6.1 GRAPHS OF NORMAL PROBABILITY DISTRIBUTIONSChapter 6:

Normal Curves and Sampling Distributions

Random Variables

Remember 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

Normal 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.

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Important Features of a Normal Curve

1. The curve is bell-shaped, with the highest point over the mean μ

2. The curve is symmetrical about a vertical line through μ

3. The curve approaches the horizontal axis but never 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 1.

Figure 6.1

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 standard deviation σ.

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.

Important Features of a Normal Curve

Normal curves with the same mean but different standard deviations

The Area Under the Normal Curve 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.

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The Area Under the Normal 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.

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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

The Area Under the Normal Curve The Empirical Rule

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). The empirical 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

Figure 6.3

Area Under a Normal Curve

Using the Empirical Rule

1. 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!

Example 1 – 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?

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600

700 800 900500400300The probability that a radio selected at random will last from 600 to 700 hours is 34%.

Example 1 Extension – 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 400 to 700 hours?

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600

700 800 900500400300The probability that a radio selected at random will last from 600 to 700 hours is 81.5%.

13.5 + 34 + 34 = 81.5%

Assignment

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