sadc course in statistics basic principles of hypothesis tests (session 08)

SADC Course in Statistics

Basic principles of hypothesis tests

(Session 08)

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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


From Objectives to HypothesesConsider 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


Testing a given claimQuestion: 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 H0.

We now recast the claims on slide 3 in the formof a series of null hypotheses.


Formulating the null hypothesis

H0: The average under-five mortality rate in

year 2000 in sub-Saharan Africa is 185 deaths per 1000 live births

H0: Mean years of education, of the heads of

households in Tanzania, are equal in male headed and female headed HHs

H0: 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!


What if H0 is untrue?

Need also to set up alternative hypothesis H1 which must be preferred if H0 is rejected

H1: The average under-five mortality rate in year 2000 in sub-Saharan Africa is not equal to 185 deaths per 1000 live births

H1: Mean years of education, of the heads of households in Tanzania, are unequal across the gender of the household head

H1: There is a relationship between level of access to clean water and the number of episodes of diarrhoea in the household


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

H0: =185 versus H1: 185, and

H0: 1 = 2 versus H1 : 1 2


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 1st 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.


The second exampleFor our 2nd 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.


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?


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?


Two types of errors

In testing the null hypothesis against thealternative 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 ofthese errors are denoted by and respectively.


More formally…

= Prob(Rejecting H0| H0 true)

= Prob(Failing to reject H0| H1 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.


Power of test

Note that 1 - is called the power of the test, i.e.

Power = Prob (Rejecting H0 | H1 true)

= 1 – Prob(Type II error)

It is often used in sample size calculations where testing is involved.


Some practical work follows…

sadc course in statistics basic principles of hypothesis tests (session 08)

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