Question 1

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 2

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 probability that z is between 0.0 and 2.0 in a standard normal distribution is 0.4772.

Question 3

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

Both the uniform and normal distributions have the property that their mean and median are equal.

Question 4

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 standard deviation of a uniform distribution is equal to the difference between the maximum and minimum values divided by the square root of 12. 
Therefore, the standard deviation is (150-120)/sqrt(12) = 8.66 minutes.

Question 5

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 95.44% rule, or the empirical rule for a normal distribution, which states that approximately 95.44% of the data falls within two standard deviations of the mean.

Question 6

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 7

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 known for its characteristic bell-shaped curve. It is symmetrical and not skewed or rectangular.

Question 8

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 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 more than 140 minutes?
A)1.00
B)0.50
C)0.333
D)0.00

Accepted Answer

To find the probability, we need to find the area under the distribution curve for the time interval when the flight is more than 140 minutes.

The total area under any uniform distribution curve is 1. Therefore, the probability of any event on this curve will be the ratio of the area of that event to the total area under the curve.

The length of the interval when the flight is more than 140 minutes is (150-140) = 10 minutes, which is 1/3 of the total length of the distribution curve (150-120 = 30). Therefore, the probability of a flight being more than 140 minutes is 1/3 or 0.333.

Question 10

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

Using the properties of the normal distribution, we calculate the z-score for 15 days: z = (15 - 20) / 5 = -1. A z-score of -1 corresponds to the 15.87th percentile, meaning 15.87% of invoices were paid within 15 days.

Question 11

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 12

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 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 probability that a flight is less than 135 minutes?
A)1.00
B)0.50
C)0.25
D)0.00

Accepted Answer

The probability is calculated using the formula for uniform distribution: (x - min) / (max - min). Here, x = 135, min = 120, and max = 150. So, (135 - 120) / (150 - 120) = 15 / 30 = 0.50.

Question 14

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

The mean plus and minus two standard deviations include about 95% of the observations in a normal distribution. This is also known as the 95% confidence interval.

Question 15

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 16

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 17

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

The area under the standard normal curve between z = -1.0 and z = -2.0 can be found using a standard normal table, which gives a value of 0.1359. Therefore, the correct answer is C.

Question 18

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)99.74%

Accepted Answer

Using the empirical rule for normal distributions, 99.74% of the data falls within three standard deviations from the mean. Here, 600 hours is three standard deviations below the mean (750 - 3*50), and 900 hours is three standard deviations above the mean (750 + 3*50).

Question 19

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 distribution with a bell-shaped curve. It has only one mode, and the mean, median, and mode are all equal. The points of the curve meet the x-axis at z = -1 and z = 1, not -3 and 3.

Question 20

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.

Quiz 7: Continuous Probability Distributions

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?

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

What is an important similarity between the uniform and normal 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?

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?

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?

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

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?

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 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 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?

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?

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?

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

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 = −1.0 and z = −2.0?

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?

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