6.1 GRAPHS OFNORMAL PROBABILITYDISTRIBUTIONSChapter 6:Normal Curves and Sampling Distributions
Random VariablesRemember from 5.1 A discrete random variable can take on only a finite number of values or a countable number of values A continuous random variable can take on any of the countless number of values in an interval It is important to distinguish between discrete and continuous variables because there are different techniques for each
Page 250Normal Distributions One of the most important examples of a continuous probability distribution is the normal distribution. Can be used for both discrete and continuous random variables. The graph of a normal distribution is called a normal curve. The normal curve is also called a bell-shaped curve.
Important Features of a Normal Curve1. The curve is bell- shaped, with the highest point over the mean μ2. The curve is symmetrical about a vertical line through μ3. The curve approaches the A Normal Curve horizontal axis but never Figure 6.1 touches or crosses it.4. The inflection (transition) points between cupping upward and downward occur above μ + σ and μ – σ.5. The area under the curve is
Important Features of a Normal Curve The parameters that control the shape of a normal curve are the mean and the standard deviation. When both are specified, a specific normal curve is determine. The notation N(μ,σ) stands for a normal model with mean μ and Normal curves with the same standard deviation σ. mean but different standard deviations μ locates the balance point and σ determines the extent of the spread. The curve is close to the horizontal axis at μ + 3σ and μ – 3σ. If the is σ large, then the curve will be more spread out.
PageThe Area Under the Normal 252Curve The area beneath the curve and above the axis is exactly 1. The graph of the normal distribution is important because the portion of the area under the curve in a given interval represents the probability that a measurement will lie in that interval.
Page The Area Under the Normal 252 Curve Recall Chebyshev’s Theorem (Section 3.2) Can be applied to any distribution. Gives the smallest portion of data that lies within k standard deviations of the mean. At least 75% of the data lie within 2 standard deviations of the mean At least 88.9% of the data lie within 3 standard deviations of the mean At least 93.75% of the data lie within 4 standard deviations of the mean
For normal distributions we can get a much more precise result. The empirical rule describes the area of a distribution that is symmetrical and bell- shaped (normal). Theempirical rule is stronger than Chebyshev’s theorem because it gives definite percentages, not just lower limits.
The Empirical Rule Approximately 68% of the data values will lie within 1 standard deviation of the mean Approximately 95% of the data values will lie between 2 standard deviations of the mean Approximately 99.7% of the data values will lie between 3 standard deviation of the mean
The Empirical Rule Area Under a Normal Curve Figure 6.3
Using the Empirical Rule1. Sketch the bell curve above the horizontal axis.2. Set up the horizontal axis using the mean and standard deviation given.3. Shade the area that you are looking for.4. Use Figure 6-3 to calculate the area/percentage Always Draw a Picture!
Page Example 1 – Empirical Rule 253 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? 300 400 500 600 700 800 900 The probability that a radio selected at random will last from 600 to 700 hours is
Page Example 1 Extension – Empirical 253 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 400 to 700 hours? 13.5 + 34 + 34 = 81.5% 300 400 500 600 700 800 900 The probability that a radio selected at random will last from 600 to 700 hours is