Question 1

Find the z value such that .
A) z = -1.645
B) z = -1.96
C) z = 1.645
D) z = 1.96

Accepted Answer

The answer of Find the z value such that .
A)...

Question 2

Suppose the average price of gasoline for a city in the United States follows a continuous uniform distribution with a lower bound of $3.50 per gallon and an upper bound of $3.80 per gallon.What is the probability a randomly chosen gas station charges more than $3.70 per gallon?
A) 0.3000
B) 0.3333
C) 0.6667
D) 1.0000

Accepted Answer

Since the average price of gasoline follows a continuous uniform distribution, the probability of a gas station charging more than $3.70 per gallon is equal to the proportion of the distribution to the right of $3.70. This proportion can be calculated as:

$$\frac{3.80-3.70}{3.80-3.50} = \frac{0.10}{0.30} = \frac{1}{3}$$

So the probability is 0.3333, which is closest to choice B.

Question 3

The probability that a normal random variable is less than its mean is ___.
A) 0.0
B) 0.5
C) 1.0
D) Cannot be determined

Accepted Answer

By the definition of a normal distribution, 50% of the area under the curve is to the left of the mean and 50% is to the right of the mean. Therefore, the probability that a normal random variable is less than its mean is 0.5.

Question 4

Find the z value such that =0.9082.
A) z = -1.33
B) z = 0.1814
C) z = 0.8186
D) z = 1.33

Accepted Answer

Using a standard normal distribution table, we can find the z value that corresponds with a cumulative area of 0.9082 to be approximately 1.33. Therefore, choice D is the best answer.

Question 5

You work in marketing for a company that produces work boots.Quality control has sent you a memo detailing the length of time before the boots wear out under heavy use.They find that the boots wear out in an average of 208 days,but the exact amount of time varies,following a normal distribution with a standard deviation of 14 days.For an upcoming ad campaign,you need to know the percent of the pairs that last longer than six months-that is,180 days.Use the empirical rule to approximate this percent.
A) 2.5%
B) 5%
C) 95%
D) 97.5%

Accepted Answer

According to the empirical rule, for a normally distributed dataset, approximately 68% of the data falls within one standard deviation of the mean, approximately 95% falls within two standard deviations of the mean, and approximately 99.7% falls within three standard deviations of the mean. In this case, we want to know the percentage of pairs that last longer than six months (180 days), which is approximately two standard deviations below the mean (208-2*14=180). Therefore, we can expect that approximately 2.5% of the pairs will last longer than six months (100% - 95%). The closest answer choice is D) 97.5%, which is the complement of the percentage we found.

Question 6

How many parameters are needed to fully describe any normal distribution?
A) 1
B) 2
C) 3
D) 4

Accepted Answer

A normal distribution is fully described by its mean and standard deviation, which are two parameters. Therefore, the best choice is B, with two parameters required.

Question 7

Find the probability P(-1.96 ≤ Z ≤ 1.96).
A) 0.0500
B) 0.9500
C) 0.9750
D) 1.9500

Accepted Answer

By the standard normal distribution table, the area to the left of 1.96 is 0.9750 and the area to the left of -1.96 is also 0.0250. Therefore, the area between -1.96 and 1.96 is the difference between these two areas: 0.9750 - 0.0250 = 0.9500. This represents the probability that a standard normal random variable falls between -1.96 and 1.96.

Question 8

The time to complete the construction of a soapbox derby car is normally distributed with a mean of three hours and a standard deviation of one hour.Find the probability that it would take more than five hours to construct a soapbox derby car.
A) 0
B) 0.0228
C) 0.4772
D) 0.9772

Accepted Answer

We need to find the probability of taking more than 5 hours to construct the car. Let X be the random variable for the time to complete the construction.
P(X > 5) = 1 - P(X ≤ 5)
To find the probability of X ≤ 5, we need to standardize the value using z-score:
z = (X - μ) / σ = (5 - 3) / 1 = 2
Using the standard normal distribution table, we find that P(Z ≤ 2) = 0.9772
Therefore, P(X ≤ 5) = P(Z ≤ 2) = 0.9772
So, P(X > 5) = 1 - P(X ≤ 5) = 1 - 0.9772 = 0.0228
Hence, the correct choice is B.

Question 9

The probability P(Z > 1.28)is closest to ____.
A) -0.10
B) 0.10
C) 0.20
D) 0.90

Accepted Answer

The probability P(Z < −1.28) refers to the area under the standard normal distribution curve to the left of Z = -1.28. Consulting standard normal (Z) tables or using statistical software, we find this area is approximately 0.10, indicating that the probability is closest to 0.10.

Question 10

It is known that the length of a certain product X is normally distributed with $\mu$ = 20 inches.How is the probability related to ?
A) is greater than
B) is smaller than
C) is the same as .
D) No comparison can be made with the given information.

Accepted Answer

The probability is related to the area under the normal distribution curve that is greater than .

Question 11

What does it mean when we say that the tails of the normal curve are asymptotic to the x axis?
A) The tails get closer and closer to the x axis but never touch it.
B) The tails gets closer and closer to the x axis and eventually touch it.
C) The tails get closer and closer to the x axis and eventually cross this axis.
D) The tails get closer and closer to the x axis and eventually become this axis.

Accepted Answer

Asymptotic means getting closer and closer without actually touching or crossing. Therefore, option A is the best choice as it accurately describes the behavior of the tails of a normal curve.

Question 12

An analyst is forecasting net income for Excellence Corporation for the next fiscal year.Her low-end estimate of net income is $250,000,and her high-end estimate is $350,000.Prior research allows her to assume that net income follows a continuous uniform distribution.The probability that net income will be greater than or equal to $337,500 is _______.
A) 12.5%
B) 29.6%
C) 87.5%
D) 96.4%

Accepted Answer

The answer of An analyst is forecasting net income for...

Question 13

The probability P(Z < -1.28)is closest to ____.
A) -0.10
B) 0.10
C) 0.20
D) 0.90

Accepted Answer

Using a standard normal distribution table or calculator, we find that the area to the left of -1.28 is 0.1003, which is closest to 0.10 (answer B).

Question 14

For any normally distributed random variable with mean μ and standard deviation σ,the percent of the observations that fall between [μ - 2σ,μ + 2σ] is the closest to ______.
A) 68%
B) 68.26%
C) 95%
D) 95.44%

Accepted Answer

For a normal distribution, approximately 95.44% of the data falls within two standard deviations (μ ± 2σ) from the mean.

Question 15

A hedge fund returns on average 26% per year with a standard deviation of 12%.Using the empirical rule,approximate the probability the fund returns over 50% next year.
A) 0.5%
B) 1%
C) 2.5%
D) 5%

Accepted Answer

The answer of A hedge fund returns on average 26%...

Question 16

Find the probability P(-1.96 ≤ Z ≤ 0).
A) 0.0250
B) 0.0500
C) 0.4750
D) 0.5250

Accepted Answer

The area under the standard normal curve between -1.96 and 0 represents the probability that a standard normal random variable falls between these two values. Using a standard normal table, the area between -1.96 and 0 is 0.4750.

Question 17

The time to complete the construction of a soapbox derby car is normally distributed with a mean of three hours and a standard deviation of one hour.Find the probability that it would take between 2.5 and 3.5 hours to construct a soapbox derby car.
A) 0.3085
B) 0.3830
C) 0.6170
D) 0.6915

Accepted Answer

To solve this problem, we need to use the standard normal distribution, with a mean of 0 and a standard deviation of 1. We can convert the given values to z-scores using the formula:
z = (x - μ) / σ

For x = 2.5:
z = (2.5 - 3) / 1 = -0.5

For x = 3.5:
z = (3.5 - 3) / 1 = 0.5

Using a standard normal distribution table or calculator, we can find the area between these two z-scores:
P(-0.5 < Z < 0.5) = 0.3830

Therefore, the probability that it would take between 2.5 and 3.5 hours to construct a soapbox derby car is 0.3830 or approximately 38.3%. The closest choice to this answer is B.

Question 18

Alex is in a hurry to get to work and is rushing to catch the bus.She knows that the bus arrives every six minutes during rush hour,but does not know the exact times the bus is due.She realizes that from the time she arrives at the stop,the amount of time that she will have to wait follows a uniform distribution with a lower bound of 0 minutes and an upper bound of six minutes.What is the probability that she will have to wait more than two minutes?
A) 0.1667
B) 0.3333
C) 0.6667
D) 1.0000

Accepted Answer

The probability that Alex will have to wait more than two minutes is calculated by finding the proportion of the interval from 2 to 6 minutes out of the total interval from 0 to 6 minutes. This is (6 - 2) / (6 - 0) = 4 / 6 = 2 / 3, which is 0.6667 or 66.67%.

Question 19

Sarah's portfolio has an expected annual return at 8%,with an annual standard deviation at 12%.If her investment returns are normally distributed,then in any given year Sarah has approximately ______________________.
A) A 50% chance that the actual return will be greater than 8%
B) About a 68% chance that the actual return will fall within 4% and 20%
C) About a 68% chance that the actual return will fall within -20% and 20%
D) About a 95% chance that the actual return will fall within -4% and 28%.

Accepted Answer

The answer of Sarah's portfolio has an expected annual return...

Question 20

For any normally distributed random variable with mean μ and standard deviation σ,the proportion of the observations that fall outside the interval [μ - σ,μ + σ] is the closest to _____.
A) 0.0466
B) 0.3174
C) 0.8413
D) 0.1687

Accepted Answer

According to the Empirical Rule, approximately 68% of the observations fall within one standard deviation of the mean for a normal distribution. Therefore, approximately 32% of the observations fall outside this interval. Half of this 32% falls outside the interval [μ - σ,μ] while the other half falls outside the interval [μ,μ + σ]. Therefore, the proportion of observations that fall outside the interval [μ - σ,μ + σ] is approximately 0.32/2 = 0.16, which is closest to 0.3174 (choice B).

Quiz 6: Continuous Probability Distributions

Find the z value such that .

Suppose the average price of gasoline for a city in the United States follows a continuous uniform distribution with a lower bound of $3.50 per gallon and an upper bound of $3.80 per gallon.What is the probability a randomly chosen gas station charges more than $3.70 per gallon?

The probability that a normal random variable is less than its mean is ___.

Find the z value such that =0.9082.

How many parameters are needed to fully describe any normal distribution?

Find the probability P(-1.96 ≤ Z ≤ 1.96) .

The time to complete the construction of a soapbox derby car is normally distributed with a mean of three hours and a standard deviation of one hour.Find the probability that it would take more than five hours to construct a soapbox derby car.

The probability P(Z > 1.28) is closest to ____.

It is known that the length of a certain product X is normally distributed with $\mu$ = 20 inches.How is the probability related to ?

What does it mean when we say that the tails of the normal curve are asymptotic to the x axis?

The probability P(Z < -1.28) is closest to ____.

For any normally distributed random variable with mean μ and standard deviation σ,the percent of the observations that fall between [μ - 2σ,μ + 2σ] is the closest to ______.

A hedge fund returns on average 26% per year with a standard deviation of 12%.Using the empirical rule,approximate the probability the fund returns over 50% next year.

Find the probability P(-1.96 ≤ Z ≤ 0) .

The time to complete the construction of a soapbox derby car is normally distributed with a mean of three hours and a standard deviation of one hour.Find the probability that it would take between 2.5 and 3.5 hours to construct a soapbox derby car.

Sarah's portfolio has an expected annual return at 8%,with an annual standard deviation at 12%.If her investment returns are normally distributed,then in any given year Sarah has approximately ______________________.

For any normally distributed random variable with mean μ and standard deviation σ,the proportion of the observations that fall outside the interval [μ - σ,μ + σ] is the closest to _____.