hypothesis testing lecture 4. examples of various hypotheses the sodium content in furresøen is x...
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Examples of various hypotheses
• The sodium content in Furresøen is x
• Sodium content in Furresøen is equal to the content in Madamsø
• The proportion of Turks in Aalborg is x %
• Proportion of Turks in Århus is the same as in Aalborg
• Average height of men in Sweden is the same as in Denmark
Basics
• Null hypothesis
• Alternative hypothesis
• Type I errors: Rejecting falsely
• Type II errors: Accepting falsely
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
Critical Region
Assume
• We want to test if the sodium content 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 3.4 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 3.4 and standard error SE = 0.1 and the true concentration is 3.8
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)? OR
5. Calculate p-value
Turks
1. Are data binomial?
2. Calculate proportion p and standard error
3. Calculate
4. Is t bigger than 2 (numerically)?
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