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Goal of Hypothesis Test Terms & Notation Chi-Square Test Goodness-of-Fit Testing Example
Goal of Hypothesis Test
To examine statistical evidence, and to determine whether it supports or contradicts a claim The life of lamps is more than 10,000 hours The data are from normal distribution
To reduce the directly-relevant data to a “level of suspicion” based purely on the data
Terms & Notation
Null Hypothesis (H0) vs. Alternative hypothesis (H1 or HA) Type I Error vs. Type II Error
Parametric Test vs. Non-Parametric Test Significance level (α) and Critical Region
“Reject H0” vs. “Do not reject H0“
Central Limit Theorem Sampling distribution of the sample mean
Test Statistic vs. Table Value P-value
Null Hypothesis vs. Alternative hypothesis
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Type I Error vs. Type II Error
Type I error H0 is true but reject H0
Pr(reject H0 | H0) = α
Type II error H1 is true but do not reject H0
Pr(do not reject H0 | H1) = β
Parametric Test vs. Non-Parametric Test Parametric Test
Parameters of population Mean test, variance test, etc.
Non-Parametric Test Make no assumptions about the frequency
distributions of the variables being assessed Independent test, distribution test, etc.
Central Limit Theorem
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Test Statistic vs. Table Value
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Chi-Square Test
Non-Parametric Test T. S. ~χ2(ν)
Goodness-of-Fit Test Also known as “Pearson's chi-square test”
Independent Test Homogeneity Test
Goodness-of-Fit Testing
Used to test if a sample of data came from a population with a specific distribution
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Observations and Expected FrequenciesInterval Obs t F(t) = p(T < t) C.F. Frequency
0 ~ < 1 14 1 0.218078 12.86659 12.86659
1 ~ < 2.5 12 2.5 0.459359 27.10219 14.2356
2.5 ~ < 5 18 5 0.707707 41.75474 14.65255
5 ~ < 7.5 5 7.5 0.841975 49.67651 7.921768
7.5 ~ < 10 5 10 0.914565 53.95934 4.282832
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Test Statistic and P-value
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Observations and Expected Frequencies - Paper
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Re-Grouping
ID lower upper Freq.
1 0.3 3.3 30
2 3.3 6.3 17
3 6.3 9.3 6
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6 15.3 18.3 1
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Obs t F(x) C. F. E. F.
30 3.3 0.556 32.812 32.813
17 6.3 0.788 46.486 13.673
6 9.3 0.899 53.020 6.534
6 ≧9.3 1 59 5.980
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# of groups = 1+3.322*log(n)
