated using the probability density.
C) is always equal to 0.
D) is always greater than 0.
E) None of these is correct.

Question 6

The mean of the exponential distribution is the inverse of that of the normal distribution.

Question 7

In the normal distribution,the curve is asymptotic but never intercepts the horizontal axis either to the left or right.

Question 8

Which of the following statements are true regarding the areas beneath any normal curve?

A) About 68.3% of the area is in the interval $\mu - \sigma \text { to } \mu + \sigma .$
B) About 95.5% of the area is in the interval $\mu - 2 ? \text { to } \mu + 2 \sigma$
C) Nearly all of the area (about 99.7%)is in the interval $\mu - 3 \sigma \text { to } \mu + 3 \sigma$
D) All of these are true.
E) None of these is true.

Question 9

Any normal distribution is a symmetrical,bell-shaped curve,with mean $\mu$ = 0.0 and standard deviation $\text { ? }$ = 1.0.

Question 10

The standard deviation of the binomial distribution with n = 25 and $\pi$ = .50 is 6.25.

Question 11

Under certain conditions,the normal distribution can be used to approximate the binomial distribution.

Question 12

Which of the following is not a characteristic for a normal distribution?

A) It is symmetrical distribution
B) The mean is always zero
C) The mean,median,and mode are all equal
D) It is a bell-shaped distribution
E) The area under the curve always equals 1.0

Question 13

The exponential distribution is a family of discrete distributions,all of which are negatively skewed.

Question 14

There exists a different normal curve for every possible pair of $\mu$ and $\text { ?}$ .

Question 15

According to the law of large numbers,frequency distributions generated by statistical packages will more closely approach the theoretical distribution from which they were drawn whenever the number of observations becomes larger.

Question 16

Regardless of the shape of a particular normal curve,about 75% of the area is within the interval $\mu \pm \sigma$ ,where $\mu$ and $\text { ? }$ are the mean and the standard deviation,respectively.

Question 17

Continuous probability distributions describe probabilities associated with random variables that can take on any value along a given range or continuum and for which there are no gaps between these possible values.

Question 18

Computer statistical packages,such as Excel and Minitab,can determine the exact area beneath a specified portion of the continuous distribution curve,or probability density function,and eliminate the need for calculations or table references.

Question 19

In the normal distribution,the total area beneath the curve represents the probability for all possible outcomes for a given event.

Question 20

For any specified interval of values,the probability that a continuous random variable x will assume a value within the interval is the area beneath the curve between the two points describing the interval.

Question 21

Given that z is a standard normal random variable,a negative value of z indicates that:

A) The value z is to the left of the mean.
B) The value z is to the right of the median.
C) The standard deviation of z is negative.
D) The area between zero and z is negative.
E) The probability associated with z is negative.

Question 22

For a normal curve,if the mean is 20 minutes and the standard deviation is 5 minutes,then the area between 22 and 25 minutes is:

A) 0.3413.
B) 0.1554.
C) 0.4967.
D) 0.1859.
E) 0.2248.

Question 23

Using the standard normal table,the total area between z = -1.82 and z = -1.38 is:

A) 0.0494
B) 0.4656
C) 0.1554
D) 0.4162
E) 0.1005

Question 24

For a normal curve,if the mean is 20 minutes and the standard deviation is 5 minutes,then the area between 11 and 19 minutes is:

A) 0.4641.
B) 0.0359.
C) 0.3848.
D) 0.8848.
E) 0.9641.

Question 25

The average labor time to sew a pair of denim jeans is 4.2 hours with a standard deviation of 30 minutes.If the distribution is normal,then the probability of a worker finishing a pair of jeans in less than 3.5 hours is:

A) 0.0808.
B) 0.4192.
C) 0.5808.
D) 0.9192.
E) 0.9808.

Question 26

Using the standard normal table,the total area between z = 0.45 and z = 1.05 is:

A) 0.1736
B) 0.3531
C) 0.3264
D) 0.6469
E) 0.1795

Question 27

The z-score representing the 75th percentile of the standard normal distribution is:

A) 0.67.
B) -0.67.
C) 1.28.
D) -1.28.
E) 1.645.

Question 28

A retailer finds that the demand for a very popular board game averages 100 per week with a standard deviation of 20.If the seller wishes to have adequate stock 95% of the time,how many of the games must she keep on hand?

A) 139.2
B) 100.0
C) 132.9
D) 195.0
E) 125.6

Question 29

Given that z is a standard normal random variable,and that the area to the right of z is 0.9066,then the value of z must be:

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

Question 30

A bakery firm finds that its average weight of the most popular package of cookies is 32.06 ounces with a standard deviation of 0.1 ounces.What portion of cookie packages will weigh less than 31.9 ounces or more than 32.3 ounces?

A) 0.8000
B) 0.4452
C) 0.4918
D) 0.9370
E) 0.0630

Question 31

The z-score representing the 90th percentile of the standard normal distribution is:

A) 1.28.
B) -1.28.
C) 0.67.
D) -0.67.
E) 1.645.

Question 32

Given that z is a standard normal random variable and that the area to the right of z is 0.1949,then the value of z must be:

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

Question 33

The proportion of the data from a standard normal distribution that falls within two standard deviations from the mean is:

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

Question 34

A manufacturer of tow chains finds that the average breaking point is at 3,500 pounds and the standard deviation is 250 pounds.If you pull a weight of at least 4200 pounds with this tow chain,what percentage of the time would you expect the chain to break?

A) 2.8%
B) 0.26%
C) 49.74%
D) 99.74%
E) 50%

Question 35

A salesman who uses his car extensively finds that his gasoline bills average $125.32 per month with a standard deviation of $49.51.The probability that his bill will be less than $50 a month or more than $150 for a single month is:

A) 0.4357.
B) 0.1915.
C) 0.6272.
D) 0.3728.
E) 0.6915.

Question 36

Using the standard normal table,the total area between z = -0.75 and z = 1.21 is:

A) 0.2734
B) 0.3869
C) 0.3397
D) 0.2266
E) 0.6603

Question 37

A videotape store has an average weekly gross of $1,158 with a standard deviation of $120.Let x be the store's gross during a randomly selected week.If this is a normally distributed random variable,then the number of standard deviations from $1,158 to $1,360 is:

A) 0.4535.
B) 0.0465.
C) 20.98.
D) 1.683.
E) none of these.

Question 38

The z-score representing the 10th percentile of the standard normal distribution is:

A) 0.255.
B) -0.255.
C) -1.645.
D) 1.28.
E) -1.28.

Question 39

If the area to the right of a positive z₁ is 0.0869,then the value of z₁ must be:

A) -1.36.
B) 1.36.
C) 1.71.
D) 1.80.
E) 0.22.

Question 40

Let z₁ be a z score that is unknown but identifiable by position and area.If the area to the right of z₁ is 0.8413,then the value of z₁ must be:

A) 1.00.
B) -1.00.
C) 0.00.
D) 0.41.
E) -0.41.

Question 41

If the mean of an exponential distribution is 2,then the value of the parameter $\lambda$ is:

A) 4.0.
B) 2.2.
C) 1.0.
D) 0.5.

Question 42

Like the normal distribution,the exponential density function f(x):

A) is bell-shaped.
B) is symmetrical.
C) approaches infinity as x approaches zero.
D) approaches zero as x approaches infinity.

Question 43

If the z-value for a given value x of the random variable x is z = 1.96,and the distribution of x is normally distributed with a mean of 60 and a standard deviation of 6,then the x-value that this z-value corresponds to is:

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

Question 44

If the random variable x is exponentially distributed with parameter $\lambda$ = 4,then P(x ? 0.25),up to 4 decimal places,is:

A) 0.6321.
B) 0.3679.
C) 0.2500.
D) 0.5000.

Question 45

Assume x is normally distributed with mean = 15 and standard deviation = 3.Use the approximate areas beneath the normal curve,as discussed in this section,to find P(x 15).

Question 46

A dispatcher for an airport shuttle averages sending a van to the airport 2 times every 60 minutes.The distribution is Poisson,and the driver must take a 15-minute lunch break.The probability that he can finish lunch before getting a call is:

A) 0.1350.
B) 0.6065.
C) 0.3935.
D) 1.6490.
E) 0.8650.

Question 47

Your high school graduating class had 564 members.Thirty-three percent of these are expected to attend college.If the number of students who attend college follows the normal distribution,the probability that less than 160 will attend college is:

A) 0.3610.
B) 0.4900.
C) 0.0090.
D) 0.0832.
E) 0.2500

Question 48

In your college campus,30% of the students use tobacco in some form.In your statistics class of 40 students,the probability of finding at least 15 students who use tobacco is:

A) 0.3051.
B) 0.1949.
C) 0.8051.
D) 0.1515.
E) 0.3485.

Question 49

Which of the following is not true for an exponential distribution with parameter $\lambda$ ?

A) Mean = $1 / \lambda$
B) Standard deviation = $1 / \lambda$
C) The distribution is completely determined once the value of $\lambda$ is known.
D) The distribution is a two-parameter distribution since the mean and standard deviation are equal.

Question 50

A local radio station maintains a very popular phone service,which provides callers with current weather reports.The incoming calls have a Poisson distribution with an average of 5 calls for each 15 minute-period.If x = time between calls,then the probability of receiving four or fewer calls in the next 15 minutes is:

A) 0.2341.
B) 0.7659.
C) 0.2636.
D) 0.7364.
E) 0.9502.

Question 51

Which of the following distributions is suitable to model the length of time that elapses before the first telephone call is received by a switchboard?

A) Exponential
B) Normal
C) Poisson
D) Binomial
E) Hypergeometric

Question 52

A large mail house which mails such items as catalogues,magazines,and other bulk mailings guarantees that there will be no more than a 3% error rate on its mailing labels.A customer who contracted a mailing to 190,000 individuals experienced a return of 5,900 items,which had incorrect addresses.Which of the following statements is true?

A) There is a 3% chance of an incorrect return.
B) There is a 0.0054 probability that a return of 5900 or more incorrect addresses could occur if the true error rate is 3%.
C) There is a 0.4964 probability that a return of 5900 or more incorrect addresses could occur if the true error rate is 3%.
D) There is a 2.69 percent possibility that a return of 5900 incorrect addresses could occur if the true error rate is 3%.
E) None of these statements are correct.

Question 53

Which of the following distributions is appropriate to measure the length of time between arrivals at a grocery checkout counter?

A) Binomial
B) Normal
C) Exponential
D) Poisson
E) Hypergeometric

Question 54

Which of the following is not a correct statement?

A) The exponential distribution describes the Poisson process as a continuous random variable.
B) The exponential distribution is a family of curves,which are completely described by the mean.
C) The mean of the exponential distribution is the complement of the mean of the Poisson.
D) The Poisson is a probability distribution or a discrete random variable while the exponential distribution is continuous.
E) All of these are correct.

Question 55

Which of the following statements are true regarding the normal distribution curve?

A) It is symmetrical.
B) It is bell-shaped.
C) It is asymptotic in that each end approaches the horizontal axis,but never reaches it.
D) Its mean,median,and mode are located at the same position.
E) All of these statements are true.

Question 56

Assume x is normally distributed with mean = 15 and standard deviation = 3.Use the approximate areas beneath the normal curve,as discussed in this section,to find P(12 x 18).

Question 57

A zipper manufacturer has found that 1.5% of its zippers are defective.The company averages 4982 zippers per day.The probability of finding between 90 and 100 defects in a given day is:

A) 0.4981.
B) 0.0310.
C) 0.0360.
D) 0.5310.
E) 0.4671.

Question 58

If the random variable x is exponentially distributed with parameter $\lambda$ = 1.5,then P(2 ? x ? 4),up to 4 decimal places,is:

A) 0.6667.
B) 0.0473.
C) 0.5000.
D) 0.2500.

Question 59

The mean of the exponential distribution equals the mean of the Poisson distribution only when the former distribution has a mean equals:

A) 1.00.
B) 0.50.
C) 0.25.
D) 2.00.

Question 60

A very large logging operation has serious problems keeping their skidders operating properly.The equipment fails at the rate of 3 breakdowns every 48 hours.Assume that x is time between breakdowns and is exponentially distributed.The probability of two or less breakdowns in the next 48-hour period is:

A) 0.9672.
B) 0.0307.
C) 0.2231.
D) 0.7769.

Question 61

In 2000,the average charge of tax preparation was $95.Assuming a normal distribution and a standard deviation of $10,use the approximate areas beneath the normal curve,as discussed in this section,to answer: What proportion of tax preparation fees were between $75 and $115?

Question 62

Determine P(z 1.58)for the standard normal curve.

Question 63

Assume the standard normal curve and determine the probability or area between z = -1.28 and z = 1.28.

Question 64

Assume x is normally distributed with mean = 15 and standard deviation = 3.Use the approximate areas beneath the normal curve,as discussed in this section,to find P(9 x 21).

Question 65

Find the z-score that determines that the area to the left of z is 0.9292.

Question 66

For the standard normal curve,determine the area to the right of z = 0.34.

Question 67

Using the standard normal curve,find the area between z = 0 and z = 3.3. ____________________.

Question 68

Determine P(-2.45 z 0),assuming a standard normal distribution.

Question 69

Assuming a standard normal distribution,determine P(z -0.75).

Question 70

In 2000,the average charge of tax preparation was $95.Assuming a normal distribution and a standard deviation of $10,use the approximate areas beneath the normal curve,as discussed in this section,to answer: What proportion of tax preparation fees were more than $95?

Question 71

Find z₁ if the area to the left of a positive z₁ is 0.9131.

Question 72

In 2000,the average charge of tax preparation was $95.Assuming a normal distribution and a standard deviation of $10,use the approximate areas beneath the normal curve,as discussed in this section,to answer: What proportion of tax preparation fees were between $80 and $90?

Question 73

Find the z-score that determines that the area to the right of z is 0.8264.

Question 74

Assume x is normally distributed with mean = 15 and standard deviation = 3.Use the approximate areas beneath the normal curve,as discussed in this section,to find P(x 12).

Question 75

Using the standard normal curve,find the area to the left of z = -2.26.

Question 76

Assume x is normally distributed with mean = 15 and standard deviation = 3.Use the approximate areas beneath the normal curve,as discussed in this section,to find P(x 9).

Question 77

In 2000,the average charge of tax preparation was $95.Assuming a normal distribution and a standard deviation of $10,use the approximate areas beneath the normal curve,as discussed in this section,to answer: What proportion of tax preparation fees were between $85 and $105?

Question 78

In 2000,the average charge of tax preparation was $95.Assuming a normal distribution and a standard deviation of $10,use the approximate areas beneath the normal curve,as discussed in this section,to answer: What proportion of tax preparation fees were more than $115?

Question 79

Determine P(-2.45 z 1.69),assuming a standard normal distribution.

Question 80

Assume x is normally distributed with mean = 15 and standard deviation = 3.Use the approximate areas beneath the normal curve,as discussed in this section,to find P(x = 20).