Question 1

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) 500,000
B) 808,500
C) 191,500
D) 308,500

Accepted Answer

The answer of The mean life of a pair of...

Question 2

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

Accepted Answer

The exponential distribution is commonly used to model the time between events in a Poisson process, where events occur randomly and independently over time. In this case, customers arriving at a grocery store could be considered a Poisson process, and the time between their arrivals would follow an exponential distribution.

Question 3

If the random variable X 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.

Accepted Answer

The formula for standardizing a random variable X with mean µ and standard deviation σ is given by (X - µ)/σ. 
Therefore, the mean of (X - µ)/σ is (µ - µ)/σ = 0/σ = 0.
The standard deviation of (X - µ)/σ is the standard deviation of X divided by σ, which is 1. 
Hence, the answer is D, 0 and 1.

Question 4

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 the choices is correct.

Accepted Answer

The standard normal random variable has a mean of 0 and a standard deviation of 1. Therefore, its values are measured in terms of how many standard deviations they are away from the mean. For example, a value of 1.5 represents 1.5 standard deviations above the mean.

Question 5

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.

Accepted Answer

The uniform distribution is characterized by all possible values of x having about the same frequency, so D is the best choice. Choices A and B describe properties that apply to all continuous probability distributions, not just the uniform distribution. Choice C describes a property that applies to a nonsymmetric distribution, but the uniform distribution is by definition symmetric.

Question 6

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

Accepted Answer

This is because as n gets closer and p gets closer to 0.5, the graph of a binomial distribution tends to have the symmetrical, bell shaped appearance of a normal curve.

Question 7

Which of the following is a continuous distribution?
A) Poisson
B) hypergeometric
C) exponential
D) binomial

Accepted Answer

The exponential distribution is a continuous probability distribution that deals with the amount of time between two independent events. The other options (Poisson, hypergeometric, and binomial) are all discrete distributions.

Question 8

A standard normal distribution has a mean of ____________ and standard deviation of ____________.
A) zero,zero
B) zero,one
C) one,one
D) one,zero

Accepted Answer

A standard normal distribution has a mean of zero and a standard deviation of one. This is what defines it as "standard" – it is the reference distribution against which other distributions can be compared and standardized. The other choices are not correct – a mean of zero and a standard deviation of zero would mean that all values are identical, and a mean of one and a standard deviation of one would indicate a different distribution than the standard normal.

Question 9

The ___________ is a graphic that is used to check visually whether data come from a normal population.
A) exponential plot
B) normal probability plot
C) box-and-whiskers plot
D) normal distribution graph

Accepted Answer

A normal probability plot, also known as a Q-Q plot (quantile-quantile plot), is a common graph used in statistics to assess whether a set of data may come from a normal distribution. It plots the data against a theoretical normal distribution, and if the points fall along a straight line, it suggests that the data may come from a normal population.

Question 10

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.

Accepted Answer

The standard normal variable z is defined as the number of standard deviations that a value of the normal random variable X is away from its mean. Both z and X are continuous random variables, and their values can be negative or positive. Therefore, options A, B, D, and E are incorrect.

Question 11

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.

Accepted Answer

Since the average is $15 and you are concerned with those making more than $13 an hour, $13 would be to the left of $15 on the graph. There would a higher proportion of those to the right of $13 than the left because most of the data would be centered around $15 since that is the average.

Question 12

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].

Accepted Answer

This is one of the standard properties of continuous distributions, where the probability of a continuous random variable taking a specific value is always zero. This is because there are infinitely many possible values for a continuous random variable, and the chance of it exactly equaling any one of those values is infinitesimally small.

Question 13

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,

Accepted Answer

The normal curve centers around the mean of zero, and more data is concentrated close to the mean than further out by the tails.

Question 14

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

Accepted Answer

A negative z value indicates that the student's score is below the mean. Therefore, the student scored below the average of the class.

Question 15

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.

Accepted Answer

The standardized normal quantile value that gives an area of i/(n − 1) to its right under the normal curve is OS1U1B1iS1U0B.

Question 16

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.

Accepted Answer

A standard normal variable value of z = 1 corresponds to a value above the mean of the normal distribution. Therefore, a firm with a z-score of 1 has an above average price-to-earnings ratio.

Question 17

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 exceeds 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.

Accepted Answer

A negative z value indicates that the observed value (fill weight in this case) is below the mean (910 grams). Therefore, the fill weight is less than 910 grams.

Question 18

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.

Accepted Answer

If p is close to 0 or 1, then n needs to be even larger.

Question 19

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.

Accepted Answer

To construct a normal probability plot, the first step is to order the values in the data. This allows for the determination of the percentiles of the data, which are then plotted on the y-axis against corresponding percentiles from a normal distribution on the x-axis.

Question 20

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

Accepted Answer

For a continuous random variable, probability at a single value = 0.

Quiz 6: Continuous Random Variables

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?

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

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

Values of the standard normal random variable are measured

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

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

Which of the following is a continuous distribution?

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

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

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

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 property of continuous distributions is that

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.

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

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.

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

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

The normal approximation of the binomial distribution is appropriate when

The first step to constructing a normal probability plot is to

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) ?

Quiz 6: Continuous Random Variables

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?

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

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

Values of the standard normal random variable are measured

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

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

Which of the following is a continuous distribution?

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

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

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

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 property of continuous distributions is that

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.

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

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

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

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

The normal approximation of the binomial distribution is appropriate when

The first step to constructing a normal probability plot is to

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

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.