Question 1

The time to fly between New York City and Chicago is uniformly distributed with a minimum of 120 minutes and a maximum of 150 minutes. What is the distribution's standard deviation?
A) 8.66 minutes
B) 75 minutes
C) 135 minutes
D) 270 minutes

Accepted Answer

The formula for the standard deviation of a uniform distribution is (b-a)/sqrt(12), where a is the minimum value and b is the maximum value. Therefore, the standard deviation in this case is (150-120)/sqrt(12) = 8.66 minutes.

Question 2

What is the area under the normal curve between z = -1.0 and z = -2.0?
A) 0.0228
B) 0.3413
C) 0.1359
D) 0.4772

Accepted Answer

We can use a standard normal distribution table to find the area under the curve between z = -1.0 and z = -2.0. The table shows us that the area between these two values is 0.1359. Therefore, the answer is (C).

Question 3

The mean of a normal distribution is 400 pounds. The standard deviation is 10 pounds. What is the probability of a weight between 415 pounds and the mean of 400 pounds?
A) 0.5000
B) 0.1932
C) 0.4332
D) 0.3413

Accepted Answer

The answer of The mean of a normal distribution is...

Question 4

For the normal distribution, the mean plus and minus two standard deviations will include about what percent of the observations?
A) 50%
B) 99.7%
C) 95%
D) 68%

Accepted Answer

This is known as the 95% rule or the empirical rule, which states that for a normal distribution, approximately 95% of the data falls within two standard deviations from the mean.

Question 5

The time to fly between New York City and Chicago is uniformly distributed with a minimum of 120 minutes and a maximum of 150 minutes. What is the probability that a flight is less than 135 minutes?
A) 1.00
B) 0.50
C) 0.25
D) 0.00

Accepted Answer

The range of the uniformly distributed flight time is 150 - 120 = 30 minutes. To find the probability of a flight being less than 135 minutes, we need to find the proportion of this range that falls below 135. Therefore, the probability is (135 - 120) / 30 = 0.5.

Question 6

An accelerated life test on a large number of type-D alkaline batteries revealed that the mean life for a particular use before they failed is 19.0 hours. The distribution of the lives approximated a normal distribution. The standard deviation of the distribution was 1.2 hours. About 95.44% of the batteries failed between what two values?
A) 8.9 and 18.9
B) 12.2 and 14.2
C) 14.1 and 22.1
D) 16.6 and 21.4

Accepted Answer

The 95.44% range in a normal distribution corresponds to approximately ±2 standard deviations from the mean. The mean is 19.0 hours, and the standard deviation is 1.2 hours. Therefore, the range is from 19.0 - 2(1.2) = 16.6 hours to 19.0 + 2(1.2) = 21.4 hours.

Question 7

The time to fly between New York City and Chicago is uniformly distributed with a minimum of 120 minutes and a maximum of 150 minutes. What is the probability that a flight is between 125 and 140 minutes?
A) 1.00
B) 0.50
C) 0.33
D) 10.00

Accepted Answer

The probability can be calculated using the formula for uniform distribution: $P(a < X < b) = \frac{b - a}{max - min}$. Here, $a = 125$, $b = 140$, $min = 120$, and $max = 150$. So, $P(125 < X < 140) = \frac{140 - 125}{150 - 120} = \frac{15}{30} = 0.50$.

Question 8

What is an important similarity between the uniform and normal probability distributions?
A) The mean, median, and mode are all equal.
B) The mean and median are equal.
C) They are negatively skewed.
D) About 68% of all observations are within one standard deviation of the mean.

Accepted Answer

The important similarity between the uniform and normal probability distributions is that the mean and median are equal. For the uniform distribution, this is because the distribution is evenly spread out between the minimum and maximum values. For the normal distribution, this is because of the symmetric bell shape of the distribution.

Question 9

For a standard normal distribution, what is the probability that z is greater than 1.75?
A) 0.0401
B) 0.0459
C) 0.4599
D) 0.9599

Accepted Answer

The area to the right of z = 1.75 in a standard normal distribution corresponds to the probability of z being greater than 1.75, which is 0.0401. This can be found using standard normal distribution tables or a calculator with statistical functions.

Question 10

For a normal distribution, what is the likelihood (expressed as a percentage) that a random variable is within plus and minus two standard deviations of the mean?
A) 68.26%
B) 99.74%
C) 34.13%
D) 95.44%

Accepted Answer

This is known as the 68-95-99.7 rule, which states that approximately 68% of the distribution 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. Therefore, the likelihood of a random variable being within plus and minus two standard deviations of the mean is 95.44%.

Question 11

What is the probability that z is between 0.0 and 2.0?
A) 1.0000
B) 0.7408
C) 0.1359
D) 0.4772

Accepted Answer

The answer of What is the probability that z is...

Question 12

The mean score of a college entrance test is 500; the standard deviation is 75. The scores are normally distributed. What percent of the students scored below 320?
A) About 50.82%
B) About 34.13%
C) About 7.86%
D) About 0.82%

Accepted Answer

To find the percentage of students who scored below 320, we calculate the z-score using the formula z = (X - μ) / σ, where X is the score (320), μ is the mean (500), and σ is the standard deviation (75). Plugging in the values, we get z = (320 - 500) / 75 = -180 / 75 = -2.4. Looking up a z-score of -2.4 in a standard normal distribution table or using a calculator, we find that the percentage of students scoring below this z-score is about 0.82%.

Question 13

A study of a company's practice regarding the payment of invoices revealed that an invoice was paid an average of 20 days after it was received. The standard deviation equaled five days. Assuming that the distribution is normal, what percent of the invoices were paid within 15 days of receipt?
A) 15.87%
B) 37.91%
C) 34.13%
D) 86.74%

Accepted Answer

To solve this problem, we need to use the formula for z-score:

z = (x - μ) / σ

where x is the value we want to find the probability for (15 days), μ is the mean (20 days), and σ is the standard deviation (5 days).

Plugging in the values, we get:

z = (15 - 20) / 5 = -1

Next, we need to find the probability of a z-score of -1 or less. We can use a standard normal distribution table or a calculator to find this probability, which is 0.1587.

So the answer is A, 15.87%.

Question 14

The time to fly between New York City and Chicago is uniformly distributed with a minimum of 120 minutes and a maximum of 150 minutes. What is the probability that a flight is more than 140 minutes?
A) 1.00
B) 0.5
C) 0.333
D) 0.00

Accepted Answer

To find the probability that a flight is more than 140 minutes, we need to find the area of the distribution curve to the right of 140. 
The total area under the curve is 1. Since the distribution is uniform, the area is represented by a rectangle with base (150-120) = 30 and height 1/30 (since the total area is 1). 
The area to the right of 140 is (150-140)/30 = 1/3. 
Therefore, the probability that a flight is more than 140 minutes is 1/3, or 0.333.

Question 15

Which of the following is a characteristic of the normal probability distribution?
A) It's positively skewed.
B) It's bell-shaped.
C) It's asymmetrical.
D) It's rectangular.

Accepted Answer

The normal probability distribution is bell-shaped, with a symmetrical distribution of data around the mean. It is not positively skewed, asymmetrical or rectangular.

Question 16

The mean amount spent by a family of four on food is $500 per month with a standard deviation of $75. Assuming that the food costs are normally distributed, what is the probability that a family spends less than $410 per month?
A) 0.2158
B) 0.8750
C) 0.0362
D) 0.1151

Accepted Answer

The answer of The mean amount spent by a family...

Question 17

The weekly incomes of a large group of executives are normally distributed with a mean of $2,000 and a standard deviation of $100. What is the z-score for an income of $2,100?
A) 1.00
B) 2.00
C) 1.683
D) -0.90

Accepted Answer

The z-score is calculated using the formula $z = \frac{(X - \mu)}{\sigma}$, where $X$ is the value in question, $\mu$ is the mean, and $\sigma$ is the standard deviation. Plugging in the values, $z = \frac{(2100 - 2000)}{100} = 1$.

Question 18

What is the area under the normal curve between z = 0.0 and z = 1.79?
A) 0.4633
B) 0.0367
C) 0.9599
D) 0.0401

Accepted Answer

The area under the normal curve between z = 0.0 and z = 1.79 is 0.4633. This represents the proportion of the data within this range on a standard normal distribution.

Question 19

A new extended-life light bulb has an average life of 750 hours, with a standard deviation of 50 hours. If the life of these light bulbs approximates a normal distribution, about what percent of the distribution will be between 600 hours and 900 hours?
A) 95%
B) 68%
C) 34%
D) 99.47%

Accepted Answer

To solve this problem, we need to use z-scores and the standard normal distribution. We can first standardize the values of 600 and 900 using the formula:

z = (x - μ) / σ

where μ is the mean and σ is the standard deviation. Substituting the values:

z1= (600 - 750) / 50 = -3.00
z2 = (900 - 750) / 50 = 3.00

We can then use a standard normal distribution table or a calculator to find the area under the curve between these z-scores.

The area between -3.00 and 3.00 is about 0.9947 or 99.47%, which means that about 99.47% of the distribution will be between 600 hours and 900 hours. Therefore, the best choice is D.

Question 20

Which of the following is true regarding the normal distribution?
A) The mean, median, and mode are all equal.
B) It has two modes.
C) It is asymmetrical.
D) The points of the curve meet the x-axis at z = -3 and z = 3.

Accepted Answer

The normal distribution is a symmetrical bell curve, meaning it has a single peak and is not asymmetrical or bimodal. Additionally, since it is symmetrical, the mean, median, and mode are all equal. The points on the curve do not directly correspond to specific values of z-scores, so choice D is also incorrect.

Quiz 7: Continuous Probability Distributions

The time to fly between New York City and Chicago is uniformly distributed with a minimum of 120 minutes and a maximum of 150 minutes. What is the distribution's standard deviation?

What is the area under the normal curve between z = -1.0 and z = -2.0?

The mean of a normal distribution is 400 pounds. The standard deviation is 10 pounds. What is the probability of a weight between 415 pounds and the mean of 400 pounds?

For the normal distribution, the mean plus and minus two standard deviations will include about what percent of the observations?

The time to fly between New York City and Chicago is uniformly distributed with a minimum of 120 minutes and a maximum of 150 minutes. What is the probability that a flight is less than 135 minutes?

The time to fly between New York City and Chicago is uniformly distributed with a minimum of 120 minutes and a maximum of 150 minutes. What is the probability that a flight is between 125 and 140 minutes?

What is an important similarity between the uniform and normal probability distributions?

For a standard normal distribution, what is the probability that z is greater than 1.75?

For a normal distribution, what is the likelihood (expressed as a percentage) that a random variable is within plus and minus two standard deviations of the mean?

What is the probability that z is between 0.0 and 2.0?

The mean score of a college entrance test is 500; the standard deviation is 75. The scores are normally distributed. What percent of the students scored below 320?

A study of a company's practice regarding the payment of invoices revealed that an invoice was paid an average of 20 days after it was received. The standard deviation equaled five days. Assuming that the distribution is normal, what percent of the invoices were paid within 15 days of receipt?

The time to fly between New York City and Chicago is uniformly distributed with a minimum of 120 minutes and a maximum of 150 minutes. What is the probability that a flight is more than 140 minutes?

Which of the following is a characteristic of the normal probability distribution?

The mean amount spent by a family of four on food is $500 per month with a standard deviation of $75. Assuming that the food costs are normally distributed, what is the probability that a family spends less than $410 per month?

The weekly incomes of a large group of executives are normally distributed with a mean of $2,000 and a standard deviation of $100. What is the z-score for an income of $2,100?

What is the area under the normal curve between z = 0.0 and z = 1.79?

A new extended-life light bulb has an average life of 750 hours, with a standard deviation of 50 hours. If the life of these light bulbs approximates a normal distribution, about what percent of the distribution will be between 600 hours and 900 hours?

Which of the following is true regarding the normal distribution?