SAA 2023 COMPUTATIONALTECHNIQUE FOR BIOSTATISTICS Semester 2 Session 2009/2010 ASSOC. PROF. DR. AHMED MAHIR MOKHTAR BAKRI Faculty of Science and Technology Room 44, 3 rd Floor FKP Building, Nilai 2. Hypothesis Testing

Hypothesis Testing Fundamentals of Hypothesis Testing Testing a Claim about a Mean: Large Samples Testing a Claim about a Mean: Small Samples Testing a Claim about a Proportion

Definition  Hypothesis In statistics, a hypothesis is a claim or statement about a property of a population.

Rare Event Rule for Inferential Statistics If, under a given assumption, the probability of an observed event is exceptionally small, we conclude that the assumption is probably not correct.

Central Limit Theorem The Expected Distribution of Sample Means Assuming that  = 98.6 z = x = or z = 1.96 x = or Sample data: x = or z = µ x = 98.6 Likely sample means

Components of a Formal Hypothesis Test

Null Hypothesis: H 0 CHI SQUARE TEST  Must contain condition of EQUALITY: =  Observed ratio = Expected ratio  It fits the ratio 3:1  Test the Null Hypothesis directly  Reject H 0 or fail to reject H 0

Alternative Hypothesis: H 1 CHI SQUARE TEST  Must be true if H 0 is false  Must contain condition of INEQUALITY:   It does not fit the ratio 3:1  ‘Opposite’ of Null Hypothesis

Null Hypothesis: H 0  Statement about the value of a POPULATION PARAMETER  Must contain condition of EQUALITY: =, ≤, or ≥  Test the Null Hypothesis directly  Reject H 0 or fail to reject H 0

Alternative Hypothesis: H 1  Must be true if H 0 is false  Must contain condition of INEQUALITY: ,  ‘Opposite’ of Null Hypothesis

Critical Region Set of all values of the test statistic that would cause a rejection of the null hypothesis Critical Region

Critical Region Set of all values of the test statistic that would cause a rejection of the null hypothesis Critical Regions

Significance Level  denoted by   the probability that the test statistic will fall in the critical region when the null hypothesis is actually true.  common choices are 0.05, 0.01, and 0.10

Critical Value ( z score ) Fail to reject H 0 Reject H 0 Value or values that separate the critical region (where we reject the null hypothesis) from the values of the test statistics that do not lead to a rejection of the null hypothesis

Two-tailed, Right-tailed, Left-tailed Tests The tails in a distribution are the extreme regions bounded by critical values.

Two-tailed Test H 0 : µ = 100 H 1 : µ  100 UNEQUAL means less than or greater than 100 Values that differ significantly from 100  is divided equally between the two tails of the critical region Fail to reject H 0 Reject H 0

Right-tailed Test H 0 : µ  100 H 1 : µ > 100 Values that differ significantly from Fail to reject H 0 Reject H 0

Left-tailed Test H 0 : µ  100 H 1 : µ < Values that differ significantly from 100 Fail to reject H 0 Reject H 0

Conclusions in Hypothesis Testing  Always test the NULL hypothesis:  Reject H 0  Fail to reject H 0  Be careful to include the correct wording of the final conclusion or

Wording of Final Conclusion Claim contains equality? Reject H 0 ? Yes Claim becomes H 0 No Claim becomes H 1 Yes “There is sufficient evidence to reject the claim that... (original claim).” “There is not sufficient evidence to reject the claim that (original claim).” “There is sufficient evidence to support the claim that... (original claim).” “There is not sufficient evidence to support the claim that (original claim).” Reject H 0 ? Yes No Only case in which original claim is rejected Only case in which original claim is supported Start No

“Reject” versus “Fail to Reject” Case 1: (If you reject the null hypothesis.) The data do provide significant evidence, at the 5% level, that the alternative hypothesis is true. Case 2: (If you fail to reject the null hypothesis.) The data do not provide significant evidence, at the 5% level, that the alternative hypothesis is true.

“Fail to Reject” versus “Accept”  Some texts use “accept the null hypothesis”  We are not proving the null hypothesis (can’t PROVE equality)  If the sample evidence is not strong enough to warrant rejection, then the null hypothesis may or may not be true (just as a defendant found NOT GUILTY may or may not be innocent)

Type I Error  Rejecting the null hypothesis when it is true.    (alpha) represents the probability of a type I error  Example: Rejecting a claim that the mean body temperature is 98.6 degrees when the mean really is 98.6

Type II Error  Failing to reject the null hypothesis when it is false.  β (beta) represents the probability of a type II error  Example:Failing to reject the claim that the mean body temperature is 98.6 degrees when the mean really isn’t 98.6

Type I and Type II Errors Reject the null hypothesis Fail to reject the null hypothesis TRUE FALSE Type I error α Rejecting a true null hypothesis Type II error β Failing to reject a false null hypothesis CORRECT NULL HYPOTHESIS CORRECT

Controlling Type I and Type II Errors  , , and n are interrelated. If one is kept constant, then an increase in one of the remaining two will cause a decrease in the other.  For any fixed , an increase in the sample size n will cause a ??????? in   For any fixed sample size n, a decrease in  will cause a ??????? in .  Conversely, an increase in  will cause a ??????? in .  To decrease both  and , ??????? the sample size n.