Transcript Slide 1

SP 225
Lecture 9
The Normal Curve
The Normal Curve
 ‘Bell’ Shaped
 Unimodal in center
 Tails extend to infinity in either direction
The Ideal Normal Curve
Naturally Normal
 IQ tests
 Standardized tests
 Ergometric Measures
 Manufacturing Processes
Normal Curve as an
Approximation
Approximation
 No data set is perfectly normal
 Many are very close
 Allows us to ‘sum up’ distributions with
two pieces of information: mean and
standard deviation
 Allows us to draw conclusions based on
calculations vs. data inspection
Empirical Rule for Data with a
Bell-Shaped Distribution
Empirical Rule for Data with a
Bell-Shaped Distribution (2)
Empirical Rule for Data with a
Bell-Shaped Distribution (3)
Number of Standard
Deviations
 Use the z-score
 Z-score represents the number of
standard deviations from the mean of a
single data point
 Use it no matter what the mean or
standard deviation is.
x
x
z= s
Properties of Z-score
 + if above the mean
 - if below the mean
 Units are number of standard deviations
Example
 Einstein had an IQ of 160 points.
Amongst the population, IQ has a mean
value of 110 with a standard deviation of
16.
 What is the difference between Einstein
and the average person?
 What is the difference in number of
standard deviations?
 What is the z-score for Einstein’s IQ?
Areas under the Normal Curve
 Areas correspond to proportions of data.
 Areas are looked up in the z-score table
Special Properties of the
Normal Distribution
 Total area under the curve is 1
 Partial areas represent proportions
 Symmetric
Areas Example
 Suppose your weight fluctuates each
week with a mean of 0 and a standard
deviation of 1 and is normally distributed.
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What is the probability you lose 1.5 lbs or
more?
What is the probability you gain less than
1.5 lbs?
What is the probability you gain 1.5 lbs or
more?
What is the probability your weight
fluctuates by less than 1.5 lbs?
Areas Example (2)
 Find a value, c, where:
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You gain c or less pounds 75% or the time
You lose c or more lbs 20% of the time
Your weight fluctuates by c or less with
80% probability
Z-score Example
 GPAs of SUNY Oswego freshman biology
majors have approximately the normal
distribution with mean 2.87 and standard
deviation .34.
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In what range do the middle 90% of all freshman
biology majors’ GPAs lie?
Students are thrown out of school if their GPA falls
below 2.00. What proportion of all freshman biology
majors are thrown out?
What proportion of freshman biology majors have
GPA above 3.50?
http://www.oswego.edu/~srp/stats/normal_wk_4.htm
Z-score Example (2)
 The length of elephant pregnancies from
conception to birth varies according to a
distribution that is approximately normal with
mean 525 days and standard deviation 32
days.
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What percent of pregnancies last more than 600
days (that’s about 20 months)?
What percent of pregnancies last between 510 and
540 days (that’s between 17 and 18 months)?
How short do the shortest 10% of all pregnancies
last?
http://www.oswego.edu/~srp/stats/normal_wk_4.htm
Probabilities
 Areas under the curve represent
probabilities as well as proportions
 Proportions make statements about
existing data
 Probabilities make statements about
future actions