Transcript Chapter 2: The Normal Distributions

CHAPTER 2: THE NORMAL
DISTRIBUTIONS
DENSITY CURVES
A
density curve describes the overall pattern of
a distribution.

Has an area of exactly 1 underneath it.
(Theoretical)
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DENSITY CURVES
The median (M) of a density curve is the point
that divides the area under the curve in half.
 The mean (x̅) of a density curve is the “balance
point”. Point that the curve would balance at if
made of solid material.
Examples:

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MATHEMATICAL MODEL
A
density curve is an idealized description
of the distribution of data.


Values calculated from a density curve are
theoretical and use different symbols.
Mean


Greek letter mu

(x̅ for data)
Standard deviation

Greek letter sigma  (s for data)
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Uniform Density Curves
Find the proportion of observations within the given interval
P(X < 2)
 P(X < 2)
 P(X > 6)
 P(2 < X < 7)
 P(x = 2)

= 2 (1/8) = 2/8
= 2/8
= 2/8
= 5/8
≈0
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What would be the median? M = 4
Find the proportion of observations
within the given interval
1.0
.75
.5
.25
0
0
.25
.5
P(0 < X < 2)
 P(.25 < X < .5)
 P(.25 < X < .75)
 P(1.25 < X < 1.75)
 P(.5 < X < 1.5)
 P(1.75 < X < 2)

.75
1.0
1.25
= 1.0
= .125
= .25
= .25
= .46875
= .15625
1.5
1.75
2.0
What would be the
median?
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DENSITY CURVE MODELED BY AN EQUATION
A


density curve fits the model y = .25x
Graph the line.
Use the area under this density curve to
find the proportion of observations within
the given interval
P(1 < X < 2)
 P(.5 < X < 2.5)




If the curve starts at x = 0, what value of x
does it end at?
What value of x is the median?
What value of x is the 62.5th percentile?
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ANSWERS
 Graph
1.0
.5
1.0
P(1 < X < 2) = .375
1
A  bh
 P(.5 < X < 2.5) = .75
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 Max Value of X
1  .5( x)(.25 x)
 Median
2
1

.125x
 62.5th percentile
8  x2
2.0

x  8  2.83
3.0
2.24
2.83
.5  .5( x)(.25 x) .625  .5( x)(.25 x)
.5  .125x 2
.625  .125x 2
4  x2
5  x2
x 4 2
x  5  2.24
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DENSITY CURVE
 Homework
p128
9 -13 (not 13a)
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MEASURING RELATIVE STANDING
PERCENTILES:
 Percentile
Percent of observations less than or equal
to a particular observation.
Example: scores 92, 91, 85, 77, 79, 88, 99,
69, 73, 84
A score of _____ is the ______ percentile?
40th
a) 79
70th
b) 88
100th
c) 99

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NORMAL DISTRIBUTIONS:
 Normal



curves
Curves that are symmetric, single-peaked,
and bell-shaped. They are used to describe
normal distributions.
The mean is at the center of the curve.
The standard deviation controls the spread
of the curve.
The bigger the St Dev, the wider the curve.
 There are roughly 6 widths of standard
deviation in a normal curve, 3 on one side of
center and 3 on the other side.

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NORMAL DISTRIBUTIONS:
 Normal
curves all have the same overall
shape described by mean μand standard
deviation σ .
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NORMAL CURVE
3  2  1 

1  2  3 
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THE 68-95-99.7 RULE OR
EMPIRICAL RULE:
 68%
of the observations fall within 1σof theμ.
(approx.!!! Really .6827)
 95%
of the observations fall within 2σof theμ
 99.7%
of the observations fall within 3σof theμ
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Example: The distribution of the heights of women is normal
with mean of 64.5 and a standard deviation of 2.5.
Draw a normal curve.
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59.5 62
64.5 67
69.5 72
What percent of women are in the following ranges?
1) P(x < 64.5)
2) P(x < 69.5)
3) P(x > 62)
4) P(x > 57)
5) P(57 < x < 67)
6) P(59.5 < x < 67)
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and one obtains
Notations:
N(μ,σ); example of the women N(64.5, 2.5)
FYI Area under
a normal
curve
This integral
is 1 if and
only if a = 1/(c√(2π)), and in t
of a normally distributed random variable with expecte
dx
These Gaussians are plotted in the accompanying figur
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Gaussian functions centered at zero minimize the Four
Homework
p137 23 - 26
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SECTION 2.2: STANDARD NORMAL
CALCULATIONS
 All



normal distributions
Share many common properties,
Are the same if measured in the same units.
Use the notation N(mean, standard deviation).
 The

Has a mean of 0 and standard deviation of 1.


standard normal distribution
N(0,1)
Taking any normal distribution and converting it
to have a mean of 0 and StDev of 1 is called
standardizing.
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STANDARDIZING AND Z-SCORES.
A
standardized value is called a z-score.
 A z-score tells us how many standard
deviations the original observation falls away
from the mean, and in which direction.

Observations larger than the mean have
positive z-scores, while observations smaller
than the mean have negative z-scores.
 To
standardize a value, subtract the mean of
the distribution and then divide by the
standard deviation.
x
z

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Example: Stat class test scores are: 92, 91, 85, 77, 79, 88, 99, 69, 73, 84
If you scored 91, how did you do relative to the class?
a) 91:
b) 88:
c) 73:
Example: A student took a math test and got an 80. He took a Latin test and got
a 90. If the math scores had a mean of 70 with a standard deviation of 8 and
Latin had a mean of 95 with a standard deviation of 3, in which class did he do
relatively better?
z
x

COMPLETE RESPONSE TO A NORMAL
DISTRIBUTION QUESTION
1. Check
normality – you can only use z-scores if
approximately normal!!!!
2. State in terms of x and draw a picture with
shading the area. Label with x, μ,σ
3. Standardize x to a z-score. On the picture label
the Z-score.
4. use table A or calculator: (2nd vars) normalcdf
(lowerbound, upperbound, μ,σ)
5. Write your conclusion in the context of the
problem. Remember you have approximate
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probability
LET’S GO THOUGH AN EXAMPLE OF HOW
TO USE THE TABLE.
 What
proportion of all young women are greater
than 68 inches tall, given that the distribution of
heights for all young women follow N(64.5, 2.5)?
• Step one – Find P(x > 68) on N(64.5, 2.5)
  2.5
64.5
68
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• Step two – standardize x and label picture with zscore
  2.5
z
x

68  64.5
z
2.5
Z-scores
64.5
68
0
1.4
z  1.4
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• Step three – find the probability
by using Table A, and the fact
that the total area is equal to 1.
z  1.40
• This value is for area to
left of z-score, we need
area to right of z-score in
this problem.
• P(x > 1.4) = 1 - .9192
• P(x > 1.4) = .0808
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• Step four – write the conclusion in context of the
problem.
  2.5
8.08%
64.5
68
• The proportion of young women that are taller than 68
inches is 8.08%
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Using Table A, convert the following z-scores to probability. Draw a picture!
1) Find P(z < 2.3)
1) Find P(z < -1.52)
1) Find P(z > -0.43)
1) Find P(z > 3.1)
1) Find P(-1.52 < z < 2.3)
1) Find P(-3 < z < 3)
Example: For 14 year old boys, cholesterol levels are ≈ N(170, 30)
a) What percent of boys have a level of 240 or more?
a) What percent of boys are between 170 and 240?
Homework
p. 118 1 – 4
p. 121 6 – 8 (not 7d)
FINDING A DATA VALUE FROM A ZSCORE
 To calculate a value in which x% fall
above or below the point


Use table A to find the z-score.
Substitute z, μ and σ into the equation and
solve for x.
z
x

x = μ+ zσ
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Example: SAT-V scores are ≈ N(505, 110)
1) How high must a student score to be the 30th percentile?
1) How high must a student score to get in the top 10%?
1) What scores contain the middle 50% of scores?
SECTION 2.2
 Homework:
p142
29 – 30
p 147 31 – 36 (not 32b)
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WEIGHTS OF MOUNDS CANDY BARS
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Computer output of a normal probability plot shows lines as boundaries –
if the data falls within the lines, it is approximately normal.
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In this example, the histogram and the normal
probability plot both show that this data is not
approximately normal.
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Homework: p154 37 – 39 (not 39a)
p162 51 – 61 (not 61c)
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