Download AP Stats Chapter 2 Student

Chapter 2
THE NORMAL
DISTRIBUTIONS
Z-Scores and Density Curves
A Question
 Last year, Eunice had Mr. Allen for math and
received a 87% in the class, while Irene had Mr.
Merlo for the same math class and received a 80%.
It has been mathematically proven that Mr. Merlo is
a much harder teacher. In fact, his class average was
15% lower than Mr. Allen last year. Who is smarter?
 Why can you argue that Irene is smarter?
 What extra piece of information might prove that
Eunice is actually smarter?
The Standardized Value
 The Standardized Value (z-score) is a measure
of the number of standard deviations a piece of data
is away from the mean in a normal distribution.
 If a test or other measure has been standardized, z-
scores can be used to determine whether or not
individuals are better.
A More Detailed Question
 Last year, Eunice had Mr. Allen for math and
received a 87% in the class, while Irene had Mr.
Merlo for the same math class and received a 80%.
It has been mathematically proven that Mr. Merlo is
a much harder teacher. In fact, Mr. Allen’s class
average was 15 points higher than Mr. Merlo’s 70%
average. If we know that Mr. Allen’s class had a
standard deviation of 2% and Mr. Merlo’s class had a
standard deviation of 10%, Who is smarter?
Density Curves
 A density curve is what you get when you collect a
lot of data and you get a fluid shaped graph.
 It has an area of exactly 1 underneath it.
 That’s because it represents 100% of your data.
 The median cuts the area in half.
 The mean is the balance point.
Two Different Density Curves
What Is the Most Common Density Curve?
Normal Distribution
 This is the standard bell-shaped curve.
 The mean and median are always the same in a
normal distribution.
 Although different normal distributions are similar,
they might have different shapes.


Some are “taller” or “wider” than others.
What determines how “tall” or “wide” a normal distribution is?

The standard deviation.
One Up, One Down
 Although the shape may change, the proportion of
the data between the two standard deviations
remains the same.


68% of the outcomes are between one standard deviation
above and below the average.
Notice one standard deviation away is at the inflection point.
The Empirical Rule
 The Empirical Rule (68-95-99.7) Rule tells you the
proportion of the data that is in the middle when you
move 1-2-3 standard deviations away from the mean.
Standard Normal Calculations
AND WHAT THE AP GRADERS ARE LOOKING
FOR
Finding a Probability
 If a population is known to have a normal
distribution of ages with an average of 16 and a
standard deviation of 1.2, what is the probability that
a randomly chosen individual will be older than 18?
 N(µ, σ)N(16, 1.2)
P(x>18)
= P(z>(18-16)/1.2)
= P(z>1.67)
= 1-.9525
= .0475
Know how to find the probability of an event
occuring
 Using the same information from the previous slide,
what proportion of the population is between the
ages of 16 and 17?
N(16, 1.2)
P(16 < X < 17)
= P((16-16)/1.2)< Z < (17-16)/1.2)
= P(0 < Z < 0.83)
= 0.7967 – 0.5
=0.2967
Know what a percentile is and how to a value
at a certain percentile
 Using the same information from the previous two
slides, what age does an individual have to be in
order to be above the 35th percentile?
N(16, 1.2)
P(Z < -0.39) = 0.35
-0.39 = (x – 16)/1.2
-.47 = x – 16
X = 15.53
Calculator
 Normalcdf(lowerbound, upperbound, avg, s.d.)
Example
Find P(x>18)=normalcdf(18, 999999999, 16, 1.2)
 InvNorm(percent behind, avg, s.d.)
Example
P(x < ___) = 0.35InvNorm(0.35, 16, 1.2)
Know how to find a population average if
you know the probability of an event and s.d.
 In a certain baseball league 20% of the individuals
have more than 60 RBIs. If the standard deviation of
all the players’ RBIs is 15 and the distribution is
known to be approximately normal, what is the
average number of RBIs in this league?
 The league average is 47.4 RBIs
Know how to find the middle n% of a normal
distribution
 Looking at a distribution that is N(10, 2), what
interval contains the 20% of the population with the
shortest interval?
Solution
In any normal distribution, the n% with the will be in
the middle, because that is where your largest
percent of data is. So, this question is really just,
“where is the middle 20%?”
Solution Continued
 Since we’re looking for the middle 20% of a N(10, 2),
we will look for the z-score that have 40% above and
40% below.
 A similar method using z=0.25 would give us an x
value of 10.5. So, the smallest interval contain 20%
of the data is between 9.5 and 10.