1. SADC Course in Statistics Basic principles of hypothesis tests (Session 08)

2. To put your footer here go to View > Header and Footer 2 Learning Objectives By the end of this session, you will be able to explain what is meant by a null hypothesis and an alternative hypotheses write down a null hypothesis that would enable a claim about some event to be tested statistically write down the alternative hypothesis corresponding to the null hypothesis describe clearly the two types of errors that arise when testing the null against the alternative hypothesis

3. To put your footer here go to View > Header and Footer 3 From Objectives to Hypotheses Consider the following claims made by (say) a local NGO… The under-five mortality rate in year 2000 in sub-Saharan Africa is significantly lower than its value in 1990 of 185 per 1000 live births Mean years of education, of the heads of households in Tanzania, differ according to the gender of the household head There is a relationship between level of access to clean water and the number of episodes of diarrhoea in the household

4. To put your footer here go to View > Header and Footer 4 Testing a given claim Question:Is there an evidence-based approach to test these claims? Question:If so, how can the claim be tested? Answer:Set up a hypothesis in a very precise way and use data to reject, or fail to reject the hypothesis. This hypothesis is called the null hypothesis It is usually denoted by H 0. We now recast the claims on slide 3 in the form of a series of null hypotheses.

5. To put your footer here go to View > Header and Footer 5 Formulating the null hypothesis H 0 : The average under-five mortality rate in year 2000 in sub-Saharan Africa is 185 deaths per 1000 live births H 0 : Mean years of education, of the heads of households in Tanzania, are equal in male headed and female headed HHs H 0 : There is NO relationship between level of access to clean water and the number of episodes of diarrhoea in the household Note that the null is very exactly stated!

6. To put your footer here go to View > Header and Footer 6 What if H 0 is untrue? Need also to set up alternative hypothesis H 1 which must be preferred if H 0 is rejected H 1 : The average under-five mortality rate in year 2000 in sub-Saharan Africa is not equal to 185 deaths per 1000 live births H 1 : Mean years of education, of the heads of households in Tanzania, are unequal across the gender of the household head H 1 : There is a relationship between level of access to clean water and the number of episodes of diarrhoea in the household

7. To put your footer here go to View > Header and Footer 7 Mathematical formulation Let be the mean under-five mortality rate in year 2000 Let the mean number of years of education of male and female heads of HHs in Tanzania, be 1 and 2 respectively Then the first two hypotheses above may be written as H 0 : =185 versus H 1 : 185, and H 0 : 1 = 2 versus H 1 : 1 2

8. To put your footer here go to View > Header and Footer 8 Data needed for tests about means Since the null and alternative hypotheses concern the unknown population means, the test is based on the sample means. For our 1 st example, find that in year 2000, the mean under-5 mortality from results of 30 countries gives mean = 138.1, std. error=14.03 Do you think this provides evidence against the null hypothesis? Discuss this in small groups, using intuitive arguments.

9. To put your footer here go to View > Header and Footer 9 The second example For our 2 nd example, find: mean = 6.62 years for males mean = 6.46 years for females Thus difference in mean = 0.16 The 95% confidence interval for this difference is: (-0.174, 0.495) Do you think this provides evidence against the null hypothesis? Again, discuss this in small groups, using intuitive arguments.

10. To put your footer here go to View > Header and Footer 10 Discussion of findings… What are your conclusions from the discussions above? Did you pay attention to the change that had occurred and considered whether (from an intuitive point of view) this change constituted a large change? Did the value of the standard error help? Did knowledge of the confidence limits help?

11. To put your footer here go to View > Header and Footer 11 What sample statistics to use? In both examples above, we used the sample mean because the claim concerned one or more means Suppose we were in the year 2015, and want to test the claim by donors that the proportion of people living below the poverty line is less than half? What is the sample statistic you would use in this case? Can you write down the null and alternative hypotheses here?

12. To put your footer here go to View > Header and Footer 12 Two types of errors In testing the null hypothesis against the alternative hypothesis, two errors can arise… 1.Rejecting the null hypothesis when it is actually true 2.Failing to reject the null hypothesis when the alternative is true Probabilities associated with the occurrence of these errors are denoted by and respectively.

13. To put your footer here go to View > Header and Footer 13 More formally… = Prob(Rejecting H 0 | H 0 true) = Prob(Failing to reject H 0 | H 1 true) is called the Type I error, while is called the Type II error. Of course we want to minimise these errors. This is not usually possible simultaneously. So in practice, is pre-set, usually to a value < 0.05, with the hope that would be relatively small.

14. To put your footer here go to View > Header and Footer 14 Power of test Note that 1 - is called the power of the test, i.e. Power = Prob (Rejecting H 0 | H 1 true) = 1 – Prob(Type II error) It is often used in sample size calculations where testing is involved.

15. To put your footer here go to View > Header and Footer 15 Some practical work follows…