Normal Distributions & the Empirical Rule Unit 4 Normal Distributions & the Empirical Rule How do we describe normal distributions and use the empirical rule? M2 Unit 4: Day 2

Normal distribution: modeled by a bell shaped curve called a normal curve that is symmetric about the mean. Ex: * The area under the curve is 1 (100%)

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

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

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

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

Give the percent of area under the curve

Give the percent of area under the curve

You try: Give the percent of area under the curve

Using Probability with the Normal curve. Find the probability that a randomly selected x-value is between and 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 NOTE: For Probability, change percent to decimal when adding them together.

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

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? 68% of white oak trees have heights between 86.5 feet and 93.5 feet 79.5 83 86.5 90 93.5 97 100.5

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. 97.5% of the students scores were Less than a 95. 80 83 86 89 92 95 98

The weight of 2 month old puppies averaged 15 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? 81.5% of the puppies Weigh between 13.6 and 16 pounds. 12.8 13.6 14.4 15.2 16 16.8 17.6

The weight of newborn baby averaged 7 pounds with a standard deviation of 1 pound. 84% What percentage of babies weigh more than 6 pounds? What percentage of babies weigh less than 6 pounds? 16% 81.5% What percentage of babies weigh between 5 and 8 pounds? 4 5 6 7 8 9 10

A normal distribution has a mean of 18 and a standard deviation of 2. 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 12 14 16 18 20 22 24

Suppose driving speeds on the interstate show a normal distribution with a mean of 78 and a standard deviation of 3. 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? 69 72 75 78 81 84 87

Homework: pg 266 (#1-11, 18) Pg 267 (#1-8)