1. Graphs of Normal Probability Distributions
Section 8.1

2. Objectives
Graph a normal curve and summarize its important properties.
Apply the empirical rule to solve real-world problems.

3. In general the normal curve is smooth and symmetrical about the vertical line extending upward from the mean .
The highest point of the curve occurs at .
The curve tends to level out and approach the horizontal (x axis).

4. A Normal Curve

5. Properties of Normal Curve
1) Curve is bell-shaped, with high-point over the mean
2) Curve is symmetrical about a vertical line through the mean
3) Curve approaches horizontal axis
5) Area under the curve is 1

6. Empirical Rule
1) 68% of data is within 1 σ ofμ
2) 95% of data is within 2 σ ofμ
3) 99.7% of the data is within 3 σ ofμ

7. The preceding statement is called the empirical rule because, for symmetrical, bell-shaped distributions, the given percentages are observed in practice.
Area Under a Normal Curve

8. Example–Empirical rule
The playing life of a Sunshine radio is normally distributed with mean  = 600 hours and standard deviation  = 100 hours.
What is the probability that a radio selected at random will last from 600 to 700 hours?

9. Example– Solution
Distribution of Playing Times
The probability a Sunshine radio will last between 600 and 700 playing hours is about 0.34.

10. Example–Empirical rule
The scores on the statistics exam are normally distributed with a mean  = 85 percent and a standard deviation  = 5 percent. Find the probability that an exam selected at random will be a score between 75 and 85?
34% % = 47.5%
There is a 47.5% chance the score will be between 75 and 85.

11. Example–Empirical rule
At a fair, 37 pumpkins were entered for judging under the normal pumpkin division. If the masses are normally distributed and the average weight is 12.5 kg, with a standard deviation for 1.5 kg. What is the probability that a pumpkin has a mass between 14 kg and 17kg?
13.5%+2.35% = 15.85%
There is a 15.85% chance a pumpkin will weigh between 14 and 17 kilograms.

12. 8.1 Graphs of Normal Probability Distributions
Summarize Notes
Read section 8.1
Homework
Worksheet