1 STATISTICAL HYPOTHESIS Two-sided hypothesis: H 0 :  = 50H 1 :   only here H 0 is valid all other possibilities are H 1 One-sided hypothesis: H 0 :   50H 1 :   50 H 0 is valid 50 H 1 is valid (H 0 :   50H 1 :   50) Statistical hypothesis is a certain assumption about POPULATION. NULL hypothesis X ALTERNATIVE hypothesis

2 TEST CRITERION, CRITICAL POINT Test criterion (TC) is a random variable, distribution of which is known both for H 0 and H 1 distribution of test criterion for H 0 (it is the same distribution as in the case of H 0 but with different parameters) Critical point (CP) - boundary between intervals in which we accept or reject H0. CP we reject H 0 we accept H 0

General concept of statistical test 3 We test H 0 :   50 against H 1 :   50. Population has „infinitely“ data points with  ≤ 50 – one-sided test. H0H0 50 H1H1 Our question: Can we (on the basis of sample) reject the hypothesis that population has   50 with sufficient probability?

General concept of statistical test 4

5

6

7 Now there is a question: Is difference between theoretical (blue) and experimental (red) distributions so big that we can reject null hypothesis with sufficient probalility (in this case we reject null hypothesis) or Is difference between theoretical (blue) and experimental (red) distributions so small that we cannnot reject the possibility that population has mean  50 with sufficient probalility (in this case we cannot reject null hypothesis)

General concept of statistical test 8 ACCEPTED CRITICAL POINT REJECTED

General concept of statistical test 9 TEST CRITERION

General concept of statistical test 10

11 TWO SIDED TEST a  /2 b  lower critical point upper critical point interval where H 0 is accepted interval where H 0 is rejected  /2  interval where H 0 is rejected

12 p-VALUE p-value - the probability of observing a value of TC as extreme or more extreme as the one we observed. It's also referred to as the observed significance level. p <   is rejected p >   is not rejected 1-p – probability of rejecting H 0 here we reject H 0 here we accept H 0 critical point

EVALUATION THE RESULTS OF THE TEST (decision rules) 13 TC < CP  H 0 IS ACCEPTED TC > CP  H 0 IS REJECTED p <   H 0 IS REJECTED p >   H 0 IS ACCEPTED

14 EXAMPLE Accuracy of hypsometer was exemined. Known height (  0 = 20 m). was 15 x measured. We know sample mean = 19,2 m and sample SD = 1,1 m. We want to know whether the hypsometer is OK. formulation of H 0 and H 1 :  formulation of H 0 and H 1 : H 0 : Measurements of the hypsometer are correct. H 0 :  =  0 (sample with statistics and S 2 is taken from population with normal distribution N( ,  2 ) where parametr  is equal to known value  0 ).

15 EXAMPLE H 1 : Measurements of the hypsometer are not correct. H 0 :    0 (sample with statistics and S 2 does not take from population with normal distribution N( ,  2 ) where parametr  is equal to known value  0 ).   = 0,05: for rejecting of H 0 we need probability at least 1 -  = for rejecting of H 0 we need probability at least 1 -  = 0.95.

16 EXAMPLE  selection of test test of mean for one sample

17 EXAMPLE  critical point =TINV (0,05;14) = 2,145 (t quantile for  and df = 15-1= 14  because of symmetry of t-distribution we have 2 CP - –2,145 a +2,145.

18 POSTUP TESTOVÁNÍ NA PŘÍKLADU test criterion - 2,72 interval of rejecting H 0 interval of NOT rejecting of H 0 for interval of rejecting H 0 interval of NOT rejecting of H 0 for interval of rejecting H 0

19 Type I error (  ) error of rejecting a null hypothesis when it is actually true TYPE I ERROR AND TYPE II ERROR, THE POWER OF A STATISTICAL TEST

20 Type II error (  ) the error of failing to reject („accept“) a null hypothesis when in fact we should have rejected it TYPE I ERROR AND TYPE II ERROR, THE POWER OF A STATISTICAL TEST

21 The power of a statistical test (1-  ) is the probability that the test will reject a false null hypothesis. TYPE I ERROR AND TYPE II ERROR, THE POWER OF A STATISTICAL TEST

22 TYPE I ERROR AND TYPE II ERROR, THE POWER OF A STATISTICAL TEST

23 TYPE I ERROR AND TYPE II ERROR, THE POWER OF A STATISTICAL TEST

FACTORS INFLUENCING POWER 24 o Type I error (position of critical value) o Effect size (difference between H 0 and H 1 ) o Variability o Sample size

FACTORS INFLUENCING POWER 25 Type I error

26 Effect size FACTORS INFLUENCING POWER

27 FACTORS INFLUENCING POWER variability

PRACTICAL IMPORTANCE OF POWER OT THE TEST 28

PRACTICAL IMPORTANCE OF POWER OT THE TEST 29

Power analysis - example 30 After introduction of a new method of water treatment in water company, the content of chlorine in drinking water was monitored. According to the standard, 0,3 mg.l -1 is the allowance of chlorine in drinking water. Assess whether a real content of chlorine meets the requirements of the standard. Moreover, we need to know how many samples is necessary to take in order that test error with possible serious consequences was not higher then 5 %. Preliminary 23 samples were taken (content of Cl in water v mg.l -1 ):

Power analysis - example 31 This is an example of one-sided test (we are interested only in exceeding of standard, all mean values below standard are OK) H 0 : content of Cl  0,3 mg.l -1 H 1 : content of Cl  0,3 mg.l -1

Power analysis - example 32

Power analysis - example 33 test criterion t degrees of freedom p-value >  (0,05) We can also compare test criterion (1,494) with critical value of t-distribution (1,717): One sample t-test – one sided alternative – in R:

Power analysis - example 34 When null hypothesis is not rejected, we need to compute real power od the test Power is only 42%. It means that cca 58% of tests incorrectly „accept“ the null hypothesis – this test is very unreliable

Power analysis - example 35 Now we need to know how many samples we should take if we want to keep required Type II error – 0,05. We can adjust „delta“ – effect size – according to our best knowlege. Required sample size is 92

Power analysis - example 36 If this sample size is unacceptable we must to raise „errors“ of test. We start with TypeI error – we evaluated its practical consequences as not so problematic Now we need 61 samples

Power analysis - example 37 If this sample size (62 sapmles) is still too big sample, we can slightly raise also TypeII error – eg. from 0,05 to 0,10. After increasing of TypeII error we need „only“ 46 samples