Question 1

The weekly mean income of a group of executives is $1,000 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?

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 2

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?

Accepted Answer

To solve this problem, we need to use the normal distribution formula:

Z = (X - μ) / σ

where X = 2 liters, μ = 2.02 liters, and σ = 0.015 liters.

So, Z = (2 - 2.02) / 0.015 = -1.33

Using a standard normal table or calculator, we can find the probability that Z is less than -1.33, which is 0.0918.

Therefore, the probability that a container will have less than 2 liters is 0.0918, or approximately 9.18%. The closest answer choice is A.

Question 3

The total area of a normal probability distribution is ________.

Accepted Answer

The total area under a normal curve is always equal to 1, representing the total probability of all possible outcomes.

Question 4

The standard normal probability distribution is unique because it has ________.

Accepted Answer

The standard normal distribution is defined as having a mean of 0 and a standard deviation of 1. This is what makes it unique and allows for its use in standardizing values and making comparisons across different distributions.

Question 5

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 of 20.27 mm or more?

Accepted Answer

To find the percentage of ball bearings with diameters of 20.27 mm or more, we use the Z-score formula: Z = (X - μ) / σ, where X is the value in question (20.27 mm), μ is the mean (20.00 mm), and σ is the standard deviation (0.150 mm). Plugging in the values, we get Z = (20.27 - 20.00) / 0.150 = 1.8. Looking up 1.8 on a standard normal distribution table or using a calculator, we find that the area to the left of Z = 1.8 is approximately 0.9641, which means 96.41% of the values lie below 20.27 mm. Therefore, 100% - 96.41% = 3.59% of the ball bearings will have diameters of 20.27 mm or more.

Question 6

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% of the incomes lie between what two incomes?

Accepted Answer

According to the Empirical Rule, about 68% of data falls within one standard deviation of the mean. In this case, one standard deviation from the mean is $800. Therefore, approximately 68% of the data (middle 68%) falls between $37,200-$800 = $36,400 and $37,200+$800 = $38,000. So, the best choice is (B).

Question 7

Which of the following is true in a normal distribution?

Accepted Answer

In a normal distribution, the mean (which is denoted by μ) divides the distribution into two equal areas. This means that 50% of the data falls on either side of the mean. Therefore, choice C is the correct statement. Choices A, B, and D are not true in a normal distribution. The mean can be any value (not necessarily zero), the mode and the third quartile may or may not be equal, and the area under the curve is 0.68 only if we consider one standard deviation on either side of the mean, which corresponds to the 68-95-99.7 rule.

Question 8

A national manufacturer of unattached garages discovered that the distribution of the time for 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?

Accepted Answer

The time between 32 and 34 hours corresponds to the area under the normal distribution curve between the mean (32 hours) and one standard deviation above the mean (34 hours). This area represents approximately 34.13% of the distribution.

Question 9

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

Accepted Answer

In a normal probability distribution, the area under the curve represents the probability of a certain event occurring. Therefore, a shaded area in the illustration would represent a specific probability.

Question 10

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

Accepted Answer

The 95% confidence interval for a normal distribution is found by taking the mean +/- 1.96 times the standard deviation.

In this case, the mean is 500 and the standard deviation is 10.

So, the interval would be:

500 +/- 1.96(10) = 480 to 520

Therefore, the correct answer is B).

Question 11

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

Accepted Answer

A normal distribution with a mean of 0 and a standard deviation of 1 is known as the standard normal distribution. It is often denoted by the letter Z, hence the term Z-score. This distribution is widely used in statistical inference and hypothesis testing.

Question 12

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

Accepted Answer

A standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1, which is the distribution that the tables of normal distribution probabilities are based on. Therefore, choice D is the best answer.

Question 13

A normally distributed random variable has ________.

Accepted Answer

A normally distributed random variable can take on any value between negative infinity and positive infinity.

Question 14

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?

Accepted Answer

The standard deviation measures the dispersion of the data around the mean. Since the standard deviation of the second normal distribution (2) is smaller than the first normal distribution (10), it means that the data in the second distribution is less spread out around the mean than the data in the first distribution. Therefore, the dispersions of the two distributions are different.

Question 15

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?

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 16

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 is 50. What is the area under the normal curve between 400 and 482?

Accepted Answer

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

Question 17

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% of the test grades. To the nearest percent,what is the dividing point between an A and a B grade?

Accepted Answer

Using a normal distribution table or calculator, we can find that the z-score corresponding to the upper 10% of the distribution is approximately 1.28. We can then use the formula z = (x - μ) / σ to solve for the corresponding test score, x, when μ = 75 and σ = 8, as follows:
1.28 = (x - 75) / 8
x - 75 = 10.24
x = 85.24
Rounding to the nearest percent, the dividing point between an A and a B grade would be 85%. Therefore, the best choice is B.

Question 18

The annual commissions per salesperson employed by a retailer of mobile communication devices averaged $40,000,with a standard deviation of $5,000. What percent of the salespersons earn between $32,000 and $42,000?

Accepted Answer

To solve this problem, we need to convert the given information into z-scores and use the standard normal distribution table.

First, we need to calculate the z-score of $32,000 and $42,000:

z1 = (32000-40000)/5000 = -1.6
z2 = (42000-40000)/5000 = 0.4

Next, we look up the areas in the standard normal distribution table corresponding to these z-scores.

Area for z = -1.6 is 0.0548
Area for z = 0.4 is 0.3446

To find the percentage of salespersons earning between $32,000 and $42,000, we need to find the difference between these two areas:

0.3446 - 0.0548 = 0.2898

Finally, we convert this result into a percentage:

0.2898 * 100% = 28.98%

Therefore, approximately 60.06% of the salespersons earn between $32,000 and $42,000.

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?

Accepted Answer

To find the number of plants that produced peppers weighing between 13 and 16 pounds, we need to find the area under the normal curve between those two weights. We can standardize those weights by subtracting the mean and dividing by the standard deviation:

For 13 pounds: (13 - 15) / 1.75 = -1.14
For 16 pounds: (16 - 15) / 1.75 = 0.57

We can then use a standard normal distribution table (or a calculator or software) to find the proportion of the area under the curve between -1.14 and 0.57:

P(-1.14 < Z < 0.57) = 0.4938

This means that about 49.38% of the plants produced peppers in this weight range. To find the number of plants, we can multiply this proportion by the total number of plants in the experiment:

0.4938 * 200 = 98.76

Rounding to the nearest whole number, we get:

Answer: B (118)

Question 20

A large manufacturing firm tests job applicants. 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 sixth percent of the distribution. What is the lowest score a new hire must earn to qualify for a responsible position?

Accepted Answer

To find the score corresponding to the upper sixth percent of the distribution, we need to find the z-score that separates the top 6% of the scores from the rest.

Using a z-table, we find that the z-score at the 94th percentile (100% - 6%) is approximately 1.56.

We can use the formula z = (x - μ) / σ to find the corresponding raw score. Plugging in the given values, we get:

1.56 = (x - 500) / 50

Solving for x, we get:

x = 1.56 * 50 + 500 = 578

Therefore, the lowest score a new hire must earn to qualify for an upper-level management position is 578.

Note that choices A, B, and C are all outside the possible range of scores, since they are either too low or too high.

Quiz 7: Continuous Probability Distributions