One

1
Unit One
Chapters 1, 2, 5
2
Review Chapters 1,2,5

• Data
• Population
• Normal Distribution
• Probability
• Standard Normal Distribution
• Z Scores

3
Data
is

• Information

4
So that it is easier to understand

• Data is often organized into frequency
  distributions, histograms, and pie charts

5
Population

• Everyone in the group in which you are interested
• Examples
• Girl Scouts in the US
• students at MDC
• teachers at MDC
• Latinos in Miami
• plumbers in Miami

6
Parameters of the Population

• Mean
• The center of the scores
• Symbol for mean µ
• Formula for mean

• Standard Deviation
• The spread of scores around the mean Symbol for
  SD s

7
Many Continuous Traits Such as Height, Weight,
Age, etc. are Normally Distributed
8
Although all normal curves have the same basic
bell shape, there are an infinite number of
normal curves.

• The mean (µ) and standard deviation (s) of the
  data determine the shape as per this formula

9
Properties of Bell Shaped Curves

• Its called a Normal Curve
• Its also called a Density Curve or a probability
  density function
• Its the graph of a continuous probability
  distribution
• The total area under the curve must equal 1
• Every point on the curve must have a vertical
  height that is 0 or greater

10
If we can identify where any particular score
lies, we can determine how many scores are above
and how many scores are below the one we have
chosen.
X
11
We can do this because we know that the entire
area under the curve 100 of the area, so any
areas that we select are a portion or percentage
of the whole area under the curve.
1s
-2s
-1s
-3s
2s
3s
µ
12
We can remember some percentages associated with
whole standard deviations from the mean
µ
1s
-2s
-1s
-3s
2s
3s
13
But how are we going to calculate the area under
the curve for ANY value?

• Fortunately, these values have been calculated
  for us and are listed in a table in our book
  (table A2)

14
Table A-2 Standard Normal (z) Distribution
(Negative)
1
-2.575 (.005)
-1.645 (.05)
15
Table A-2 Standard Normal (z) Distribution
(Positive)
1
-2.575 (.005)
-1.645 (.05)
2.575 (.005)
1.645 (.05)
16
The values across the top and in the far left
column are the scores (which are also the
standard deviations)
Table A-2 Standard Normal (z) Distribution
(Positive)
1
-2.575 (.005)
-1.645 (.05)
2.575 (.005)
1.645 (.05)
17
The values in the center of the table are the
areas and also the percentages or probabilities
Table A-2 Standard Normal (z) Distribution
(Positive)
1
-2.575 (.005)
-1.645 (.05)
2.575 (.005)
1.645 (.05)
18
The values listed in the table are for a normal
curve with mean 0 and standard deviation 1

• A distribution with a mean 0 and SD1 is called
  a Standard Normal Distribution
• The scores from this distribution are called Z
  Scores

19
If we want to find the areas/probabilities
associated with normal distributions where the
mean is not 0 or the SD is not 1

• We first have to convert the score to a Z Score

20
How is a score transformed into a standard score?

• The mean (µ) is subtracted from each score (x- µ)
• The above difference is divided by the standard
  deviation (s) of the population
• The transformed score is called a Z score

X µ s
Z
21
A Standard Normal Distribution is

• Bell shaped and symmetrical around the mean
• The mean (µ) 0
• The standard deviation (s) 1
• Scores above the mean are positive
• Scores below the mean are negative

(s) 1
1
-1
-2
2
-3
3
(µ) 0
22
Advantages to converting the data to standard (Z)
scores

• We know that the mean (µ) of this group of scores
  is always 0 and the standard deviation (s) is
  always 1 no matter what the mean and SD of the
  original scores was
• We can use tables which are found in our book or
  on the internet to calculate the areas under the
  curve

23
Good web explaination of area normal curve
(gary katz)
http//www.csun.edu/gk45683/Lecture20720-20Are
a20and20the20Normal20Curve.pdf
24
Steps for Finding Probability When You Have a Raw
Score

• Draw a picture of the Data Distribution and
  include the Raw Score
• Convert raw data to Z scores
• Use the table for Z scores to find the area
  associated with the Z score the area is the
  probability

25
Steps for Finding a Score When You Know the
Probability

• Look up the area (which is the same as
  probability) in the table for Z scores
• Convert the area to a Z score
• Convert the Z score to a raw score

26
If we take multiple samples of size n from the
population

• The mean of the samples will be the same as the
  mean of the population µ
• The SD of the samples
• ? the population SD
• n the sample size

27
If we want to find the probability of a sample we
follow the same procedure that we do for a single
measurement

• Draw a picture
• Convert the sample mean to a Z score by
  subtracting its mean and dividing by its SD
• Find the area (probability) corresponding to the
  Z Score in the table

28
Click on the link below to see a video
demonstration of how to convert raw scores to Z
Scores and calculate probability

• http//realmedia.pearsoncmg.com/aw/math/triola/sec
  tion_5.3.ram