Univariate inferences about a mean
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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

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Univariate Inferences about a Mean

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Univariate inferences about a mean

Univariate Inferences about a Mean

Shyh-Kang Jeng

Department of Electrical Engineering/

Graduate Institute of Communication/

Graduate Institute of Networking and Multimedia


Scenarios

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


Evaluating normality of univariate marginal distributions

Evaluating Normality of Univariate Marginal Distributions

3


Tests of hypotheses

Tests of Hypotheses

Developed by Fisher, Pearson, Neyman, etc.

Two-sided

One-sided


Assumption under null hypothesis

Assumption under Null Hypothesis


Rejection or acceptance of null hypothesis

Rejection or Acceptance of Null Hypothesis


Student s t statistics

Student’s t-Statistics


Student s t distribution

Student’s t-distribution


Student s t distribution1

Student’s t-distribution


Student s t distribution2

Student’s t-distribution


Origin of the name student

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


Test of hypothesis

Test of Hypothesis


Selection of a

Selection of a

  • 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


Evaluating normality of univariate marginal distributions1

Evaluating Normality of Univariate Marginal Distributions

14


Confidence interval for m 0

Confidence Interval for m0


Neyman s interpretation

Neyman’s Interpretation

m0


Statistical significance vs practical significance

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


More specific hypotheses

More Specific Hypotheses

  • Null hypothesis

  • Alternative hypothesis


Type i and type ii errors

Type I and Type II Errors


Type i and type ii errors1

Type I and Type II Errors


Power

Power

  • The probability of concluding that the sample came from the H1distribution (i.e., concluding there is a significant difference), when it really did from the H1distribution (there is a difference)


Power vs difference of means

Power vs. Difference of Means

power

1

0


Effective sizes

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


Effective sizes1

Effective Sizes


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