Describing the relationship between two data sets and using that information for prediction'

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Describing the relationship between two data sets
and using that information for prediction.
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Todays plan Review and describing two data sets

• Review z scores
• Characteristics of z scores
• Finding percentages with z scores
• Describing the relationship between two data
  sets correlation
• What is it?
• What does it mean?
• How is it calculated?
• Using the relationship between two variables for
  prediction regression
• What is it?
• What does it mean?
• How is it calculated?
• Reminder Exam will be handed out next week
  after class!

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Some practice data for review purposes
Your stem leaf diagram here
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Some practice data for review purposes
Own brand
-0
3,1
0
8,2,5,7,5,0,4,7,4,8,4,4,8
1
6,7,4,5,2,1,4,3,1,4,8,6,1,1,6
2
0,2
Median?
Denic
-0
3,1,1,1,6,2,2,1,1,1,5,3,2
0
3,4,6,6,1,3,5,3,4,2,2,2,1,3,3,2,0
1
0,4
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Median?
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Your X, SS, and s work here
X
SS
s
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Your X, SS, and s work here
Sx/N 313/32 9.8 45/32 1.4
X
Sx2- (Sx)2/N 4293 - (3132/32)
1231.5
SS
585 - (452/32)
521.7
s
SS/N-1
1231.5/31 6.3
521.7/31 4.1
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What is a z score?

• Simple data transformation.
• A score within a distribution minus the
  distributions mean and then that difference
  divided by the distributions standard deviation.
• Used to describe data in standard deviation units
  (rather than the original units of measurement)
• Can be used to compare across distributions or
  find percentile rank within distributions easily
  (assuming a normal distribution).

z
or
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Four neat things about z scores

• Mean of a set of z scores 0.0
• Standard deviation of a set of z scores 1.0
• Each z score tells that scores distance from the
  mean in standard deviation units.
• Does not change the shape of the original
  distribution (though the z distribution is a
  different distribution, as indicated by the
  different mean and standard deviation).

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How many standard deviations from the mean is a
heart rate difference of 5?
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How many standard deviations from the mean is a
heart rate difference of 5?
X - X s
z
(5 - 9.8) / 6.3 -0.76 (5 - 1.4) / 4.1 0.88
Why one positive and the other negative?
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Percentage below (or above) a given score?
Assuming a normal distribution of scores, what
percentage of scores are below a score of 5?
Notes Negative z scores are not on the
table Rely on your sketch to tell you if you
need to add 0.5 to the tabled result.
1. Sketch the problem by locating on your sketch
where the necessary z scores will be in relation
to the mean. 2. Shade in the area that you are
looking for. 3. Use Table A, Column A to find
the correct z score. 4. Use either column B or
column C to find the shaded area.
mz 0.0
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For own brand assuming a normal distribution of
scores, what percentage of scores are below a
score of 5?
mz 0.0
z -.76
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(No Transcript)
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Assuming a normal distribution of scores, what
percentage of scores are below a score of 5?
22.36
mz 0.0
z -.76
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Percentage below (or above) a given score?
Assuming a normal distribution of scores, what
percentage of scores are below a score of 5?
Notes Negative z scores are not on the
table Rely on your sketch to tell you if you
need to add 0.5 to the tabled result.
1. Sketch the problem by locating on your sketch
where the necessary z scores will be in relation
to the mean. 2. Shade in the area that you are
looking for. 3. Use Table A, Column A to find
the correct z score. 4. Use either column B or
column C to find the shaded area.
mz 0.0
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Assuming a normal distribution of scores, what
percentage of scores are below a score of 5?
mz 0.0
z 0.88
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(No Transcript)
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Assuming a normal distribution of scores, what
percentage of scores are below a score of 5?
31.06
mz 0.0
z 0.88
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Assuming a normal distribution of scores, what
percentage of scores are below a score of 5?
81.06
31.06
50.00
mz 0.0
z 0.88
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Describing the relationship between two data sets.

• In many cases, two variables (x and y) collected
  from the same source are related
• Weight (pounds) and height (inches) from the same
  person.
• Sunny days/month (days) and monthly rainfall
  (inches).
• Shoe size and annual income from the same person.
• How can we describe the degree to which these
  variables are related in a single number?
• Why helps us to understand natural
  relationships.
• Why may suggest if one variable causes another.
• Problem different scales of measurement.
• Scatterplots help us to see relationships between
  variables.
• Pearsons r uses the correlation coefficient (and
  z scores) to measure the degree of relatedness of
  two variables.

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Describing two data sets graphically the
scatterplot
(74,235)
(70,190)
(64,125)
(60,110)
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Describing two data sets graphically the
scatterplot
(10,12)
(25,3)
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Describing two data sets graphically the
scatterplot
(5,32)
(12,23)
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Pearsons r describes relatedness numerically

• Pearsons r is a correlation coefficient.
• A correlation coefficient expresses
  quantitatively the magnitude and direction of the
  relationship (between two variables).
• Pearsons r is a measure of the extent to which
  paired scores occupy the same or opposite
  positions within their own distributions.

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Characteristics of Pearsons r.

• Pearsons r must be between -1 and 1, inclusive
  (-1 lt r lt 1).
• When r 1 or -1, there is a perfect linear
  relationship between the two variables (usually
  designated x and y).
• r 1, perfect positive relationship (as x
  increases, y always increases by a constant
  ratio)
• r -1, perfect negative relationship (as x
  increases, y always decreases by a constant
  ratio).
• When r 0, there is no linear relationship
  between x and y.
• When r is not -1, 0, or 1, there is a linear
  relationship of some direction (positive or
  negative) and some magnitude.
• r S(zxzy)/N-1 (the corrected average of the z
  score cross-products)

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What is the relationship between x and y?
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What is the relationship between x and y?
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What is the relationship between x and y?
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What is the relationship between x and y?
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What is the relationship between x and y?
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What is the relationship between x and y?
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What is the relationship between x and y?
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What is the relationship between x and y?
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What is the relationship between x and y?
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Examples of various rs.
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What is the correlation between own and denic?
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What is the correlation between own (X) and denic
(Y)?
r S(zxzy)/N-1
or
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What is the correlation between own (X) and denic
(Y)?
The only new thing! Multiply each X by its paired
Y and sum them all up.
Recognize these?
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What is the correlation between own (X) and denic
(Y)?
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N 32 SX 313 SX2 4293 SSx 4293 -
(3132/32) 1231.5 SY 45 SY2 585 SSy
585-(452/32) 521.7 SXY 693
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What is the correlation between own (X) and denic
(Y)?
N 32 SX 313 SX2 4293 SSx 4293 -
(3132/32) 1231.5 SY 45 SY2 585 SSy
585-(452/32) 521.7 SXY 693
r .315
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There is a moderate positive correlation.
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Using correlation to predict scores regression

• If there is a non-zero correlation between x and
  y (r 0.0), it means that there is a linear
  relationship between the two variables, x and y.
• It also means that if you know an x score, you
  can use the correlation to help you predict what
  the y score paired with that x score would be.
• The closer the correlation is to a perfect one (r
  1 or r -1) the better the prediction will be.
• The predicted score (y or Y prime) will be based
  on a straight line
• (Y bX a) drawn through the scatterplot such
  that the distance of each point from the line is
  minimized (linear regression).
• When r 1 or r -1, all the points fall on the
  line
• When r is lt 1 or r gt -1, at least some of the
  points are not on the line.

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Fill in Eddies Y score.
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Fill in Eddies Y score.
Y 10
As X increases by 1, Y increases by 2. (rise/run
2/1 2 slope)
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Fill in Zacks Y score.
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Fill in Zacks Y score.
When X 5, Y 10
When X 0, Y 0
As X increases by 1, Y increases by 2. (rise/run
2/1 2 slope) When the line hits the Y axis
(x 0) Y 0 (Y intercept 0)
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Regression line Y bYXaY (bY slope, aY y
intercept)
Y 2X0
Note no errors (all points fall on the line)
As X increases by 1, Y increases by 2. (rise/run
2/1 2 slope bY) When the line hits the Y
axis (x 0) Y 0 (Y intercept 0 aY)
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For HR data, what Y is predicted for X 19?
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For HR data, what Y is predicted for X 19?
N 32 SX 313 SX2 4293 SSx 4293 -
(3132/32) 1231.5 SY 45 SY2 585 SSy
585-(452/32) 521.7 SXY 693
Y bYX aY
aY Y - bYX
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For HR data, what Y is predicted for X 19?
N 32 SX 313 SX2 4293 SSx 4293 -
(3132/32) 1231.5 SY 45 SY2 585 SSy
585-(452/32) 521.7 SXY 693
Y bYX aY
bY .2053
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For HR data, what Y is predicted for X 19?
N 32 SX 313 SX2 4293 SSx 4293 -
(3132/32) 1231.5 SY 45 SY2 585 SSy
585-(452/32) 521.7 SXY 693
Y .2053X aY
aY Y - bYX aY (45/32) - .2053(313/32) aY
1.406 - .20539.781 aY -0.602
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For HR data, what Y is predicted for X 19?
Y .2053X -0.602 Y .2053(19) -0.602 Y
3.3
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For HR data, what Y is predicted for X 19?
Y .2053X -0.602
(19,3.3)
(0,-0.6)
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Next week

• Explore error in prediction (Chapter 7)
• Review correlation and regression.
• Hand out exam 1 (Due Friday, 2/18 at 3 PM).

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For HR data, what Y is predicted for X 19?
Y .2053X -0.602
(19,3.3)
(0,-0.6)
Y 1.4