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Normal Distributions & the Empirical Rule

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1 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

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%)

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

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

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

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

7 Give the percent of area under the curve

8 Give the percent of area under the curve

9 You try: Give the percent of area under the curve

10 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.

11 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.

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

13 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

14 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.

15 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.

16 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?

17 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

18 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?

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


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