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Hypothesis testing
Descriptive statistics
Inferential statistics
Allow us to make statements
about a population based on
info from samples of population
Hypothesis testing
Systematic model: summarises evidence from sampling
Can now decide between possible hypotheses
Null hypothesis = HO
States: no difference between two items
Alternate hypothesis = HA
States: the two items are different
Hypotheses stated in terms of population parameters
Hypothesis testing
e.g. Is there a difference between the heights of
students at UWC and at Wits?
HO: there is no difference in the average
height of the two groups of students
HA: there is a difference in the average
height between the two groups of students.
Find: there is a difference in average height.
Two possibilities: populations are indeed different or
difference is due to random error
Hypothesis testing
Q: how much difference is
there in the sample?
Hypothesis testing
What is the probability of obtaining this much difference just
by chance if we have sampled populations that are not
different?
i.e., is HO correct?
Probability = alpha (α) probability
If α probability of the statistic is > 0.05, then fail to reject HO
If α probability of the statistic is ≤ 0.05, then reject HO.
Hypothesis testing
When rejecting, or failing to reject a HO, we could be
making one of two errors:
Type I error: conclude there is a difference when there is
not a difference
α probability
Type II error: fail to find a difference that actually exists
β probability
Only way to decrease both α and β
is to increase your sample size.
Hypothesis testing
Reasoning of hypothesis testing
1.
Make a statement (the null hypothesis) about some
unknown population parameter.
2.
Collect some data.
3.
Assuming the null hypothesis is true, what is the
probability of obtaining data such as ours? (this is the
“p-value”).
4.
If this probability is small, then reject the null
hypothesis.
Hypothesis testing
Stating hypotheses
One-sided
Two-sided
H0: µ=110
H0: µ = 110
HA: µ < 110
HA: µ ≠ 110
Hypothesis testing
Setting a criterion
Decide what p-value would be “too unlikely” (the alpha level).
The retention region.
The range of sample mean values that are “likely” if H0 is true.
If your sample mean is in this region, retain the null hypothesis
The rejection region.
The range of sample mean values that are “unlikely” if H0 is
true.
If your sample mean is in this region, reject the null hypothesis
Hypothesis testing
Setting a criterion
Z as test statistic
Z test-statistic converts a sample mean into a z-score from the null
distribution.
Zcrit is the criterion value of Z that defines the rejection region
Ztest is the value of Z that represents the sample mean you
calculated from your data
Z test 
X  H0
X
Standard error!!!!
p-value is the probability of getting a Ztest as extreme as yours under
the null distribution
Hypothesis testing
Computing sample statistics
A test statistic (e.g. Ztest, Ttest, or Ftest) is
information we get from the sample that we use to
make the decision to reject or keep the null
hypothesis.
A test statistic converts the original measurement
(e.g. a sample mean) into units of the null
distribution (e.g. a z-score), so that we can look up
probabilities in a table.
Hypothesis testing
Setting a criterion
Accept H0
Reject H0
Zcrit
Reject H0
H
0
Zcrit
Hypothesis testing
Making a decision
Accept H0
Reject H0
Zcrit
Reject H0
Zcrit
• If we want to know where our sample mean lies in the null
distribution, we convert X-bar to our test statistic Ztest
• If an observed sample mean were lower than z = -1.65 then it
would be in a critical region where it was more extreme than
95% of all sample means that might be drawn from that
population
Hypothesis testing
One-tailed tests
If HA states  is < some value, critical region occupies left tail
If HA states  is > some value, critical region occupies right tail
Hypothesis testing
Hypothesis testing
Hypothesis testing
Two-tailed hypothesis testing
HA is that µ is either greater or less than µH0
HA: µ ? µH0
 is divided equally between the two tails of the
critical region
Hypothesis testing
Hypothesis testing
Progress assessment
By now, you should be able to answer the following
questions:
• Do I understand all the terms dealt with in the chapter on definitions?
• What are the different types of data, and how are they represented?
• What is the difference between descriptive and inferential statistics?
• What is the difference between a sample and a population?
• What is the difference between design structure and treatment
structure?
• What is a measure of location, and which is the most commonly used?
• What are the most commonly used measures of dispersion, and can I
use the formulas in order to calculate them?
• What is the normal curve, and which parameters define it?
Progress assessment
Q’s cont.
• How is the normal curve used in order to determine probability?
• What is a Z score and Z dispersion?
• What are we doing when we are hypothesis testing?
• What is the difference between Type I and Type II errors?
• How do we use Z scores in order to reject or fail to reject the null
hypothesis?
• What is the difference between a one-tailed and a two-tailed
test?