Normal Distributions & the Empirical Rule

How do we describe normal distributions and use the

empirical rule?

Unit 4

M2 Unit 4: Day 2

* The area under the curve is 1 (100%)

Ex:

Normal distribution: modeled by a bell shaped curve called a normal curve that is symmetric about the mean.

Empirical rule: (68-95-99.7% rule) 68% of the data will be located within one standard

deviation symmetric to the mean

3 2 2 3 3 2 2 3x x x x x x x

Empirical rule: (68-95-99.7% rule) 95% of the data will be located within 2 standard deviations

symmetric to the mean

3 2 2 3

3 2 2 3x x x x x x x

Empirical rule: (68-95-99.7% rule) 99.7% of the data will be located within 3 standard deviations

symmetric to the mean

3 2 2 3

3 2 2 3x x x x x x x

Because the data percentages are symmetric about the mean with respect to ’s, we can break up the percentages further

34 13.5 47.5% x

Give the percent of area under the curve

x34 34 13.5 2.35 0.15 84%

Give the percent of area under the curve

You try: Give the percent of area under the curve

13.5 2.35 15.85% 2.35 34 2.35 38.7%

Using Probability with the Normal curve.

Find the probability that a randomly selected x-value is between and x 2x

2P x x x

0.34 .0135 0.475

NOTE: For Probability, change percent to

decimal when adding them together.

Using Probability with the Normal curve.

Find the probability that a randomly selected x-value is less than or equal to x

NOTE: For Probability, change percent to

decimal when adding them together.

P x x

0.34 0.34 0.135 0.0235 0.015

0.84

You try: Find the probability of selecting a random x-value:

)a P x x )b P x x x

) 2c P x x ) 2 3d P x x x

0.16 0.34

0.025 0.9735

The heights of fully grown white oak trees are normally distributed with a mean of 90 feet and standard deviation of 3.5 feet. About what percent of white oak trees have heights between 86.5 feet and 93.5 feet?

3.5x x 3.5x

79.5 83 86.5 90 93.5 97 100.5

68% of white oak trees have heights

between 86.5 feet and 93.5 feet

A math 2 student makes a mean score of 89 on tests with a standard deviation of 3. Assuming a normal distribution, estimate the percent of grades the student had that were less than a 95.

80 83 86 89 92 95 98

97.5% of the students scores were

Less than a 95.

The weight of 2 month old puppies averaged 15.2 pounds with a standard deviation of .8 pounds. What percentage of puppies weigh between 13.6 and 16 pounds?

12.8 13.6 14.4 15.2 16 16.8 17.6

81.5% of the puppies Weigh between 13.6

and 16 pounds.

The weight of newborn baby averaged 7 pounds with a standard deviation of 1 pound.

4 5 6 7 8 9 10

What percentage of babies weigh more than 6 pounds?

What percentage of babies weigh less than 6 pounds?

What percentage of babies weigh between 5 and 8 pounds?

84%

16%

81.5%

A normal distribution has a mean of 18 and a standard deviation of 2.

12 14 16 18 20 22 24

What is the probability that a randomly selected x-value from the distribution is between 12 and 20?

What is the probability that a randomly selected x-value from the

distribution is less than 24?

.8385 .9985

Suppose driving speeds on the interstate show a normal distribution with a mean of 78 and a standard deviation of 3.

69 72 75 78 81 84 87

Within what range do about 68% of the speeds fall?

Within what range do about 95% of the speeds fall?

Within what range do about 99.7% of the speeds fall?

75 81x

72 84x

69 87x

Homework:

• pg 266 (#1-11, 18)

• Pg 267 (#1-8)