Question 1

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

The 95.44% coverage corresponds to approximately ±2 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 from 19.0 - 2(1.2) = 16.6 hours to 19.0 + 2(1.2) = 21.4 hours.

Question 2

i. Disaster Airlines determined that the mean number of passengers per flight is 152 with a standard deviation of ten passengers. Practically all flights have between 142 and 162 passengers. ii. The total area under the normal curve is less than 1.
iii. The mean of a normal probability distribution is 500 and the standard deviation is 10.
About 68 percent of the observations lie between 480 and 520.
A) (i), (ii), and (iii) are all correct statements.
B) (i) is a correct statement but not (ii) or (iii).
C) (i) and (iii) are correct statements but not (ii).
D) (i) and (ii) are correct statements but not (iii).
E) (i), (ii), and (iii) are all false statements.

Accepted Answer

The answer of i. Disaster Airlines determined that the mean...

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 normal distribution follows the 68-95-99.7 rule, which states that for a normal distribution, about 68% of the observations fall within one standard deviation of the mean, about 95% fall within two standard deviations of the mean, and about 99.7% fall within three standard deviations of the mean. 
Since the question is asking for the percentage of observations within plus or minus 1.96 standard deviations of the mean, we can assume that the answer is close to 95%. Therefore, the best choice is C, 95%.

Question 4

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 probability that z is greater than 1.75 in a standard normal distribution is 0.0401. This can be found using standard normal distribution tables or a calculator with statistical functions.

Question 5

A sample of 500 evening students revealed that their annual incomes were normally distributed with a mean income of $50,000 and a standard deviation of $4,000. How many students earned between $47,000 and $53,000?
A) 137
B) 273
C) 113
D) 387

Accepted Answer

First, we need to convert the given values into standard units.

For $47,000: Z = \frac{47,000-50,000}{4,000} = -0.75$
For $53,000: Z = \frac{53,000-50,000}{4,000} = 0.75$

Next, we need to find the area under the normal curve between these two standard units using a standard normal table or calculator.

Using a standard normal table: 
Area between -0.75 and 0.75 = 0.7734 - 0.2266 = 0.5468

Finally, we need to convert this area back to the original units (number of students) by multiplying it by the total number of students (500).

Number of students between $47,000 and $53,000 = 0.5468 x 500 = 273
Therefore, the answer is B.

Question 6

A sample of 500 evening students revealed that their annual incomes were normally distributed with a mean income of $50,000 and a standard deviation of $4,000. How many students earned between $45,000 and $53,000?
A) 170
B) 273
C) 334
D) 387

Accepted Answer

To find the number of students who earned between $45,000 and $53,000, we need to find the z-scores for each of these values and use them in the standard normal distribution table.

z-score for $45,000: 
z = (45,000 - 50,000) / 4,000 
z = -1.25

z-score for $53,000: 
z = (53,000 - 50,000) / 4,000 
z = 0.75

Using the standard normal distribution table, we can find the area between these two z-scores:

Area between z = -1.25 and z = 0.75 
= 0.7734 - 0.1056 
= 0.6678

To find the number of students, we need to multiply this area by the total number of students:

0.6678 x 500 = 334

Therefore, the answer is C.

Question 7

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 answer of What is the area under the normal...

Question 8

The mean amount of gasoline and services charged by Key Refining Company credit customers is $70 per month. The distribution of amounts spent is approximately normal with a standard deviation of $10. What is the probability of selecting a credit card customer at random and finding the customer charged between $70 and $83?
A) 0.1962
B) 0.4032
C) 0.3413
D) 0.4750

Accepted Answer

To find the probability that the amount charged is between $70 and $83, we need to standardize both values using the formula:

z = (x - μ) / σ

where x is the value we want to standardize (either $70 or $83), μ is the mean amount charged ($70), and σ is the standard deviation ($10).

For $70:
z = (70 - 70) / 10 = 0

For $83:
z = (83 - 70) / 10 = 1.3

We can use a standard normal distribution table or calculator to find the probability of z being between 0 and 1.3. The probability is approximately 0.4032.

Question 9

i. For a normal probability distribution, about 95 percent of the area under normal curve is within plus and minus two standard deviations of the mean and practically all (99.73 percent) of the area under the normal curve is within three standard deviations of the mean. ii. The total area under the normal curve is 1.
iii. The mean of a normal probability distribution is 500 and the standard deviation is 10. About 68 percent of the observations lie between 480 and 520.
A) (i), (ii), and (iii) are all correct statements.
B) (i) is a correct statement but not (ii) or (iii).
C) (i) and (iii) are correct statements but not (ii).
D) (i) and (ii) are correct statements but not (iii).
E) (i), (ii), and (iii) are all false statements.

Accepted Answer

Statement (i) is correct as it describes the empirical rule for normal distributions. Statement (ii) is correct because the total area under a probability distribution curve is always 1. Statement (iii) is incorrect because 68 percent of the observations lie within one standard deviation of the mean, which would be between 490 and 510, not 480 and 520.

Question 10

The employees of Cartwright Manufacturing are awarded efficiency ratings. The distribution of the ratings approximates a normal distribution. The mean is 400, the standard deviation 50. What is the area under the normal curve between 400 and 482?
A) 0.5000
B) 0.4495
C) 0.3413
D) 0.4750

Accepted Answer

The area under the normal curve between the mean (400) and a score (482) can be found by calculating the Z-score of 482 and then looking up the corresponding area in a standard normal distribution table. The Z-score is calculated as (X - μ) / σ, where X is the score, μ is the mean, and σ is the standard deviation. For 482, the Z-score is (482 - 400) / 50 = 1.64. Looking up the Z-score of 1.64 in a standard normal distribution table gives an area of 0.4495 to the left of 482. Since we are looking for the area between the mean and 482, and the total area to the left of the mean (400) is 0.5000 (50% of the distribution), the area between 400 and 482 is approximately 0.4495.

Question 11

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 can find the area between z = -1.0 and z = -2.0 to be approximately 0.1359. Therefore, the best choice is C.

Question 12

What is the proportion of the total area under the normal curve within plus and minus two standard deviations of the mean?
A) 68%
B) 99.7%
C) 34%
D) 95%

Accepted Answer

Two standard deviations from the mean in either direction covers approximately 95% of the total area under the normal curve.

Question 13

The mean of a normal probability distribution is 500 and the standard deviation is 10. About 95 percent of the observations lie between what two values?
A) 475 and 525
B) 480 and 520
C) 400 and 600
D) 350 and 650

Accepted Answer

In a normal distribution, 95% of the observations lie within two standard deviations of the mean. Therefore, the range would be between 500-2(10) = 480 and 500+2(10) = 520. The best choice is B, which includes these values.

Question 14

A national manufacturer of unattached garages discovered that the distribution of the lengths of time it takes two construction workers to erect the Red Barn model is approximately normally distributed with a mean of 32 hours and a standard deviation of 2 hours. What percent of the garages take between 32 and 34 hours to erect?
A) 16.29%
B) 76.71%
C) 3.14%
D) 34.13%

Accepted Answer

To find the percentage of garages that take between 32 and 34 hours to erect, we use the properties of the normal distribution. The Z-score for 34 hours is calculated as (34 - 32) / 2 = 1. The percentage of values between the mean and a Z-score of 1 in a normal distribution is approximately 34.13%.

Question 15

The annual commissions per salesperson employed by a manufacturer of light machinery averaged $40,000 with a standard deviation of $5,000. What percent of the sales persons earn between $32,000 and $42,000?
A) 60.06%
B) 39.94%
C) 34.13%
D) 81.66%

Accepted Answer

The answer of The annual commissions per salesperson employed by...

Question 16

The distribution of the annual incomes of a group of middle management employees approximated a normal distribution with a mean of $37,200 and a standard deviation of $800. About 68 percent of the incomes lie between what two incomes?
A) $30,000 and $40,000
B) $36,400 and $38,000
C) $34,800 and $39,600
D) $35,600 and $38,800

Accepted Answer

We know that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean. Therefore, we can calculate the incomes that are one standard deviation below and one standard deviation above the mean and use that to determine the range of incomes where 68% of the data falls.

One standard deviation below the mean is:
$37,200 - $800 = $36,400

One standard deviation above the mean is:
$37,200 + $800 = $38,000

Therefore, about 68% of the incomes lie between $36,400 and $38,000. 
So, the correct answer is B.

Question 17

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 found using standard normal distribution tables or a calculator designed for statistics. The value corresponds to the probability of a z-score being between these two values. The correct answer is 0.4633, which represents the proportion of the data within this range on a standard normal distribution.

Question 18

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 19

The seasonal output of a new experimental strain of pepper plants was carefully weighed. The mean weight per plant is 15.0 pounds, and the standard deviation of the normally distributed weights is 1.75 pounds. Of the 200 plants in the experiment, how many produced peppers weighing between 13 and 16 pounds?
A) 100
B) 118
C) 197
D) 53

Accepted Answer

To find out how many plants produced peppers weighing between 13 and 16 pounds, we need to calculate the z-scores for these weights and then use a standard normal distribution table to find the proportion of plants within this range.

The z-score for a pepper weighing 13 pounds is: 
z = (13 - 15) / 1.75 = -1.14

The z-score for a pepper weighing 16 pounds is: 
z = (16 - 15) / 1.75 = 0.57

Using a standard normal distribution table, we can find the proportion of plants with z-scores between -1.14 and 0.57: 
P(-1.14 < z < 0.57) = P(z < 0.57) - P(z < -1.14) 
= 0.7186 - 0.1271 
= 0.5915

To find the number of plants in this range, we multiply this proportion by the total number of plants: 
0.5915 x 200 = 118.3

Rounding to the nearest whole number, we get: 
Answer: B (118)

Therefore, approximately 118 plants produced peppers weighing between 13 and 16 pounds.

Question 20

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

Accepted Answer

The area between 415 kg and the mean of 400 kg 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.

Quiz 6: Continuous Probability Distributions

For the normal distribution, the mean plus and minus 1.96 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?

A sample of 500 evening students revealed that their annual incomes were normally distributed with a mean income of $50,000 and a standard deviation of $4,000. How many students earned between $47,000 and $53,000?

A sample of 500 evening students revealed that their annual incomes were normally distributed with a mean income of $50,000 and a standard deviation of $4,000. How many students earned between $45,000 and $53,000?

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

The employees of Cartwright Manufacturing are awarded efficiency ratings. The distribution of the ratings approximates a normal distribution. The mean is 400, the standard deviation 50. What is the area under the normal curve between 400 and 482?

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

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

The mean of a normal probability distribution is 500 and the standard deviation is 10. About 95 percent of the observations lie between what two values?

The annual commissions per salesperson employed by a manufacturer of light machinery averaged $40,000 with a standard deviation of $5,000. What percent of the sales persons earn between $32,000 and $42,000?

The distribution of the annual incomes of a group of middle management employees approximated a normal distribution with a mean of $37,200 and a standard deviation of $800. About 68 percent of the incomes lie between what two incomes?

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?

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