Hypothesis Testing

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Hypothesis Testing

• Introduction to Inductive Statistics

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Background Websites

• http//www.intuitor.com/statistics/CurveApplet.htm
  l
• http//www.intuitor.com/statistics/T1T2Errors.html
  

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Terms

• Descriptive Statistics what weve done so far
• Inductive Statistics making decisions on the
  basis of statistical evidence
• Hypothesis the relationship or proposition you
  wish to test, stated to affirm the relationship
  or proposition
• Null Hypothesis the negative form of the
  proposition to be tested

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Some propositions

• Statistics is making decisions on the basis of
  incomplete or imperfect information.
• A hypothesis or proposition can be refuted by one
  observation but not proved by many.
• We thus proceed by determining the likelihood
  that the null hypothesis can be rejected.

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The Dilemma

• There is always the possibility of making the
  wrong decision
• Rejecting a true hypothesis Type 1 Error
• Failing to reject a false hypothesis Type 2
  Error

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An Example of the Issue A Jurys Decision Making
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An Example of the Issue A Jurys Decision Making
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The Statistical Decision Making Framework
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The Statistical Decision Making Framework
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The Jury and the Researcher Compared
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Steps for Making a Decision

• Specify a hypothesis and the null hypothesis
• Specify a level of probability which you will use
  to decide whether to reject the null hypothesis.
• Specify the test statistic and the sampling
  distribution you will use to make a decision.
• Calculate the statistics and compare to the
  theoretical probability distribution, for
  example, the t distribution http//www.uwm.edu/r
  enlex/T.html
• Interpret the results.

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General Form of the Test for a Mean

• Z tests
• (Sample mean population mean) / SE of sample
  mean, or
• (Sample mean population mean)/ (s / vn)
• T tests
• (Sample mean population mean) / SE of sample
  mean, or
• (Sample mean population mean)/ (s / vn-1)

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General Form of a T Test

• t sample estimate null hypothesis/ SE
• Which simplifies to
• t sample estimate/SE
• When the null hypothesis is that the sample
  statistic is 0.

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T Distribution
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Example 1

• Hypothesis There is a difference in the average
  number of persons per household in the 18th and
  the 14th wards.
• Null Hypothesis There is no difference in the
  average number of persons per household in the
  18th and the 14th wards, or more specifically,
  any difference we measure is a matter of the
  particular sample we have.

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Example, cont.

• Level of probability 95 confidence level, so
  that only 1 in 20 times would the results be
  different.
• Test statistic Means and a T-Test of the
  difference of two groups.
• t (mean1 mean2)/ (SE of the difference of
  mean1-mean2)
• Calculate the statistics.

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Results
Two-sample t test on PERSONS grouped by WARD
Group N Mean SD
14 316 8.15 3.70
18 120 5.25 2.55
Separate Variance t 9.29 df
310.5 Prob 0.00 Difference in
Means 2.90 95.00 CI 2.29 to
3.52 Pooled Variance t
7.90 df 434 Prob 0.00
Difference in Means 2.90 95.00 CI
2.18 to 3.62
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Results, Graphically Displayed
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Interpret the Results

• Lets look at the t distribution again
  http//www.uwm.edu/renlex/T.html and p. 135 of
  text.
• We can reject the null hypothesis that the two
  means are the same in the underlying population
  (the unknown truth).
• We say that there is a statistically significant
  difference between the average number of persons
  in the two wards.

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Example 2

• Hypothesis There is a difference in the average
  number of persons per household in the 18th and
  the 20th wards.
• Null Hypothesis There is no difference in the
  average number of persons per household in the
  18th and the 20th wards, or more specifically,
  the difference is a matter of the particular
  sample we have.

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Results
TEST PERSONS WARD Data for the following
results were selected according to (WARDltgt
14) AND (wardltgt 22) Two-sample t test on
PERSONS grouped by WARD Group N
Mean SD 18 120
5.25 2.55 20 342
5.72 2.62 Separate
Variance t -1.73 df 213.5 Prob
0.08 Difference in Means
-0.47 95.00 CI -1.01 to 0.07
Pooled Variance t -1.71 df 460
Prob 0.09 Difference in Means
-0.47 95.00 CI -1.02 to
0.07
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Results, Graphically Displayed
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Interpret the Results

• We cannot reject the null hypothesis that the two
  means are the same in the underlying population
  (the unknown truth).
• We say that there is not a statistically
  significant difference between the average number
  of persons in the two wards.

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Additional T tests

• Tests whether regression coefficients and
  correlation coefficients are statistically
  significantly different from 0.
• Test of regression slope (b)
• t b / SE(b)
• Test of correlation coefficients
• t r / SE(r)

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Testing a Regression Coefficient The Impact of
Year Built on Size, 14th Ward, 1905
Dep Var SIZE N 101 Multiple R 0.23
Squared multiple R 0.05 Adjusted squared
multiple R 0.04 Standard error of estimate
588.27 Effect Coefficient Std Error
Std Coef Tolerance t P(2 Tail) CONSTANT
785.26 128.85 0.00 .
6.09 0.00 YEAR 24.72
10.67 0.23 1.00 2.32
0.02 Effect Coefficient Lower 95
Upper 95 CONSTANT 785.26 529.59
1040.92 YEAR 24.72 3.56
45.88
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An example of the issue A jurys decision making