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.
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.
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.
Do an example using the normal distribution and the standardization.
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.
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
X−0.5X−0.5 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 the Practice in class to be done in groups.
Assign Homework. Suggested problems: 1 - 11 odds, 8, 10, 12 - 19.