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CH. 6: Normal Distribution

Module by: Susan Dean, Barbara Illowsky, Ph.D.. E-mail the authors

Summary: This module is the complementary teacher's guide for the "Normal Distribution" chapter of the Collaborative Statistics collection (col10522) by Barbara Illowsky and Susan Dean.

A fair number of students are familiar with the "bell-shaped" curve. Stress that the normal is a continuous distribution like the uniform and exponential. However, the left and right tails extend indefinitely but come infinitely close to the xx size 12{x} {}-axis. It is not necessary to show the probability distribution function for the normal (it is in the book) because there are normal probability tables and technology available for probability and percentile calculations.

Visualize the Data

Draw a picture of the normal graph and explain that it is symmetrical about the mean. The shape of the graph depends on the standard deviation. The smaller the standard deviation, the skinnier and taller the graph. A change in the mean shifts the graph to the right or left. The notation for the normal is X~N(μ,σ)X~N(μ,σ). Draw several normal curves (superimposed upon each other). Have students determine how the means and standard deviations are changing.

Figure 1
Normal distributions curve that is vertically steep, about twice as high as it is wide
Figure 2
Normal distributions curve that is horizontally wide, about twice as wide as it is tall

The Normal Distribution Notation

The standard normal distribution is of special interest. Notation: Z~N(0,1)Z~N(0,1) where ZZ = one z-score (the number of standard deviations a value is to the right or left of the mean). The mean is 0 and the variance (and standard deviation) is 1. Any normal distribution can be standardized to the standard normal by the z-score formula: z = value μ σ z= value μ σ . Do an example showing the standardization. For X~N(3,2)X~N(3,2) and Y~N(5,6)Y~N(5,6), the values x=4x=4 and y=8y=8 are each 1212 standard deviation to the right ( 12σ12σ) of their respective means. Therefore, they both have a z-score of 1212.

Example 1

Do an example using the normal distribution and the standardization.

Problem 1

Several studies have shown that the amount of time people stand in line waiting for a bank teller is normally distributed. Suppose the mean waiting time is 3 minutes and the standard deviation is 1.5 minutes. Let XX = the amount of time, in minutes, one person stands in line waiting for a teller. Notation: X~N(3,1.5)X~N(3,1.5)

Find the probability that one person waits in line for a teller less than 2 minutes. Have students draw the picture and write a probability statement. The picture should have the xx-axis.

Solution

Figure 3: Probability statement: P ( X < 2 ) = 0.2500 P(X<2)=0.2500. If you use the TI-83/84 series, the function normalcdf(0, 2, 3, 1.5) in 2nd DISTR. k = 5.47 k=5.47
Distribution curve with left third of the curve shaded
Figure 4: P ( X < k ) = 0 . 95 P ( X < k ) = 0 . 95 size 12{P \( X<k \) =0 "." "95"} {} . If you use the TI83/84 series of articles, use the function InvNorm(.95,3,1.5)InvNorm(.95,3,1.5) size 12{ ital "InvNorm" \( "." "95",3,1 "." 5 \) } {}. k = 5 . 47 k = 5 . 47 size 12{k=5 "." "47"} {}
Distribution curve with 0.95 as the mean

Note:

The normal approximation to the binomial is NOT included in this text. With graphics calculators and computer software, it is easy to draw a binomial graph with a small nn size 12{n} {} and then make nn size 12{n} {}, say, 50. Students will see the graph approach the normal. The normal approximation states that if XX size 12{X} {} follows a binomial distribution with number of trials equal to nn size 12{n} {} and probability of success for any trial equal to p(X~B(n,p))p(X~B(n,p)) size 12{p \( X "~" B \( n,p \) \) } {}, then by adding ±0.5±0.5 size 12{ +- 0 "." 5} {} to XX size 12{X} {}, you get a new random variable YY size 12{Y} {} ( YY size 12{Y} {} is either X+0.5X+0.5 size 12{X+0 "." 5} {}or X0.5X0.5 size 12{X - 0 "." 5} {}) and YY size 12{Y} {} follows a normal distribution (Y~N(np,npq))(Y~N(np,npq)) size 12{ \( Y "~" N \( ital "np", ital "npq" \) \) } {}. For the approximation to be a good one, you want np>5np>5 size 12{ ital "np">5} {}, nq>5nq>5 size 12{ ital "nq">5} {}, and n>20n>20 size 12{n>"20"} {}.

Assign Practice

Assign the Practice in class to be done in groups.

Assign Homework

Assign Homework. Suggested problems: 1 - 11 odds, 8, 10, 12 - 19.

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