Question 1

The MPG (Miles per Gallon)for a mid-size car is normally distributed with a mean of 32 and a standard deviation of .8.What is the probability that the MPG for a selected mid-size car would be: Less than 33.2?
A)43.32%
B)6.68%
C)93.32%
D)86.64%
E)13.36%

Accepted Answer

To find the probability that the MPG for a selected mid-size car is less than 33.2, we need to standardize the value using z-score formula.
z=(x-μ)/σ 
where x=33.2,μ=32, and σ=0.8
z=(33.2-32)/0.8=1.5
Using the standard normal distribution table or calculator, we find the probability that z is less than 1.5 to be 0.9332 or 93.32%. Therefore, the probability that the MPG for a selected mid-size car would be less than 33.2 is 93.32%.

Question 2

The MPG (Miles per Gallon)for a mid-size car is normally distributed with a mean of 32 and a standard deviation of .8.What is the probability that the MPG for a selected mid-size car would be: More than 33.2?
A)43.32%
B)6.68%
C)93.32%
D)86.64%
E)13.36%

Accepted Answer

We need to calculate the probability of the MPG being above 33.2, which can be done by standardizing the value using the formula z = (x - μ) / σ, where x is the value of 33.2, μ is the mean of 32 and σ is the standard deviation of 0.8. Thus, z = (33.2 - 32) / 0.8 = 1.5. Using a standard normal distribution table or calculator, the probability of getting a value beyond 1.5 is 0.0668 or 6.68%, which corresponds to choice B.

Question 3

The normal approximation of the binomial distribution is appropriate when:
A)np $\ge$ 5
B)n(1 - p) $\ge$ 5
C)np $\le$ 5
D)n(1 - p)$\le$ .5 and np $\le$ 5
E)np $\ge$ .5 and n(1 - p)$\ge$ 5

Accepted Answer

The normal approximation of the binomial distribution is appropriate when np≥5 and n(1-p)≥5. Hence, option E is the correct choice as it satisfies both these conditions.

Question 4

If the random variable of x is normally distributed,____% 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 the above

Accepted Answer

The rule for normal distribution is that approximately 68% of all possible observed values will be within one standard deviation of the mean, approximately 95% will be within two standard deviations, and approximately 99.7% will be within three standard deviations. Therefore, the answer is option C - 99.73%.

Question 5

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 counts the number of standard deviations that the value of the normal random variable X is away from its mean. This relationship is given by the formula Z = (X - μ) / σ, where μ and σ are the mean and standard deviation of the normal random variable X. The standard normal variable Z is continuous, and its values can be negative or positive. While the normal random variable X is also continuous, its values can be negative or positive.

Question 6

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. If we select one box of cereal at random from this population,what is the probability that it will weigh less than 900 grams?
A)4772
B).9772
C).9544
D).0228
E).0456

Accepted Answer

We need to find the probability that a box of cereal weighs less than 900 grams. Using the given mean and standard deviation, we can find the z-score:

z = (x - μ) / σ = (900 - 910) / 5 = -2

Using a standard normal distribution table or calculator, we can find the probability that a z-score is less than -2:

P(z < -2) = 0.0228

Therefore, the probability that a box of cereal weighs less than 900 grams is 0.0228 or 2.28%. The best choice is D.

Question 7

Which of the following statements is not a property of the normal probability distribution?
A)The normal distribution is symmetric.
B)95.44% 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 the above answers are properties of the normal distribution.

Accepted Answer

95.44% of all possible observed values of the random variable x are within plus or minus two standard deviations of the population mean, not three.

Question 8

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. If we select one box of cereal at random from this population,what is the probability that it will weigh more than 904 grams?
A).8849
B).3849
C).1151
D).7698
E).2302

Accepted Answer

The probability that a randomly selected box of cereal will weigh more than 904 grams is 0.8849. This is because the z-score for 904 grams is (904 - 910) / 5 = -1.2, and the area under the normal curve to the right of -1.2 is 0.8849.

Question 9

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 describe the distribution of time between successive events, such as the arrival of customers at a grocery store. It is a continuous probability distribution that has a single parameter, lambda (λ), which represents the rate at which events occur.

Question 10

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 makes it "standard" - it can be used to compare and analyze other normal distributions with different means and standard deviations.

Question 11

If the random variable X has a mean of µ and a standard deviation  $\sigma$ then (X - µ)/ $\sigma$ has a mean and standard deviation respectively:
A) 
B) 
C)1 and 0
D)0 and 1

Accepted Answer

The answer of If the random variable X has a...

Question 12

A study shows that employees that begin their work day 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%
B)20%
C)10%
D)30%
E)16.7%

Accepted Answer

The total range of arrival times is 50 minutes (from 8:40 to 9:30). The time interval from 9:00 to 9:10 is 10 minutes. The probability is the ratio of the interval to the total range: 10/50 = 0.20 or 20%.

Question 13

The price-to-earning ratio for firms in a given industry is distributed according to normal distribution.In this industry,a firm with a standard normal variable value of Z = 1:
A)Has an above average price-to-earning 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

Since the standard normal distribution has a mean of 0 and a standard deviation of 1, a Z-score of 1 corresponds to one standard deviation above the mean. Therefore, a firm with a Z-score of 1 would have a price-to-earnings ratio that is above average compared to the rest of the industry.

Question 14

Which of the following is a continuous distribution?
A)Poisson
B)Hypergeometric
C)Exponential
D)Binomial

Accepted Answer

The only option in the list that represents a continuous distribution is the Exponential distribution. The other three, Poisson, Hypergeometric, and Binomial, are discrete distributions.

Question 15

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%.
D)Is less than the proportion earning more than the mean wage.

Accepted Answer

Since the wage distribution is normal with mean $15 per hour, we can assume that it is symmetric about the mean. Therefore, the proportion of workers earning more than $13 per hour is equal to the proportion of workers earning less than $17 per hour (since the distribution is continuous). Since $17 is closer to the mean of $15 than $13 is, the proportion of workers earning more than $13 per hour is greater than the proportion earning less than $13 per hour. Therefore, choice A is true. The other options are not necessarily true based on the information given.

Question 16

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)A,B,or C above dependent on the value of the mean
E)A,B,or C above dependent on the value of the standard deviation

Accepted Answer

The area under the normal curve between z = 0 and z = 1 is greater than the area between z = 1 and z = 2 because the normal distribution is symmetric and the density decreases as you move away from the mean (z = 0).

Question 17

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 above

Accepted Answer

Values of the standard normal random variable are measured in the number of standard deviations from the mean. This is because the standard normal distribution has a mean of 0 and a standard deviation of 1, so its values are expressed in terms of how many standard deviations away from the mean they are.

Question 18

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 true for continuous random variables due to the infinite number of possible values between any two values. Therefore, the probability of getting exactly one specific value is zero.

Question 19

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 (in this case, fill weight) is below the mean. Therefore, the fill weight is less than 910 grams. Option A is incorrect because z can take negative values, and options B, D, and E are incorrect because a negative z value does not indicate that the fill weight exceeds, equals or may be more or less than 910 grams depending on the standard deviation.

Question 20

A student's grade on an examination was transformed to a z value which is negative.Therefore,we know that he scored:
A)Higher than 16% of the class
B)Higher than 45% of the class
C)Above the first quartile
D)Below the mean
E)Lower than 16% of the class

Accepted Answer

A negative z value indicates that the student's score is below the mean. Therefore, choice D is correct. None of the other options can be determined from the given information.

Quiz 6: Continuous Random Variables

The MPG (Miles per Gallon) for a mid-size car is normally distributed with a mean of 32 and a standard deviation of .8.What is the probability that the MPG for a selected mid-size car would be: Less than 33.2?

The MPG (Miles per Gallon) for a mid-size car is normally distributed with a mean of 32 and a standard deviation of .8.What is the probability that the MPG for a selected mid-size car would be: More than 33.2?

The normal approximation of the binomial distribution is appropriate when:

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

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

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. If we select one box of cereal at random from this population,what is the probability that it will weigh less than 900 grams?

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

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. If we select one box of cereal at random from this population,what is the probability that it will weigh more than 904 grams?

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

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

If the random variable X has a mean of µ and a standard deviation $\sigma$ then (X - µ) / $\sigma$ has a mean and standard deviation respectively:

A study shows that employees that begin their work day 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:

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

Which of the following is a continuous distribution?

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:

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.

Values of the standard normal random variable are measured

A property of continuous distributions is that:

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 student's grade on an examination was transformed to a z value which is negative.Therefore,we know that he scored: