Graphs of Normal Probability Distributions Section 28

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

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).

A Normal Curve

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

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

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

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?

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

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% + 13.5% = 47.5% There is a 47.5% chance the score will be between 75 and 85.

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.

28 Graphs of Normal Probability Distributions Summarize Notes Homework Worksheet Quiz