Hypothesis TestingHypothesis Testing

Lecture 3

Examples of various hypotheses

• Average salary in Copenhagen is larger than in Bælum

• Sodium content in Furresøen is equal to the content in Madamsø

• Proportion of Turks in Århus is the same as in Aalborg

• Average height of men in Sweden is the same as in Denmark

• The average temperature is increasing over time

Formulation of hypothesis

Assume we are interested in a parameter Θ (e.g. the mean of the data). Let Θ0 be a number.

There are three different kinds of hypotheses:

H0: Θ = Θ0 H0: Θ ≥ Θ0 H0: Θ ≤ ΘHA: Θ ≠ Θ0 HA: Θ < Θ0 HA: Θ > Θ0

H0 is called the null hypothesis.HA is called the alternative hypothesis.

Examples of various hypotheses

• Average salary in Copenhagen is larger than in Bælum

H0: μC ≥ μB. HA: μC < μB.

• Sodium content in Furresøen is equal to the content in Madamsø

H0: μF = μM. HA: μF ≠ μM.

• Proportion of Turks in Århus is the same as in Aalborg

H0: PÅ = PA. HA: PÅ ≠ PA.

• Average height of men in Sweden is the same as in Denmark

H0: μS = μD. HA: μS ≠ μD.

• The average temperature is increasing over time

H0: μtime 1 ≥ μtime 2. HA: μtime 1 < μtime 2 if time 1 ≥ time 2.

COMPARE

SMALL DIFFERENCE

BIG DIFFERENCEE NOT EQUAL MEANS

EQUAL MEANS

NORMAL DISTRIBUTION(average height in Sweden and Denmark)

BINOMIAL DISTRIBUTION(Proportion of Turks in Århus and Aalborg)

BIG OR NOT?

The Test Procedure

Formulate a HYPOTHESIS!

Numerically bigger than

Does the data support the hypothesis or not?

Types of errors•Type I error: Rejecting falsely.•Type II error: Accepting falsely.

Decision H0 is true H0 is false

Reject H0 Type I error No error

Accept H0 No error Type II error

Ideally we would like a test where it is difficult to make errors.

Unfortunately

If you make a test where

• it is difficult to make a Type I error

• it is easy to make a Type II error

• and the other way around

Level of significance

So we want to construct a way to decide to

• ACCEPT or

• REJECT

the hypothesis based on data in a way such that

This sounds really technical!!!

Hmm

I don’t like this at all!

Critical Region

Assume

• We want to test if the sodium contest here is approx 3.8 units

• We have data y1, …, yn

• We have calculated average and SE.Support that content is 3.8

Support that content is 3.8

Support that content is < 3.8

Support that content is < 3.8

Support that content is > 3.8

Support that content is > 3.8

What do we know?If the content is 3.8 then the average is normally distributed with mean 3.8

With probability of 95% is the average less than 2*SE from 3.8

If the true content is 3.8 then the average

is in the red area with prob 5%

Test:• The hypothesis is that the true

content is 3.8• Estimate mean and SE.• The critical region is

• If the average is in the critical area then reject the hypothesis else accept

Significance level

Prob(Type I error) = 5 %

Alternative approach

Can we give a number telling us to what extend the observations support the hypothesis?

Yes, of course!

Why do you think I asked?

Hmmm

Supports hypothesis

Here we should definitely reject

If the true content is 3.8 then

and

Assume that we observe an average of 3.8 and SE = 0.1

Then what?

What is the probability of observing this???

What is the probability of observing this???

95% of data sets will have an average in this area (mean +/- 2 SE)

95% of data sets will have an average in this area (mean +/- 2 SE)

Assume we obtain an average of 3.8 and standard error SE = 0.1 and the true concentration is 3.8

P-value

Summing Up

A Statistical test can be

1.On a 5% significance level

2.By calculating the p-value

Hypothesis about the Mean

1. Is the concentration 3.8?

2. Is the proprotion of Turks in Århus 7.5%

Normal Distribution

Binomial Distribution

Sodium

1. Are data normal?

2. Estimate average and standard error

3. Calculate

4. Is t bigger than 2 (numerically)? OR5. Calculate p-value

Turks

1. Are data binomial?

2. Calculate proportion p and standard error

3. Calculate

4. Is t bigger than 2 (numerically)?

Last slide before the end• Are 3.8 in the 95% CI ?

• Accept the hypothesis (mean = 3.8) on a 5% significance level

That’s the same!!

The End