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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Your X, SS, and s work here
X
SS
s
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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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Percentage below (or above) a given score?
Assuming a normal distribution of scores, what
percentage of scores are below a score of 5 for
own brand?
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.
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.
mz 0.0
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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 for
denic?
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.
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.
mz 0.0
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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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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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For own brand 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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For denic 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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For denic 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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For denic assuming a normal distribution of
scores, what percentage of scores are below a
score of 5?
50.00
31.06
31.06
50.00
81.06
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 may be related
• Weight (pounds) and height (inches) from the same
  person.
• Sunny days/month (days) and monthly rainfall
  (inches).
• A persons shoe size and their annual income.
• 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 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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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 YbYXaY (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 in class on Monday, 2/26 at
  200 PM).