Transcript 7.1 Normal Curves

7.1 Normal Curves
•
Normal curve
1. Curve is bell shaped with highest point over mean
2. Curve is symmetrical about a vertical line through 
3. Curve approaches horizontal axis but never touches or
crosses it.
4. Inflection point (were concave up part of curve meets
concave down part) is one  (standard dev.) from the mean
( is measure of spread)
5. Total area under the curve = 1
6. Portion of area under curve between an interval represents
the probability that that measurement lies in the at interval
7. Normal curve can be represented by N(, )
Draw Normal curve and examples of curves that are not normal
7.1 Normal Curves
• Empirical Rule – For Normal Distributions
– Approximately 68% of the data values lie within one
 (standard dev.) on each side of mean
– Approximately 95% of the data values lie within
two  on each side of the mean
– Approximately 99.7% of the data values lie within
three  on each side of the mean.
7.1 Normal Curves
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Ex: If males have a mean height of 69” (=
69”) with a  = 3”
What is the probability that a randomly
selected male is..
a.
b.
c.
d.
e.
f.
g.
h.
Between 66” and 72”
Between 66” and 69”
Greater than 69”
Greater than 72”
Between 69” and 75”
Between 69” and 78”
Between 66” and 75”
Between 66” and 78”
7.1 Normal Curves
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Why are normal distributions important?
1. Good description (approximation) for some
distributions of real data. (SAT scores; heights of
people etc.)
2. Good approximations to results of many kinds of
chance observations (coin tossing)
3. Many statistical inference procedures (inference is
predicting something about a population by taking
a sample) are based on normal distributions
* Many sets of real data do not follow distribution (e.g.,
income)
7.2 Standard Units and Areas Under the
Standard Normal Distributions
Z - SCORE
z – score z = x-

x – observation of interest
 - mean of data for population
 - std. dev. of data for population
z – number of std. dev. the
observation of interest is away
from the mean. + z is above
mean, -z is below mean
N(69”,3”) find z –scores for:
a) HT = 74” b) HT = 65” c) HT= 71.2”
Find HT corresponding to (use x = z  + )
a) z = 1.2
b)z = -0.8
7.2 Standard Normal distribution
• z – score: z = x-

• To find the raw score x = z  + 
(point of interest)
• Standard Normal Distribution – “normalizes data by using zscores.”
=0
 = 1 uses the z-score from any normal
distribution
(draw Std. Normal Distribution)
Areas under Normal Curve
Can use empirical rule
from –1< z < 1 = 68%
from –2 < z < 2 = 95%
from –3 < z < 3 = 99.7%
7.2 Standard Normal distribution
• Transformation from
From  = 10  = 2
Normal Curve
Standard Normal Curve
Since any normal distribution can be transformed into a STANDARD
NORMAL DISTRIBUTION, we can utilize
the STANDARD NORMAL DISTRIBUTION TABLE for calculating
probabilities
7.2 Standard Normal distribution
• Standard Normal Distribution Table
– Always reports area to left of z-score
– To get probabilities below a certain z value, read
directly off the table
– To get probabilities above a certain value, take
value of table and subtract from 1
– Example of z = -1.0 P(z < -1.0) =
example of z = 2.18 P(z < 2.19) =
example of z = 0.91 P(z >.91) = 1-
7.2 Standard Normal distribution
To solve
1. Always sketch Normal curve!!!!
2. Find region of Interest and shade that region
3. Use probability table to find
a) area if z-score is known
b) z-score if area is known
4. Determine final answer for probability
a. Area on left of z-score answer is area directly from table
b. Area on right of z-score answer is ( 1 – area) in table
c. Area in the middle answer is area(z2) – area(z1)
For Raw Score use z-score from table and calc x
x=z+
Examples
7.3 More practice
• Actual normal curve parameters
–  = mean
–  = standard deviation
– x = raw data point of interest
• Standardized value
– z number of standard deviations from mean
• Probabilities of normal curves in
– z chart
• Since normal curves always contain the same probabilities
(Emperical Rule), z values help us to easily convert to the
nearest probability.
• Conversion is x value to z to probability
• Or probability to z value to x value
7.3
• Practice converting z values in ranges to
probabilities
• Working with word problems
– Always draw normal curve and shade area of interest
– Find prob from x values (x to z to %)
– Find x values from probability ( % to z to x)
Conversion problems
– Less than
– Greater than
– Range
• Warranty problem
• Guaranteed time (pizza, oil change)
7.4 Sampling Distributions
• Using samples to make estimates
• Population – a set of measurements
• Sample – a subset of measurements of
population. (random is best type)
• Parameter – numerical descriptive measure
of a population
• Statistic – numerical descriptive measure of
a sample
7.4 Sampling Distributions
• Statistic description of sample
• Parameter description of a population
Measure Statistic Parameter
Mean
Variance
Std. Dev
Probability
x(x bar)
s2
s
p (p hat)
 (mu)
2
 (sigma)
p
7.4 Sampling Distributions
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Statistics (sample values) are used to make
inferences about parameters (population values)
Inferences
1. Estimation – estimate value of a population parameter
2. Testing - formulate decision about the value of the
population parameter
3. Regresssion – make predictions or forecasts about
the value of the statistical variable.
7.4 Sampling Distributions
• Sampling distribution - probability
distribution of a sample statistic based on all
possible simple random samples of the
same size from the same population.
http://onlinestatbook.com/stat_sim/sampling_dist/index.html
7.4 Sampling Distribution
Pop heights
76
78
79
81
86
Sample n=3
76,78,79
76,78,81
76,78,86
76,79,81
76,79,86
76,81,86
78,79,81
78,79,86
78,81,86
79,81,86
x
77.7
78.3
80
78.7
80.3
81
79.3
81
81.7
82
x only
76
80
82
84
86
P(79  x  81) = 0.50
n=3
76
78
80
82
84
86
P(79  x  81) = 0.80
n =4
76
 = x/n =80
78
78
80
82
84
86
Frequency
Cholesterol levels of 38-41 year old men
14
12
10
8
6
4
2
0
0
100
200
Cholesterol Level
300
400
7.5 Central Limit Theorem
• x distribution, given x distribution is Normal
• Theorem – for a normal Probability distribution
– The x distribution is a normal distribution
– The mean of x distribution is 
– The standard deviation of the x distribution is / n
x = 
x= / n
z = x - x = x - 
x
/ n
ex  = 10.2,  = 1.4, n = 5
Standard error is the standard deviation of a sampling
distribution
7.5 Central Limit Theorem
• Central Limit Theorem – Given any
distribution of the random variable x, as the
sample size increases, the x bar distribution
approaches a normal distribution ( n  30)
• (draw example)