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Hypothesis Testing

Hypothesis 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

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Hypothesis Testing

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  1. Hypothesis Testing Lecture 3

  2. 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

  3. 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≤Θ HA: Θ≠Θ0 HA: Θ < Θ0 HA: Θ > Θ0 H0 is called the null hypothesis. HA is called the alternative hypothesis.

  4. 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.

  5. NORMAL DISTRIBUTION(average height in Sweden and Denmark) COMPARE BIG DIFFERENCEE NOT EQUAL MEANS SMALL DIFFERENCE EQUAL MEANS

  6. BINOMIAL DISTRIBUTION(Proportion of Turks in Århus and Aalborg) BIG OR NOT?

  7. The Test Procedure Formulate a HYPOTHESIS!

  8. Numerically bigger than Does the data support the hypothesis or not?

  9. Types of errors • Type I error: Rejecting falsely. • Type II error: Accepting falsely. Ideally we would like a test where it is difficult to make errors.

  10. 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

  11. 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

  12. This sounds really technical!!! Hmm I don’t like this at all!

  13. 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

  14. 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%

  15. 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 %

  16. Alternative approach Hmmm Supports hypothesis Here we should definitely reject Can we give a number telling us to what extend the observations support the hypothesis? Yes, of course! Why do you think I asked?

  17. If the true content is 3.8 then and Assume that we observe an average of 3.8 and SE = 0.1 Then what?

  18. Assume we obtain an average of 3.8 and standard error SE = 0.1 and the true concentration is 3.8 95% of data sets will have an average in this area (mean +/- 2 SE) What is the probability of observing this???

  19. P-value

  20. Summing Up • A Statistical test can be • On a 5% significance level • By calculating the p-value

  21. Hypothesis about the Mean • Is the concentration 3.8? • Is the proprotion of Turks in Århus 7.5% Normal Distribution Binomial Distribution

  22. Sodium • Are data normal? • Estimate average and standard error • Calculate • Is t bigger than 2 (numerically)? OR • Calculate p-value

  23. Turks • Are data binomial? • Calculate proportion p and standard error • Calculate • Is t bigger than 2 (numerically)?

  24. 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!!

  25. The End

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