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Name _______________________________
AP STATISTICS CHAPTER 2:
DENSITY CURVES AND THE NORMAL
DISTRIBTION
The histogram to the right represents the final grades
for Mr. E’s Calculus classes.
1. What proportion of students had scores below 70?
2. What proportion of students had scores between
75 and 100?
As data sets become larger, we can draw a curve
which describes the overall pattern of the distribution.
1. About what proportion of this curve lies below
70?
2. About what proportion of this curve lies between
70 and 75?
3. If the proportions represented by each bar in the
histogram were equal to the area of each bar, then
what is the total area under this curve?
Steps in exploring quantitative data:
Density Curves:
SUMMARY/QUESTIONS TO ASK IN CLASS
Name _______________________________
AP STATISTICS CHAPTER 2:
SYMMETRIC AND SKEWED DISTRIBUTIONS
Locate the mean and median in these density curves:
NOTATION FOR MEAN AND STANDARD
DEVIATION
Mean
Standard
Deviation
NORMAL DISTRIBUTIONS
Words which describe a normal distribution:
The density curve for a normal distribution is
described by giving
The Empirical Rule: also known as
SUMMARY/QUESTIONS TO ASK IN CLASS
Name _______________________________
AP STATISTICS CHAPTER 2:
The distribution of Iowa Test of Basic Skills (ITBS)
vocabulary scores for 7th grade students in Gary,
Indiana, is close to Normal. Suppose the distribution
is N(6.84, 1.55).
a) Sketch the Normal density curve for this
distribution.
b) What percent of ITBS vocabulary scores are
less than 3.74?
c)
What percent of the scores are between 5.29
and 9.94?
The height of a giraffe can be described by a normal
distribution. Mean,  = 204 inches with a standard
deviation,  = 5.5 inches. Draw a normal curve to
represent this information.
 What percent of giraffes are taller than 215
inches?
 What percent of giraffes are shorter than 204
inches?
 What percent of giraffes are taller than 198.5
inches?
 A height of 209.5 inch corresponds to what
percentile of giraffe height?
SUMMARY/QUESTIONS TO ASK IN CLASS
Name _______________________________
AP STATISTICS CHAPTER 2:
A student states that he got a 640 on both the verbal
and math sections of the SAT. The student stated, “I
feel stronger in math, so I don’t understand what
went wrong.”
After researching the College Board’s website the
following was found:
Math:  = 455,  = 100
Verbal:  = 500,  = 85.
Does this imply that the math score is actually better
or worse? Explain.
STANDARD NORMAL CALCULATIONS
The standard normal distribution is ________
What if we know that data is distributed normally,
but has a mean and standard deviation other than 0
and 1?
allows us to
change the units so that they can be compared with
the standard normal distribution.
Given: 1)
2)
3)
The standardized value of x is given by:
This new value is called the :
Ex: Former NBA star Michael Jordan is 78 in. tall,
while WNBA player Rebecca Lobo is 76 tall. Men’s
heights have a mean of 69 in, and a standard
deviation of 2.8 in. Women’s heights have a mean of
63.6 in, and a standard deviation of 2.5 in. Which
player is relatively taller? In other words, which
player is taller in comparison to their gender?
Former NBA player Mugsy Bogues is 63 in. tall.
Whose height is more “unusual”, Mugsy’s or
Michael’s?
SUMMARY/QUESTIONS TO ASK IN CLASS
Name _______________________________
AP STATISTICS CHAPTER 2:
Find the proportion of observations from the standard
Normal distribution that are between -1.25 and 0.81.
How to solve problems involving normal
distributions:
 State:
 Plan:
 Do :
 Conclude :
When Tiger Woods hits his driver, the distance the
ball travels can be described by N(304, 8). What
percent of Tiger’s drives travel between 305 and 325
yards?
SUMMARY/QUESTIONS TO ASK IN CLASS
Name _______________________________
AP STATISTICS CHAPTER 2:
NORMAL PROBABILITY PLOTS
A normal probability plot is a special graph which
allows us to assess the normality of a given data set.
Steps for constructing a normal probability plot:
Weight
56
65
Percentile
Z-score
78
79
83
85
88
90
91
105
1.
2.
For each data point, compute the
corresponding percentile.
Use these percentiles to find the
corresponding z-score for each data point.
Constructing the plot:
x-axis:
y-axis:
What to look for:
SUMMARY/QUESTIONS TO ASK IN CLASS
Name _______________________________
AP STATISTICS CHAPTER 2:
NORMAL DISTRIBUTION CALCULATIONS
Consider babies born in the “normal” range of 37 to
43 weeks of gestation. Extensive data supports the
assumption that for such babies born in the U.S.,
birth weight is normally distributed with a mean of
3432 g, and a standard deviation of 482 g.
Notation:
we call the distribution of scores
say that
has the
distribution.
, and
In the following problems, and problems of this type:
1)
2)
3)
Ex 1: What is the probability that the birth weight of
a randomly chosen baby of this type is less than 4000
g?
Ex 2: What is the probability that the birth weight of
a randomly chosen baby of this type is more than
3000 g?
Ex 3: What is the probability that the birth weight
will be between 3000 and 4000 g?
Ex 4: What is the probability that the birth weight
will be between 2500 and 3500 g?
Ex 5: What is the 30% percentile of babies’ weights?
What is the interquartile range?
SUMMARY/QUESTIONS TO ASK IN CLASS