Question 1

An analysis of the grades on the first test in History 101 revealed that they approximate a normal curve with a mean of 75 and a standard deviation of 8. The instructor wants to award the grade of A to the upper 10 percent of the test grades. What is the dividing point between an A and a B grade?
A) 80
B) 85
C) 90
D) 95

Accepted Answer

To find the dividing point between an A and a B grade, we need to find the score that corresponds to the top 10% of the distribution.

Using a z-table or calculator, we find that the z-score that corresponds to the top 10% of the distribution is approximately 1.28.

We can use the formula:

z-score = (X - mean) / standard deviation

to solve for the corresponding test score (X).

1.28 = (X - 75) / 8

Solving for X, we get:

X = 85

Therefore, the dividing point between an A and a B grade is 85.

Question 2

The standard normal probability distribution is unique because it has:
A) A mean of 1 and any standard deviation
B) Any mean and a standard deviation of 1
C) A mean of 0 and any standard deviation
D) A mean of 0 and a standard deviation of 1

Accepted Answer

The standard normal probability distribution is defined as having a mean of 0 and a standard deviation of 1. This unique definition allows for the calculation of probabilities and standard scores (z-scores), which can be used for comparisons and statistical inference.

Question 3

What is a normal distribution with a mean of 0 and a standard deviation of 1 called?
A) Frequency distribution
B) Z-score
C) Standard normal distribution
D) Binomial probability distribution

Accepted Answer

A normal distribution with a mean of 0 and a standard deviation of 1 is called a standard normal distribution. This distribution is widely used in statistical analysis and can be used to calculate Z-scores.

Question 4

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 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

The time between 32 and 34 hours is one standard deviation above the mean in a normal distribution. Approximately 34.13% of the data falls within this range.

Question 5

A cola-dispensing machine is set to dispense a mean of 2.02 liters into a container labeled 2 liters. Actual quantities dispensed vary and the amounts are normally distributed with a standard deviation of 0.015 liters. What is the probability a container will have less than 2 liters?
A) 0.0918
B) 0.3413
C) 0.1926
D) 0.8741

Accepted Answer

We need to find P(x < 2), where x is the amount of cola dispensed. We standardize the variable by finding the z-score:
z = (2 - 2.02) / 0.015 = -1.33
Using a standard normal table or calculator, we find that the probability of a standard normal variable being less than -1.33 is approximately 0.0918. Therefore, the probability that a container will have less than 2 liters is 0.0918.

Question 6

Tables of normal distribution probabilities are found in many statistics books. These probabilities are calculated from a normal distribution with
A) a mean of 1 and a standard deviation of 1
B) a mean of 100 and a standard deviation of 15
C) a mean of 0 and a standard deviation of 15
D) a mean of 0 and a standard deviation of 1

Accepted Answer

Tables of normal distribution probabilities are calculated based on a standard normal distribution, which has a mean of 0 and a standard deviation of 1. Therefore, choice D is the correct answer.

Question 7

Two normal distributions are compared. One has a mean of 10 and a standard deviation of 10. The second normal distribution has a mean of 10 and a standard deviation of 2. Which of the following is true?
A) The locations of the distributions are different
B) The distributions are from two different families of distributions
C) The dispersions of the distributions are different
D) The dispersions of the distributions are the same

Accepted Answer

The dispersions of the distributions are different because the standard deviation of the first distribution is larger than the standard deviation of the second distribution. The mean of both distributions is the same, so the locations of the distributions are the same. Both distributions belong to the normal family of distributions, so B is incorrect.

Question 8

The weekly mean income of a group of executives is $1000 and the standard deviation of this group is $100. The distribution is normal. What percent of the executives have an income of $925 or less?
A) About 15%
B) About 85%
C) About 50%
D) About 23%

Accepted Answer

To find the percentage of executives with an income of $925 or less, we calculate the Z-score: $Z = \frac{X - \mu}{\sigma} = \frac{925 - 1000}{100} = -0.75$. Looking up a Z-score of -0.75 in a standard normal distribution table or using a calculator gives us a value of about 0.2266, which translates to about 22.66% of the executives. This is closest to 23%, making option D the correct answer.

Question 9

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 answer of The employees of Cartwright Manufacturing are awarded...

Question 10

Which of the following is true in a normal distribution? A) The mean equals zero. B) Mode and the 3^rd quartile are equal. C) Mean divides the distribution into two equal parts. D) The area under the curve is 0.68.

Accepted Answer

In a normal distribution, the mean divides the distribution into two equal parts. Option A is incorrect as the mean may not always be zero, Option B is incorrect as the mode and quartiles vary in a normal distribution and Option D is incorrect as the area under the curve is 0.68 only between one standard deviation on either side of the mean, not for the entire distribution.

Question 11

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 area under the normal curve between these two weights. This can be done using the standard normal distribution table or a calculator with a normal distribution function.

Using a calculator, we can find that the z-score for a pepper weighing 13 pounds is:

z = (13 - 15) / 1.75 = -1.14

Similarly, 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 that the area to the left of a z-score of -1.14 is 0.1271 and the area to the left of a z-score of 0.57 is 0.7123. Therefore, the area between these two z-scores (i.e., the proportion of plants producing peppers weighing between 13 and 16 pounds) is:

0.7123 - 0.1271 = 0.5852

To find the number of plants that produced peppers in this weight range, we need to multiply this proportion by the total number of plants:

0.5852 x 200 = 117.04

Rounding to the nearest whole number, we get:

Answer: B (118)

Question 12

The total area of a normal probability distribution is
A) between -3.0 and 3.0.
B) 1.00.
C) dependent on a value of 'z'.
D) approximated by the binomial distribution.

Accepted Answer

The total area under a normal probability distribution curve is always equal to 1.00.

Question 13

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

Using the properties of the normal distribution, we calculate the z-scores for $32,000 and $42,000, then find the area between these z-scores. This area corresponds to approximately 60.06% of the distribution.

Question 14

In an illustration of a normal probability distribution, a shaded area represents
A) a permutation
B) a combination
C) a probability
D) a standard deviation

Accepted Answer

In a normal probability distribution, the shaded area represents a probability. The area under the curve represents the total probability, with specific shaded areas representing specific probabilities or probabilities within a certain range. The other choices (permutation, combination, standard deviation) do not apply to normal probability distributions.

Question 15

A random variable from an experiment where outcomes are normally distributed
A) can have any value between - $\infty$ and + $\infty$.
B) can have only a few discrete values.
C) can have a mean of 0 and a standard deviation of 1.
D) can have no values.

Accepted Answer

The range of values for a normally distributed random variable can be from negative infinity to positive infinity, but values beyond 3 standard deviations from the mean are rare. Therefore, the range can be approximated as between -3σ and +3σ, where σ is the standard deviation.

Question 16

The weight of cans of fruit is normally distributed with a mean of 1,000 grams and a standard deviation of 50 grams. What percent of the cans weigh 860 grams or less?
A) 0.0100
B) 0.8400
C) 0.0026
D) 0.0001

Accepted Answer

To find the percentage of cans that weigh 860 grams or less, we use the Z-score formula: Z = (X - μ) / σ, where X is the value in question, μ is the mean, and σ is the standard deviation. Plugging in the values: Z = (860 - 1000) / 50 = -140 / 50 = -2.8. Looking up a Z-score of -2.8 in a standard normal distribution table or using a calculator gives us a value of approximately 0.0026, indicating that about 0.26% of the cans weigh 860 grams or less.

Question 17

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

Since it's a normal distribution, we can use the empirical rule which states that approximately 68% of the data falls within 1 standard deviation of the mean. Thus, the incomes between $37,200- $800 = $36,400 and $37,200 + $800 = $38,000 will contain about 68% of the incomes. Therefore, the correct option is B, $36,400 and $38,000.

Question 18

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 fall within 1.96 standard deviations of the mean. Therefore, the range of values between which 95% of the data lies is:
500 - (1.96 x 10) and 500 + (1.96 x 10)
= 480 and 520
Hence, the best choice is B which includes the range 480 and 520.

Question 19

Ball-Bearings, Inc. produces ball bearings automatically on a Kronar BBX machine. For one of the ball bearings, the mean diameter is set at 20.00 mm (millimeters). The standard deviation of the production over a long period of time was computed to be 0.150 mm. What percent of the ball bearings will have diameters 20.27 mm or more?
A) 41.00%
B) 12.62%
C) 3.59%
D) 85.00%

Accepted Answer

First, we need to calculate the z-score:
z = (20.27 - 20.00) / 0.150 = 1.80

Using a z-table, we find that the area to the right of the z-score of 1.80 is 0.0359 or 3.59%. Therefore, 3.59% of the ball bearings will have diameters 20.27 mm or more.

Question 20

A large manufacturing firm tests job applicants who recently graduated from college. The test scores are normally distributed with a mean of 500 and a standard deviation of 50. Management is considering placing a new hire in an upper level management position if the person scores in the upper 6 percent of the distribution. What is the lowest score a college graduate must earn to qualify for a responsible position?
A) 50
B) 625
C) 460
D) 578

Accepted Answer

To find the score that falls in the upper 6 percent of the distribution, we need to find the z-score associated with that percentage.

Using a standard normal distribution table or calculator, we find that the z-score corresponding to the upper 6 percent is approximately 1.64.

We can then use the formula:

z = (x - μ) / σ

where z is the z-score, x is the raw score we're looking for, μ is the mean, and σ is the standard deviation.

Plugging in the values we know, we get:

1.64 = (x - 500) / 50

Solving for x, we get:

x = 1.64 * 50 + 500 = 578

Therefore, the lowest score a college graduate must earn to qualify for a responsible position is 578.

Quiz 7: Continuous Probability Distributions

An analysis of the grades on the first test in History 101 revealed that they approximate a normal curve with a mean of 75 and a standard deviation of 8. The instructor wants to award the grade of A to the upper 10 percent of the test grades. What is the dividing point between an A and a B grade?

The standard normal probability distribution is unique because it has:

What is a normal distribution with a mean of 0 and a standard deviation of 1 called?

A cola-dispensing machine is set to dispense a mean of 2.02 liters into a container labeled 2 liters. Actual quantities dispensed vary and the amounts are normally distributed with a standard deviation of 0.015 liters. What is the probability a container will have less than 2 liters?

Tables of normal distribution probabilities are found in many statistics books. These probabilities are calculated from a normal distribution with

Two normal distributions are compared. One has a mean of 10 and a standard deviation of 10. The second normal distribution has a mean of 10 and a standard deviation of 2. Which of the following is true?

The weekly mean income of a group of executives is $1000 and the standard deviation of this group is $100. The distribution is normal. What percent of the executives have an income of $925 or less?

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?

Which of the following is true in a normal distribution?

The total area of a normal probability distribution is

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?

In an illustration of a normal probability distribution, a shaded area represents

A random variable from an experiment where outcomes are normally distributed

The weight of cans of fruit is normally distributed with a mean of 1,000 grams and a standard deviation of 50 grams. What percent of the cans weigh 860 grams or less?

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?

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?