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Math Coach Amy
Amy Ferguson Moncure

H Algebra 2/Trig · Unit 8

Normal Probabilities

Find how many standard deviations a value is from the mean, or use normalcdf for a probability.

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Key Ideas Easy Medium Hard

z = value − meanstandard deviation

normalcdf(lower, upper, mean, standard deviation)

Round a probability to the nearest thousandth.

What to Remember

Standard deviations from the mean

Subtract the mean from the value, then divide by the standard deviation.

A negative result is below the mean.

normalcdf

normalcdf(lower, upper, mean, standard deviation) gives the probability between the two bounds.

Use a large negative lower bound for “below,” and a large upper bound for “above.”

Level 1

Easy

How many standard deviations?

Example
Scores are normally distributed with mean 80 and standard deviation 10. How many standard deviations from the mean is 75?

z = value − meanstandard deviation

z = 75 − 8010

z = −510

z = −12

12 standard deviations below the mean

Answer: 12 standard deviations below the mean

1
Times are normally distributed with mean 64 minutes and standard deviation 8 minutes. How many standard deviations from the mean is 80?
Show Answer

z = value − meanstandard deviation

z = 80 − 648

z = 168

z = 2

2 standard deviations above the mean

Answer: 2 standard deviations above the mean

2
Lengths are normally distributed with mean 50 cm and standard deviation 12 cm. How many standard deviations from the mean is 44?
Show Answer

z = value − meanstandard deviation

z = 44 − 5012

z = −612

z = −12

12 standard deviations below the mean

Answer: 12 standard deviations below the mean

3
Weights are normally distributed with mean 96 ounces and standard deviation 16 ounces. How many standard deviations from the mean is 104?
Show Answer

z = value − meanstandard deviation

z = 104 − 9616

z = 816

z = 12

12 standard deviations above the mean

Answer: 12 standard deviations above the mean

4
Temperatures are normally distributed with mean 70 degrees and standard deviation 6 degrees. How many standard deviations from the mean is 61?
Show Answer

z = value − meanstandard deviation

z = 61 − 706

z = −96

z = −32

32 standard deviations below the mean

Answer: 32 standard deviations below the mean

5
Heights are normally distributed with mean 90 cm and standard deviation 4 cm. How many standard deviations from the mean is 90?
Show Answer

z = value − meanstandard deviation

z = 90 − 904

z = 04

z = 0

0. The value is the mean.

Answer: 0. The value is the mean.

Level 2

Medium

A probability to the nearest thousandth

Example
Warmup times are normally distributed with mean 18 minutes and standard deviation 4 minutes. What is the probability that a warmup lasts between 15 and 24 minutes, to the nearest thousandth?

Use normalcdf(lower, upper, mean, standard deviation).

normalcdf(15, 24, 18, 4)

Round the probability to the nearest thousandth.

0.707

Answer: 0.707

1
Quiz scores are normally distributed with mean 74 and standard deviation 8. What is the probability that a score is below 80, to the nearest thousandth?
Show Answer

Use normalcdf(lower, upper, mean, standard deviation).

normalcdf(−99999, 80, 74, 8)

Round the probability to the nearest thousandth.

0.773

Answer: 0.773

2
Heights are normally distributed with mean 66 inches and standard deviation 3 inches. What is the probability that a height is above 72 inches, to the nearest thousandth?
Show Answer

Use normalcdf(lower, upper, mean, standard deviation).

normalcdf(72, 99999, 66, 3)

Round the probability to the nearest thousandth.

0.023

Answer: 0.023

3
Package weights are normally distributed with mean 40 ounces and standard deviation 6 ounces. What is the probability that a package weighs between 34 and 52 ounces, to the nearest thousandth?
Show Answer

Use normalcdf(lower, upper, mean, standard deviation).

normalcdf(34, 52, 40, 6)

Round the probability to the nearest thousandth.

0.819

Answer: 0.819

4
Delivery times are normally distributed with mean 28 minutes and standard deviation 5 minutes. What is the probability that a delivery takes less than 22 minutes, to the nearest thousandth?
Show Answer

Use normalcdf(lower, upper, mean, standard deviation).

normalcdf(−99999, 22, 28, 5)

Round the probability to the nearest thousandth.

0.115

Answer: 0.115

5
Page counts are normally distributed with mean 55 pages and standard deviation 7 pages. What is the probability that an essay is longer than 64 pages, to the nearest thousandth?
Show Answer

Use normalcdf(lower, upper, mean, standard deviation).

normalcdf(64, 99999, 55, 7)

Round the probability to the nearest thousandth.

0.099

Answer: 0.099

Level 3

Hard

How many people

Example
Finish times are normally distributed with mean 70 seconds and standard deviation 5 seconds. Out of 200 runners, how many finish in less than 75 seconds, to the nearest whole number?

Use normalcdf(lower, upper, mean, standard deviation).

normalcdf(−99999, 75, 70, 5)

Multiply that probability by the number of people.

0.841 × 200 = 168.200

Round to the nearest whole number.

168

Answer: 168 people

1
Scores are normally distributed with mean 82 and standard deviation 9. Out of 150 students, how many score between 70 and 90, to the nearest whole number?
Show Answer

Use normalcdf(lower, upper, mean, standard deviation).

normalcdf(70, 90, 82, 9)

Multiply that probability by the number of people.

0.722 × 150 = 108.300

Round to the nearest whole number.

108

Answer: 108 people

2
Drive times are normally distributed with mean 36 minutes and standard deviation 4 minutes. Out of 120 days, how many drives take more than 41 minutes, to the nearest whole number?
Show Answer

Use normalcdf(lower, upper, mean, standard deviation).

normalcdf(41, 99999, 36, 4)

Multiply that probability by the number of people.

0.106 × 120 = 12.720

Round to the nearest whole number.

13

Answer: 13 people

3
Weights are normally distributed with mean 48 pounds and standard deviation 6 pounds. Out of 80 crates, how many weigh less than 42 pounds, to the nearest whole number?
Show Answer

Use normalcdf(lower, upper, mean, standard deviation).

normalcdf(−99999, 42, 48, 6)

Multiply that probability by the number of people.

0.159 × 80 = 12.720

Round to the nearest whole number.

13

Answer: 13 people

4
Points are normally distributed with mean 90 and standard deviation 12. Out of 180 games, how many have between 78 and 102 points, to the nearest whole number?
Show Answer

Use normalcdf(lower, upper, mean, standard deviation).

normalcdf(78, 102, 90, 12)

Multiply that probability by the number of people.

0.683 × 180 = 122.940

Round to the nearest whole number.

123

Answer: 123 people

5
Wait times are normally distributed with mean 24 minutes and standard deviation 5 minutes. Out of 175 customers, how many wait more than 30 minutes, to the nearest whole number?
Show Answer

Use normalcdf(lower, upper, mean, standard deviation).

normalcdf(30, 99999, 24, 5)

Multiply that probability by the number of people.

0.115 × 175 = 20.125

Round to the nearest whole number.

20

Answer: 20 people

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