1. Math 3 Warm Up 4/23/12
Find the probability mean and standard deviation for the following data. 2, 4, 5, 6, 5, 5, 5, 2, 2, 4, 4, 3, 3, 1, 2, 2, 3, 4, 6, 5 Hint: First create a probability distribution.

3. Unit 6: Data Analysis
Empirical Rule

4. What does a population that is normally distributed look like?

5. Empirical Rule 68-95-99.7% RULE 68% 95% 99.7%
Empirical Rule is sometimes referred to as the % Rule. Again, to use the Empirical Rule the distribution of the data must be normal bell-shaped curve.
99.7%
% RULE

6. Empirical Rule 68-95-99.7% RULE 68% 95% 99.7%
Empirical Rule is sometimes referred to as the % Rule. Again, to use the Empirical Rule the distribution of the data must be normal bell-shaped curve.
99.7%
% RULE

7. Empirical Rule 68-95-99.7% RULE 68% 95% 99.7%
Empirical Rule is sometimes referred to as the % Rule. Again, to use the Empirical Rule the distribution of the data must be normal bell-shaped curve.
99.7%
% RULE

8. Empirical Rule 68-95-99.7% RULE 68% 95% 99.7%
Empirical Rule is sometimes referred to as the % Rule. Again, to use the Empirical Rule the distribution of the data must be normal bell-shaped curve.
99.7%
% RULE

9. Empirical Rule—restated
68% of the data values fall within 1 standard deviation of the mean in either direction
95% of the data values fall within 2 standard deviation of the mean in either direction
99.7% of the data values fall within 3 standard deviation of the mean in either direction
Remember values in a data set must appear to be a normal bell-shaped histogram, dotplot, or stemplot to use the Empirical Rule!
Rule of Thumb: Range divide by 6 approximately the standard deviation.

10. Empirical Rule 34% 34% 68% 47.5% 47.5% 95% 49.85% 49.85% 99.7%
Take time to slowly click through slide.
Stress that the Empirical Rule can ONLY be used with the assumption that the distribution is normal (bell-shaped curve). Sixty-eight percent of the ordered data of a normal distribution lies within one standard deviation of the mean. Ninety-five percent of the ordered data of a normal distribution lies within two standard deviations of the mean. And, 99.7% of the ordered data of a normal distribution lies within 3 standard deviations of the mean. The normal distribution here, in this example shown, has a mean of 0 and standard deviation of 1.
49.85%
49.85%
99.7%

11. Average American adult male height is 69 inches (5’ 9”) tall with a standard deviation of 2.5 inches. What does the normal distribution for this data look like?

12. Empirical Rule-- Let H~N(69, 2.5)
What is the likelihood that a randomly selected adult male would have a height less than 69 inches?
Answer: P(h < 69) = .50
P represents Probability
Let H~N(69, 2.5) means H represents the variable height for an adult male has a normal distribution with population mean of 69 inches and a population standard deviation of 2.5 (or variance of 2.5).
Start with an easy problem using the empirical rule AND introduce proper probability notation. Explain P implies probability (likelihood of an event), H represents the random variable of height of adult male, P(h<69) is the question in statistical notation.
h represents one adult male height

13. Using the Empirical Rule
Let H~N(69, 2.5)
What is the likelihood that a randomly selected adult male will have a height between 64 and 74 inches?
P(64 < h < 74) = .95
In Calculator:
2nd Vars:
normalcdf(lower, upper, mean, std. dev.)
Let H~N(69, 2.5) means H represents the variable height for an adult male has a normal distribution with population mean of 69 inches and a population standard deviation of 2.5 (or variance of 2.5).

14. Using the Empirical Rule
Let H~N(69, 2.5)
What is the likelihood that a randomly selected adult male will have a height between 64 and 74 inches?
In Calculator:
2nd Vars
normalcdf(lower, upper, mean, std. dev.)
For this example:
Normalcdf(64, 74, 69, 2.5) = .95
Let H~N(69, 2.5) means H represents the variable height for an adult male has a normal distribution with population mean of 69 inches and a population standard deviation of 2.5 (or variance of 2.5).

15. Using Empirical Rule-- Let H~N(69, 2.5)
What is the likelihood that a randomly selected adult male would have a height of less than 66.5 inches?
There are several ways to do this problem. Some students may see it as to obtain .16.
If students obtained an answer of .16 but did it differently please ask them to discuss their method.
= .16

16. Using Empirical Rule--Let H~N(69, 2.5)
What is the likelihood that a randomly selected adult male would have a height of greater than 74 inches?
= .025

17. Using Empirical Rule--Let H~N(69, 2.5)
What is the probability that a randomly selected adult male would have a height between 64 and 76.5 inches?
= .9735

18. Assignment: Complete Normal Distribution Practice Worksheet