1 Univariate Inferences about a Mean Shyh-Kang Jeng Department of Electrical Engineering/ Graduate Institute of Communication/ Graduate Institute of Networking and Multimedia

Scenarios To test if the following statements are plausible – –A clam by a cram school that their course can increase the IQ of your children – –A diuretic is effective – –An MP3 compressor is with higher quality – –A claim by a lady that she can distinguish whether the milk is added before making milk tea 2

3 Evaluating Normality of Univariate Marginal Distributions 3

Tests of Hypotheses Developed by Fisher, Pearson, Neyman, etc. Two-sided One-sided 4

Assumption under Null Hypothesis 5

Rejection or Acceptance of Null Hypothesis 6

Student’s t-Statistics 7

8 Student’s t-distribution

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10 Student’s t-distribution

Origin of the Name “Student” Pseudonym of William Gossett at Guinness Brewery in Dublin around the turn of the 20th Century Gossett use pseudonym because all Guinness Brewery employees were forbidden to publish Too bad Guinness doesn’t run universities 11

12 Test of Hypothesis

Selection of  Often chosen as 0.05, 0.01, or 0.1 Actually, Fisher said in 1956: – –No scientific worker has a fixed level of significance at which year to year, and in all circumstances, he rejects hypotheses; he rather gives his mind to each particular case in the light of his evidence and hid ideas 13

14 Evaluating Normality of Univariate Marginal Distributions 14

Confidence Interval for  0 15

Neyman’s Interpretation 00 16

Statistical Significance vs. Practical Significance The cram school claims that its course will increase the IQ of your child statistically significant at the 0.05 level Assume that 100 students took the courses were tested, and the population standard deviation is 15 The actual IQ improvement to be statistically significant at 0.05 level is simply 17

More Specific Hypotheses Null hypothesis Alternative hypothesis 18

Type I and Type II Errors 19

Type I and Type II Errors Truth H 0 (non- effective) H 1 (effective) Study Results H 0 (non- effective) 1-  (type II error) H 1 (effective)  (type I error, false alarm) 1-  (power) 20

Power The probability of concluding that the sample came from the H 1 distribution (i.e., concluding there is a significant difference), when it really did from the H 1 distribution (there is a difference) 21

Power vs. Difference of Means power

Effective Sizes How many samples are required to validate the following claim of the cram school: – –Our course will raise IQ levels of your child by 5 points statistically significant at 0.05 level, and the type II error is 0.1 Normal IQ mean is 100, with standard deviation 15 Sample standard deviation is assumed to be 15, too 23

Effective Sizes 24