Transcript Math 10 Chapter 6 Notes: The Normal Distribution

Math 10 Chapter 6 Notes: The Normal
Distribution

Notation: X is a continuous random
variable X ~ N(, )
 Parameters:  is the mean and  is the
standard deviation
 Graph is bell-shaped and symmetrical
 The mean, median, and mode are the same
(in theory)
Math 10 Chapter 6 Notes: The Normal
Distribution

Total area under the curve is equal to 1.
Probability = Area
 P(X < x) is the cumulative distribution function or
Area to the Left.
 A change in the standard deviation, , causes the
curve to become wider or narrower
 A change in the mean, , causes the graph to shift
Math 10 Chapter 6 Notes: The Standard
Normal Distribution

A normal (bell-shaped) distribution of
standardized values called z-scores.
 Notation: Z ~ N(0, 1)
 A z-score is measured in terms of the standard
deviation.
 The formula for the z-score is
x
z

Math 10 Chapter 6 Notes: The Normal
Distribution

Bell-shaped curve
 Most values cluster
about the mean
 Area within 4 standard
deviations (+ or - 4 )
is 1
Math 10 Chapter 6 Notes: The Normal
Distribution
Ex. Suppose X ~ N(100, 5). Find the zscore (the standardized score) for x = 95 and
for 110.
x
z

x
z

= 95 – 100 = - 1
5
= 110 – 100 = 2
5
Math 10 Chapter 6 Notes: The Normal
Distribution
 The z-score lets us compare data that are scaled
differently. Ex. X~N(5, 6) and Y~N(2, 1) with x =
17 and y = 4; X = Y = weight gain
17 – 5 = 2
6
4–2 = 2
1
Math 10 Chapter 6 Notes: The Standard
Normal Distribution
 Ex. Suppose Z ~ N(0, 1). Draw pictures and
find the following.
1. P(-1.28 < Z < 1.28)
2. P(Z < 1.645)
3. P(Z > 1.645)
4. The 90th percentile, k, for Z scores.
For 1, 2, 3 use the normal cdf
For 4, use the inverse normal
Math 10 Chapter 6 Notes: The Normal
Distribution
Ex: At the beginning of the term, the amount of
time a student waits in line at the campus store is
normally distributed with a mean of 5 minutes
and a standard deviation of 2 minutes.
Let X = the amount of time, in minutes, that a
student waits in line at the campus store at the
beginning of the term.
X ~ N(5, 2) where the mean = 5 and the standard
deviation = 2.
Math 10 Chapter 6 Notes: The Normal
Distribution
Find the probability that one randomly chosen
student waits more than 6 minutes in line at the
campus store at the beginning of the term.
P(X > 6) = 0.3085.
Math 10 Chapter 6 Notes: The Normal
Distribution
Find the 3rd quartile. The third quartile
is equal to the 75th percentile.
Let k = the 75th percentile
(75th %ile).
P(X < k ) = 0.75.
The 3rd quartile or 75th percentile is
6.35 minutes (to 2 decimal places).
Seventy-five percent of the waiting
times are less than 6.35 minutes.