STAT 215 Hypothesis Testing I Colin Reimer Dawson Oberlin College - - PowerPoint PPT Presentation

Inference Goals Hypothesis Testing Philosophy of Null Hypothesis Testing Formulating Statistical Hypotheses

STAT 215 Hypothesis Testing I

Colin Reimer Dawson

Oberlin College

September 7, 2017 1 / 14

Inference Goals Hypothesis Testing Philosophy of Null Hypothesis Testing Formulating Statistical Hypotheses

Outline

Inference Goals Hypothesis Testing Philosophy of Null Hypothesis Testing Formulating Statistical Hypotheses 2 / 14

Inference Goals Hypothesis Testing Philosophy of Null Hypothesis Testing Formulating Statistical Hypotheses

Two Main Goals of Inference

• 1. Assessing strength of evidence about “yes/no” questions

(hypothesis testing)

• 2. Estimating unknown quantities in a population using a sample

(confidence intervals) 3 / 14

Inference Goals Hypothesis Testing Philosophy of Null Hypothesis Testing Formulating Statistical Hypotheses

Outline

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Inference Goals Hypothesis Testing Philosophy of Null Hypothesis Testing Formulating Statistical Hypotheses

Hypothesis Testing

Some (yes/no) questions we might want to answer with data:

• 1. Kidney stone treatment A has better outcomes than B in a

sample of cases. Is A really more effective, or did those patients just get lucky?

• 2. Do people (in the population tip more (as a %) for more

expensive restaurant meals?

• 3. Does the population of high school graduates earn more on

average than the population of GED recipients? 5 / 14

Inference Goals Hypothesis Testing Philosophy of Null Hypothesis Testing Formulating Statistical Hypotheses

In the 1920s in Cambridge, a lady claimed tea tasted different depending

• n whether the milk was added before or after the tea was poured. A

scientist in attendance proposed to put it to a blind taste test w/ 10 cups

• f tea prepared in random order.
• Is her claim plausible if she gets 5 of 10 correct? 10 of 10? 9 of 10?
• How much success is enough to believe her?

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Inference Goals Hypothesis Testing Philosophy of Null Hypothesis Testing Formulating Statistical Hypotheses

Outline

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Inference Goals Hypothesis Testing Philosophy of Null Hypothesis Testing Formulating Statistical Hypotheses

Falsification

Karl Popper: scientific theories can’t be fully verified (there is always another possible explanation), only falsified. 8 / 14

Inference Goals Hypothesis Testing Philosophy of Null Hypothesis Testing Formulating Statistical Hypotheses

Falsification With Randomness

• When sampling, we will occasionally get strange results just by
• chance. So we can’t falsify absolutely.
• But we can say a hypothesis is implausible if the data would

be very unlikely if the hypothesis were true. 9 / 14

Inference Goals Hypothesis Testing Philosophy of Null Hypothesis Testing Formulating Statistical Hypotheses

The Null Hypothesis

• R.A. Fisher: Formulate the negation of your research

hypothesis, and establish conditions under which it can be rejected.

• Fisher called this “antihypothesis” the null hypothesis, and

developed null hypothesis significance testing (NHST). 10 / 14

Inference Goals Hypothesis Testing Philosophy of Null Hypothesis Testing Formulating Statistical Hypotheses

The Alternative Hypothesis

• Jerzy Neyman and Egon Pearson added the idea of the

alternative hypothesis to Fisher’s null hypothesis formulation.

• Idea: don’t reject H0 in a vacuum — reject in favor of another

hypothesis, the alternative hypothesis (or H1).

• This is usually the one you set out to investigate: the drug is

better; the correlation is positive. 11 / 14

Inference Goals Hypothesis Testing Philosophy of Null Hypothesis Testing Formulating Statistical Hypotheses

Outline

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Inference Goals Hypothesis Testing Philosophy of Null Hypothesis Testing Formulating Statistical Hypotheses

Statistics vs. Parameters

• Summary values (like mean, median, standard deviation) can

be computed for populations or for samples.

• In a population, such a summary value is called a parameter
• In a sample, these values are called statistics, and are used to

estimate the corresponding parameter Value Population Parameter Sample Statistic Mean µ ¯ X Proportion p ˆ p Correlation ρ r Slope of a Line β1 ˆ β1 Difference in Means µ1 − µ2 ¯ X1 − ¯ X2 . . . . . . . . . 13 / 14

Inference Goals Hypothesis Testing Philosophy of Null Hypothesis Testing Formulating Statistical Hypotheses

The Null and Alternative Hypothesis

• Pairs/Threes (5 min.): Identify the relevant population

parameter for each of the following claims. What are the null and alternative hypotheses (abbreviated H0 and H1), both in words and in terms of a population parameter?

• The lady can tell the difference between cups of tea more often

than random guessing. H0: pcorrect = 0.5, H1: pcorrect > 0.5, where pcorrect is her “long run” success rate

• There is a positive linear association between pH and mercury

in Florida lakes. H0: ρ = 0, H1: ρ > 0, where ρ is the correlation coefficient between pH and Hg in all Florida lakes

• Lab mice eat more on average when the room is light. H0:

µlight − µdark = 0, H1: µlight − µdark > 0, where µ are “long run”/population means for an appropriate measure of amount

• f food consumed

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