1. STATISTICS HYPOTHESES TEST (II) One-sample tests on the mean and variance
Professor Ke-Sheng Cheng
Department of Bioenvironmental Systems Engineering
National Taiwan University

2. Tests on Mean
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Department of Bioenvironmental Systems Engineering National Taiwan University

3. (1) 2 is known
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5. (2) 2 is unknown
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Department of Bioenvironmental Systems Engineering National Taiwan University

7. (3) 2 is known
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Department of Bioenvironmental Systems Engineering National Taiwan University

8. (4) 2 is unknown
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Department of Bioenvironmental Systems Engineering National Taiwan University

9. Tests on Variance
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10. (1)  is known
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13. (2)  is unknown
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Department of Bioenvironmental Systems Engineering National Taiwan University

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15. (3)  is known
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17. (4)  is unknown
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Department of Bioenvironmental Systems Engineering National Taiwan University

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22. The p-value
The p-value is the probability that chance alone would produce a test statistic as extreme as the observed test statistic (t).
If large values of the test statistic support the alternative hypothesis (i.e., reject Ho, if T>c), the p-value is P(T>t).
6/11/2018
Department of Bioenvironmental Systems Engineering National Taiwan University

23. Test in means using t-test in R
For a two-tailed test, the p-value is twice the exceedance probability shown in the above figure.
Can also be used to find the confidence interval.
6/11/2018
Department of Bioenvironmental Systems Engineering National Taiwan University

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Department of Bioenvironmental Systems Engineering National Taiwan University

26. Test in population proportion
Prop.test in R
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27. Example
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29. The above example demonstrates that the exact calculation and the CLT approximation yield different conclusions. In the case of significance testing, when the CLT yields a borderline answer, it is best to use exact calculation.
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Department of Bioenvironmental Systems Engineering National Taiwan University

30. Paired t-test
The paired t test provides a hypothesis test of the difference between population means for a pair of random samples whose differences are approximately normally distributed.
6/11/2018
Department of Bioenvironmental Systems Engineering National Taiwan University

31. If the population from which paired differences to be analyzed by a paired t test were sampled violate one or more of the paired t test assumptions, the results of the analysis may be incorrect or misleading. For example, if the assumption of independence for the paired differences is violated, then the paired t test is simply not appropriate.
6/11/2018
Department of Bioenvironmental Systems Engineering National Taiwan University

32. Note that the two values that make up each paired difference need not be independent, and in fact are expected to be correlated, such as before and after measurements. If you treat paired data as coming from two independent samples, such as doing an inappropriate two-sample unpaired t test instead of a paired t test, then you may sacrifice power.
6/11/2018
Department of Bioenvironmental Systems Engineering National Taiwan University

33. If the assumption of normality is violated, or outliers are present, then the paired t test may not be the most powerful test available, and this could mean the difference between detecting a true difference or not. A nonparametric test or employing a transformation may result in a more powerful test. For example, if the distribution of the paired differences is not symmetric, a transformation may produce symmetry.
6/11/2018
Department of Bioenvironmental Systems Engineering National Taiwan University

34. The test statistic is calculated as:
Two-sided test
One-sided test
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Department of Bioenvironmental Systems Engineering National Taiwan University

35. Statistical Significance vs Practical Importance
A small p-value which indicates statistical significance of the effect does not necessarily imply that the effect is of practical importance.
6/11/2018
Department of Bioenvironmental Systems Engineering National Taiwan University

36. The results are statistically significant (the drug does work), but it seems unlikely that the drug company will be able to successfully market a drug that promises severely obese patients that they can expect to lose, on average, at most 5 pounds.
6/11/2018
Department of Bioenvironmental Systems Engineering National Taiwan University