Question 1

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 2

A new extended-life light bulb has an average service life of 750 hours, with a standard deviation of 50 hours. If the service 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) 99.7%

Accepted Answer

The answer of A new extended-life light bulb has an...

Question 3

For the normal distribution, the mean plus and minus 1.96 standard deviations will include about what percent of the observations?
A) 50%
B) 99.7%
C) 95%
D) 68%

Accepted Answer

The 1.96 standard deviations from the mean in the normal distribution contains approximately 95% of the observations. This is also known as the 95% confidence interval.

Question 4

Which of the following is true regarding the normal distribution?
A) 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 symmetrical and the mean, median, and mode all coincide at the center of the distribution. Therefore, option A is true. Option B is false because the normal distribution has only one mode. Option C is false because the normal distribution is symmetrical. Option D is false because the points of the curve never meet the X-axis, since the distribution extends to infinity in both directions. Instead, the curve approaches the X-axis asymptotically but never touches it.

Question 5

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 percent 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

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

Question 6

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 answer of What is an important similarity between the...

Question 7

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

The answer of A study of a company's practice regarding...

Question 8

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 minutes
D) 10 minutes

Accepted Answer

The uniform distribution has a constant probability. The range is 150 - 120 = 30 minutes. The probability of a flight being more than 140 minutes is (150 - 140) / 30 = 10 / 30 = 0.333.

Question 9

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) 0.5
C) 0.333 minutes
D) 10 minutes

Accepted Answer

The range of the flight time distribution is 30 minutes (150 - 120). Since the distribution is uniform, the probability of a flight being between 125 and 140 minutes is the proportion of the range that falls within that interval. Therefore, the probability is (140 - 125)/30 = 0.5, or 50%.

Question 10

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

Using a standard normal distribution table or calculator, we find that the area between z = -1.0 and z = -2.0 is approximately 0.1359. Therefore, the best choice is C.

Question 11

The mean of a normally distributed group of weekly incomes of a large group of executives is $1,000 and the standard deviation is $100. What is the z-score for an income of $1,100?
A) 1.00
B) 2.00
C) 1.683
D) -0.90

Accepted Answer

The z-score is calculated by subtracting the mean from the income and then dividing by the standard deviation: (1100 - 1000) / 100 = 1.00.

Question 12

The mean amount spent by a family of four on food per month is $500 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 13

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 standard deviation?
A) 8.66 minutes
B) 75 minutes
C) 135 minutes
D) 270 minutes

Accepted Answer

The standard deviation of a uniform distribution can be found using the formula sqrt((b-a)^2/12), where a is the minimum value and b is the maximum value. 
Plugging in the values, we get sqrt((150-120)^2/12) = sqrt(900/12) = 8.66. Therefore, the standard deviation is 8.66 minutes.

Question 14

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 15

What is the area under the normal curve between z = 0.0 and z = 2.0?
A) 1.0000
B) 0.7408
C) 0.1359
D) 0.4772

Accepted Answer

The area under the normal curve between z = 0.0 and z = 2.0 is 0.4772, which represents the probability or proportion of data within this range in a standard normal distribution.

Question 16

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) 0.5
C) 15 minutes
D) 270 minutes

Accepted Answer

We need to find the probability of a flight being less than 135 minutes. Since the distribution is uniform, we can find this probability by calculating the proportion of the total area under the distribution curve that lies to the left of 135.

The width of the distribution is 150 - 120 = 30 minutes. The area of a uniform distribution is given by the product of its width and height, which is 1/(max - min) in this case. So, the height of the distribution is 1/30.

To find the proportion of the area to the left of 135, we can find the area of a triangle with base 15 (from 120 to 135) and height 1/30:

(1/2)(15)(1/30) = 1/4

So, the probability that a flight is less than 135 minutes is 1/4 or 0.25. Therefore, the best choice is B.

Question 17

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

Accepted Answer

The area between 415 pounds and the mean of 400 pounds is the area under the normal distribution curve between z-scores of (415-400)/10 = 1.5 and 0. The area under the normal distribution curve between any two z-scores can be found using a standard normal distribution table or a calculator. The area between z-scores of 1.5 and 0 is 0.4332.

Question 18

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 19

Which of the following is a characteristic of the normal probability distribution?
A) Positively-skewed
B) Bell-shaped
C) Asymmetrical
D) Rectangular

Accepted Answer

A bell-shaped curve is a characteristic of a normal probability distribution.

Question 20

What is the percentage of the total area under the normal curve within plus and minus two standard deviations of the mean?
A) 68.26%
B) 99.74%
C) 34.13%
D) 95.44%

Accepted Answer

The empirical rule states that approximately 95% of the total area under a normal curve falls within plus and minus two standard deviations of the mean. Therefore, the correct answer is D.

Quiz 7: Continuous Probability Distributions

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 service life of 750 hours, with a standard deviation of 50 hours. If the service life of these light bulbs approximates a normal distribution, about what percent of the distribution will be between 600 hours and 900 hours?

For the normal distribution, the mean plus and minus 1.96 standard deviations will include about what percent of the observations?

Which of the following is true regarding the normal distribution?

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

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?

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

The mean of a normally distributed group of weekly incomes of a large group of executives is $1,000 and the standard deviation is $100. What is the z-score for an income of $1,100?

The mean amount spent by a family of four on food per month is $500 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 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 standard deviation?

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

What is the area under the normal curve between z = 0.0 and z = 2.0?

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

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?

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

What is the percentage of the total area under the normal curve within plus and minus two standard deviations of the mean?