Question. Hypothesis testing. Example. Answer: hypothesis. Test: true or not? Question. Average is not the mean! μ average. Random deviation or not?

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1 Hypothesis testing Question Very frequently: what is the possible value of μ? Sample: we know only the average! μ average. Random deviation or not? Standard error: the measure of the random deviation. Question Answer: hypothesis Test: true or not? Average is not the mean! Why? Example Question: The medicine decreases the fewer or not?. Finite no. of elements (random deviation). Systematic deviation. (the hypothesis is not true, the value of the μ is different)

2 How many trials? Sample Outcome:. Δt >0;. Δt =0; 3. Δt <0 Only? Not only the medicine influences the fewer! We suppose that other effects are random events! Population:. Effective μ 0!. Not effective μ = 0! The average deviates around the mean! The average normally is not 0! Why?. Random event, due to the finite no. of the elements.. Our base hypothesis is not true. Which case is true? Nullhypothesis Why the nullhypothesis? Population:. Effective μ 0!. Not effective μ = 0! Sample: average 0 average should be 0 Symbols. H 0 : nullhypothesis H : alternative hypothesis Alternative hypothesis μ =? Nullhypothesis! Random deviation (finite no. of elements) Nullhypothesis: the mathematical statistics is able to calculate the possible random deviation of the average from the μ!!!

3 Example Medicine decreases the fewer or not? Large difference? What is the measure of the difference? Nullhypothesis: not! μ is 0. But the average is not 0. Standard error: average deviation of the averages around the μ. sample. average -0. C If the difference is larger, ( x ± sx ) ~ 68% - confidence interval C -.5 C seems to be more sure. Base: nullhypothesis! t-value Question: x average may be 0? t = x μ s x Compare the difference to the standard error. (μ is frequently 0) Condition: Known distribution is necessary. The average deviates around the μ, so the t- value deviates around the 0. (if the nullhypothesis is true!)

4 Why is it better? We are able to calculate the random distribution of the t values!!! (Student s t-distribution) This describes the random deviation of the t from the 0!!! Shape of the distribution depends on the no. of the elements. One-sample t-test Question about the possible value of the μ. Nullhypothesis: the difference of the average is random. t = 0, the difference is due to the finite no. of elements. Base of decision t-table p is the probability, that t calc t crit randomly. Different t crit value belong to different significance level. If p is enough small, more probable that the nullhypothesis is not true. α is the significance level: the probability that t- value is randomly out an interval.

5 Base of the decision Decision. if the probability of the random event is small (p( t t crit ) is 5%) reject the nullhypothesis.. if the probability of the random event is high (p( t t crit ) is > 5%) accept the nullhypothesis. The answer is not yes or no, true or false!!! Decision using t-table Decision using computer d.f.: degree of freedom p: the probability of random event that the t t calculated. degree of freedom:

6 Possibility of the mistake Decision Decision: the nullhypothesis accepted rejected Nullhypothesis true false Right decision II. Type error (β) I. Type error (α) Right decision Conditions Test in two groups. Question about the mean of the population.. Variable has normal distribution. Conditions:. Variable has normal distribution.. The standard deviation is same in the two groups, the difference is due to the sampling. 3. The samples are independent from each other.

7 Nullhypothesis: samples derive from the same population. Alternative hypothesis: samples derive from different populations. The nullhypotesis Samples derive from the same population. But, due to the sampling: x x 0 The expected value of the difference is zero. The observed difference is due to the sampling only. The common standard error Calculation of the t-value s Q + Q x x = + n + n n n t = x x two-sample t-test s x x If the standard deviation is really common. σ = σ The degree of the freedom: n +n - Decision: look one-sample t-test!

8 σ = σ? F-test Condition: σ = σ Question: s = s? Nullhypotesis: H 0 : σ = σ. Alternative hypothesis: H : not. F = s s The expected value belonging to the nullhypotesis:. degree of freedom: nominator: n. denominator : n. F-test! (Calculation: s > s. the greater is in the nominator.) Table: (α = 0,05) (. line: d.f. of the nominator,. column: d.f. of the denominator) 6,4 8,5 0,3 7,7 6,6 99,5 9,00 9,55 6,94 5,79 F calc F table, reject. F calc < F table, accept. 3 5,7 9,6 9,8 6,59 5,4 Decision 4 4,6 9,5 9, 6,39 5,9 5 30, 9,30 9,0 6,6 5,05 on the base of the p value: p: the probability, that F calc randomly over the F table. Correlation of two variables Experiment: Data pairs: No. weight (kg) Height (cm)

9 Graphic representation Covariance A B C Q xy = i [( x x) ( y y) ] i i Cov(x, y) Q xy = n Positive tendency: Frequently: If x i > x, then y i > y and if x i < x, then y i < y. no any tendency Positive tendency Negative tendency Consequence: Q xy > 0. Negative tendency: No tendency: Correlation coefficient r Q = xy Qx Q r y Frequently: If x i > x, then y i < y and if x i < x, then y i > y. Consequence: Q xy < 0. The y values are independent from the x-values. Consequence: Q xy = 0. (if n= ) Population: r = 0 no correlation, r 0 correlation (strength is proportional to the actual value of r.

10 Testing of the r Sample: due to the finite no. of the elements we can calculate only the estimation of the r. t r Correlation t-test n r = d.f.: n - Question: Correlation or not? Decision: based on t-value. Look previous cases! Nullhypotesis: No correlation. H 0 : r = 0. Condition: at least one of the variable has normal distribution. Meaning of correlation Linear regression Not necessary being direct causality. (But may be!) In the background there may be quantities, effects that influence both measured variables. If variables have normal distribution, the dependency is linear, and we can describe with straight line.

11 Frequency data Experiment Has no normal distribution! Example: headache Effective: no headache. group: patients introducing the medicine. group: patients introducing the placebo Not effective: headache headache (a) no headache (b) headache (c) no headache (d) Contingency table Nullhypothesis If the effect is independent from the medicine, we expect:. group. group headache a c no headache b d Total a+b c+d a = b c d a d = b c total a+c b+d n Nullhypothesis: the effect is independent from the medicine.

12 χ = χ -distribution n( ad bc) ( a + b)( c + d )( a + c)( b + d ) Nullhypothesis: χ -value is 0, the difference due to the random event only. χ -distribution describes the random deviations of the χ -value. Decision Same, than in the case of t-distribution. We use χ -distribution. Expected value is 0, if the nullhypothesis is true. if calc. crit. - reject the nullhypothesis else accept. degree of freedom: in this special case =. In general: d.f.=(r-)(c-), where r no. of rows c no. of columns