Using Statistics To Make Inferences 10

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Using Statistics To Make Inferences 10

• Summary 
• To fit a straight line through data.
•  
• Goals 
• Given raw data, or the appropriate sums, to fit
  a straight line through the data.
• Practical
•  
• Recall last weeks practical. Perform scatter
  plots, evaluate correlations and add regression
  lines.
•  

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Regression
We wish to fit the straight line y  ax  b
through our data xi,yi i1,,n by estimating a
and b.
Recall these terms from lecture 9 on correlation.
Note the symbol denoting an estimate and the
for average.
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Example

• Is the height of sons related to that of their
  fathers?

Father (x) 63 68 70 64 66 72 67 71 68 62 Son (y)
65 66 72 66 69 74 69 73 65 66
Note choice of x and y variables.
First plot the data.
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Scatterplot
Father (x) 63 68 70 64 66 72 67 71 68 62 Son
(y) 65 66 72 66 69 74 69 73 65 66
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Calculation
Father (x) 63 68 70 64 66 72 67 71 68 62 Son
(y) 65 66 72 66 69 74 69 73 65 66
n 10
Sxi 63 68 62 671
Syi 65 66 66 685
Sxi2 632 682 622 45127
Sxiyi 63 65 68 66 62 66 46047
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Calculation Sxx
n 10 Sxi 671 Syi 685 Sxi2 45127 Sxiyi
46047
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Calculation Sxy
n 10 Sxi 671 Syi 685 Sxi2 45127 Sxiyi
46047
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Calculation Gradient a
n 10 Sxi 671 Syi 685 Sxx 102.9 Sxy
83.5
Gradient or slope
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Calculation Intercept b
n 10 Sxi 671 Syi 685
Sxx 102.9 Sxy 83.5
Gradient or slope
Intercept or constant
Son 14.05 0.81 Father
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Results

• Son 14.05 0.8115 Father
•  

To produce the line select two extreme values for
the x variable.
Father 72 then Son 14.05 0.8115 72
72.478
Father 62 then Son 14.05 0.8115 62
64.363
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Results
Father 72 Son 72.48 and Father 62 Son
64.36
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SPSS
Analyze gt Regression gt Linear
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SPSS
Same intercept and gradient
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Aside

• A test of extraversion was administered to two
  groups of subjects - 8 adolescent boys and 5
  adolescent girls. Assume both populations are
  normally distributed. Do the two group means
  differ significantly at a level of 0.05?
• Observed data for boys
• 119.81 127.59 126.61 139.07 152.12 140.11 155.77
  111.71
• Observed data for girls
• 111.55 108.60 118.92 122.41 114.82

What are the key words in the question?
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Aside
C C C C C C C C C C C c

• A test of extraversion was administered to two
  groups of subjects - 8 adolescent boys and 5
  adolescent girls. Assume both populations are
  normally distributed. Do the two group means
  differ significantly at a level of 0.05?
• Observed data for boys
• 119.81 127.59 126.61 139.07 152.12 140.11 155.77
  111.71
• Observed data for girls
• 111.55 108.60 118.92 122.41 114.82

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Aside

• What key words describe this data?

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Aside

• two sample
• comparison of means

Which tests might be appropriate?
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Aside
C C C C C C C c

• z or t
• Which is appropriate here?

Since s is not available we use a two sample t
test.
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Example

• On January 28, 1986, the space shuttle
  Challenger was launched at a temperature of 31F.
  The ensuing catastrophe was caused by a
  combustion gas leak through a joint in one of the
  booster rockets, which was sealed by a device
  called an O-ring. The data in the print version
  of the notes relate launch temperature to the
  number of O-rings under thermal distress for 24
  previous launches.

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Challenger's rollout from Orbiter Processing
Facility to the Vehicle Assembly Building
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The crew of the final, ill-fated flight of the
Challenger
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The Challenger breaks apart 73 seconds into its
final mission
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Debris recovered from Space Shuttle Challenger
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Example

• On January 28, 1986, the space shuttle
  Challenger was launched at a temperature of 31F.
  The ensuing catastrophe was caused by a
  combustion gas leak through a joint in one of the
  booster rockets, which was sealed by a device
  called an O-ring. The data in the print version
  of the notes relate launch temperature to the
  number of O-rings under thermal distress for 24
  previous launches.
• First plot the data.

Variables, number of rings that fail and
temperature Which is dependent?
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Scatterplot
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Calculation Sxx
Here the independent variable is Temp (x) and the
dependent variable is Ring (y)
n 24 STempi 1680 SRingi 10 STempi2
118800 STempi Ringi 627
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Calculation Sxy
Here the independent variable is Temp (x) and the
dependent variable is Ring (y)
n 24 STempi 1680 SRingi 10 STempi2
118800 STempi Ringi 627
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Calculation Gradient a
n 24 STempi 1680 SRingi 10
Sxx 1200 Sxy -73
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Calculation Intercept b
n 24 STempi 1680 SRingi 10
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Results
Prediction at temp 31 then ring 2.789, that
is at 31F, 2.789 rings might be expected to fail!
The temperature of interest is a gross
extrapolation!!
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SPSS
Same intercept and gradient
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SPSS
Scatter plots/Regression Graphs gt Legacy Dialogs
gt Scatter/Dot
Simple scatter
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SPSS
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SPSS
To fit a line 1. Open the output file 2.
Double click on the graph (the chart editor will
open) 3. Click on the reference line icon
4. Check attach label to line to add regression
equation 5. Click apply and close
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SPSS
Alternately Graphs gt Interactive gt Scatterplot
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SPSS
Use the Fit tab at the top of the window
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SPSS
As you will often find, interactive graph makes
the scatter plot, but without the regression line
requested. One way to fix this problem is to
include the type scale option for the variables.
Right click on the variable to set this.
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SPSS
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Future Lectures

• During the last two weeks of the course there
  will be no formal lectures.
• During the lecture session I will review
  examination questions as proposed by the audience.

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Future Tutorials

• During the remaining tutorial sessions we will
  not require the cluster rooms.
• I will be available in my office to deal with
  individual queries.

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Read

• Read Howitt and Cramer pages 75-82
• Read Russo (e-text) pages 202-213
• Read Davis and Smith pages 173-192