1. Normal Distributions

2. Essential Question: How do you find percents of data and probabilities of events associated with normal distributions?

3. Normal Curves (68-95-99.7 Rule)
68% of the data fall within 1 standard deviation of the mean.
95% of the data fall within 2 standard deviation of the mean.
99.7% of the data fall within 3 standard deviation of the mean.

4. Normal Curve’s Symmetry

5. Finding Areas Under a Normal Curve
Suppose the masses (in grams) of pennies minted in the United States after 1982 are normally distributed with a mean of 2.5g and a standard deviation of 0.02g.
Find the following:
Percent of pennies that have a mass between 2.46g and 2.54g.

6. Finding Areas Under a Normal Curve
Suppose the masses (in grams) of pennies minted in the United States after 1982 are normally distributed with a mean of 2.5g and a standard deviation of 0.02g.
Find the following:
The probability that a randomly chosen penny has a mass greater than 2.52g.

7. Reflect 2a.
Explain how you know that the area under the curve between 𝜇+𝜎 and 𝜇+2𝜎 represents 13.5% of the data if you know that the percent of the data within 𝜎 of the mean is 68% and the percent of the data within 2𝜎 of the mean is 95%.

8. The Standard Normal Curve
Standard Normal Distribution has a mean of 0 and a standard deviation of 1.
A data value 𝑥 from a normal distribution with a mean 𝜇 and standard deviation 𝜎 can be standardized by finding its z-score

9. The Standard Normal Curve
Areas under the standard normal curve to the left of a given z-score have be computed and appear in the standard normal table.

10. Using the Z-Score
𝑃 𝑧≤1.3 = 𝑜𝑟 90.32%

11. Example
Suppose the heights (in inches) of adult females in the United States are normally distributed with a mean of 63.8 inches and a standard deviation of 2.8 inches.
Fine each of the following:
The percent of women who are no more than 65 inches tall.
The probability that a randomly chosen woman is between 60 inches and 63 inches tall.

12. The percent of women who are no more than 65 inches tall.
Convert 65 to a z-score:
“no more than” means: ___
𝜇=63.8 𝑖𝑛𝑐ℎ𝑒𝑠
𝜎=2.8 𝑖𝑛𝑐ℎ𝑒𝑠

13. 𝑃(𝑧≤0.4)

14. Reflect 3a
Using this result, you can find the percent of females who are at least 65 inches tall without needing the table. Find the percent and explain your reasoning.

15. Convert 60 to a z-score: Convert 63 to a z-score:
The probability that a randomly chosen woman is between 60 inches and 63 inches tall.
Convert 60 to a z-score:
Convert 63 to a z-score:
𝜇=63.8 𝑖𝑛𝑐ℎ𝑒𝑠
𝜎=2.8 𝑖𝑛𝑐ℎ𝑒𝑠
𝑃( 𝑧 60 ≤𝑧≤ 𝑧 63 ) = 𝑃 𝑧≤ 𝑧 63 −𝑃(𝑧≤ 𝑧 60 ) =

16. 𝑃 𝑧≤−0.3 −𝑃(𝑧≤−1.4) =

17. Reflect 3b
How does the probability that a randomly chosen female has a height between 64.6 inches and 67.6 inches compare with your answer? Why?