Normal Distribution and Probability Worksheet

Question 1

A test score distribution is normal with mean 70 and standard deviation 10. Find the z-score for a student who scored 85.

Accepted Answer

The z-score is 1.5 because z = (85 - 70) / 10 = 1.5. The score is 1.5 standard deviations above the mean.

Question 2

The heights of a group of plants are normally distributed with mean 24 cm and standard deviation 3 cm. Find the z-score for a plant that is 18 cm tall.

Accepted Answer

The z-score is -2 because z = (18 - 24) / 3 = -2. The plant is 2 standard deviations below the mean.

Question 3

In a normal distribution, about what percent of data lies within 1 standard deviation of the mean?

Accepted Answer

About 68 percent of the data lies within 1 standard deviation of the mean. This comes from the empirical rule for normal distributions.

Question 4

In a normal distribution, about what percent of data lies between 2 standard deviations below the mean and 2 standard deviations above the mean?

Accepted Answer

About 95 percent of the data lies between 2 standard deviations below the mean and 2 standard deviations above the mean. This is part of the empirical rule.

Question 5

A normal distribution has mean 50 and standard deviation 8. What raw score corresponds to a z-score of 1.25?

Accepted Answer

The raw score is 60 because x = mean + z(standard deviation) = 50 + 1.25(8) = 60.

Question 6

A set of SAT practice scores is normally distributed with mean 500 and standard deviation 100. Using the empirical rule, estimate the probability that a randomly selected score is between 400 and 600.

Accepted Answer

The probability is about 68 percent because 400 and 600 are each 1 standard deviation from the mean of 500. About 68 percent of values in a normal distribution lie within 1 standard deviation of the mean.

Question 7

A normal distribution has mean 30 and standard deviation 5. Using the empirical rule, estimate the probability that a value is greater than 40.

Accepted Answer

The probability is about 2.5 percent because 40 is 2 standard deviations above the mean. About 95 percent of values are within 2 standard deviations, so about 5 percent are outside that range, and half of that is above 40.

Question 8

The weights of boxes are normally distributed with mean 12 pounds and standard deviation 1.5 pounds. Find the z-score for a box weighing 13.5 pounds.

Accepted Answer

The z-score is 1 because z = (13.5 - 12) / 1.5 = 1. The box is 1 standard deviation above the mean.

Question 9

A normal distribution has mean 100 and standard deviation 15. Using the empirical rule, estimate the probability that a value is less than 85.

Accepted Answer

The probability is about 16 percent because 85 is 1 standard deviation below the mean. Since about 68 percent is within 1 standard deviation, about 32 percent is outside that range, and half of that is below 85.

Question 10

A data set is normal with mean 200 and standard deviation 20. What interval contains about 99.7 percent of the data?

Accepted Answer

The interval is from 140 to 260 because 99.7 percent of the data lies within 3 standard deviations of the mean. Three standard deviations is 3 times 20, which is 60, so the interval is 200 - 60 to 200 + 60.

Question 11

The time it takes students to finish a quiz is normally distributed with mean 30 minutes and standard deviation 4 minutes. Estimate the probability that a student takes between 26 and 34 minutes.

Accepted Answer

The probability is about 68 percent because 26 and 34 are each 1 standard deviation from the mean of 30. By the empirical rule, about 68 percent of values fall within 1 standard deviation of the mean.

Question 12

A normal distribution has mean 75 and standard deviation 5. Find the raw score that is 2 standard deviations below the mean.

Accepted Answer

The raw score is 65 because 2 standard deviations below the mean is 75 - 2(5) = 65.

Question 13

A standardized exam has scores that are normally distributed with mean 600 and standard deviation 80. Find the z-score for a score of 520.

Accepted Answer

The z-score is -1 because z = (520 - 600) / 80 = -1. The score is 1 standard deviation below the mean.

Question 14

A normal distribution has mean 40 and standard deviation 6. Using the empirical rule, estimate the probability that a value is between 34 and 52.

Accepted Answer

The probability is about 81.5 percent because 34 is 1 standard deviation below the mean and 52 is 2 standard deviations above the mean. The area from -1 to 1 standard deviation is about 68 percent, and the area from 1 to 2 standard deviations on one side is about 13.5 percent, so the total is about 81.5 percent.

Question 15

The diameters of machine-made bolts are normally distributed with mean 10 mm and standard deviation 0.2 mm. Using the empirical rule, estimate the probability that a bolt has a diameter between 9.8 mm and 10.2 mm.

Accepted Answer

The probability is about 68 percent because 9.8 mm and 10.2 mm are each 1 standard deviation from the mean of 10 mm. By the empirical rule, about 68 percent of values lie within 1 standard deviation of the mean.

Normal Distribution and Probability

Normal Distribution and Probability

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