Download Statistics 03

Statistics 03
Hypothesis Testing (假设检验)
Hypothesis Testing (假设检验)
• When we have two sets of data and we want
to know whether there is any statistically
significant difference between them, we can
use hypothesis testing.
Principles of hypothesis testing
• Small probability event:
• Something that is very unlikely to take
place
• When something is a small probability
event, we usually take it as false.
• When a small probability event takes place,
we take it as true.
• e.g. Win a lottery
Null hypothesis
• As a procedure in hypothesis testing, we usually
have to establish a null hypothesis, a hypothesis
that there is no significant variability between the
two sets of data compared.
• H0: μ=μ0
• Underlying principle: the variability is caused by
random errors. Therefore small probability events
do not take place.
Hypothesis
Alternative hypothesis
• We need to establish a second hypothesis:
alternative hypothesis as an opposite to the null
hypothesis.
• H1: μ≠μ0
• In the alternative hypothesis, the small probability
event takes place. The variability between the two
sets of data compared is no longer caused by
random errors. It is the result of systematic
difference.
Case: Comparison of means
• Class A has been given a special kind of instruction.
We want to know the effect of this methodology.
• Class A took the same test as the other students.
We got the data as follows:
• Class A
mean: 72
n: 45
• The other classes
mean: 69
s: 5
Analysis
• Class A: a sample with a mean μ=72
• The others: the population where the sample is
taken with the mean μ0=69
• The special methodology should have brought
about some systematic effect on Class A. Such an
effect is not a small probability event.
• Therefore
• Null hypothesis: no significant difference between
Class A and the other classes in spite of the special
teaching methodology. i.e. H0: μ=μ0
• Alternative hypothesis: difference is significant.
i.e. H1: μ≠μ0
Computation
• 1. Establish the null hypothesis and the
alternative hypothesis
H0: μ=μ0
H1: μ≠μ0
• 2. Calculate the Z value
|Z|=|(μ-μ0)/ (σ0/√n)|
• 3. Look up in the Normal Distribution Table
for Zα, usually Zα/2=0.025=1.96
• 4. Compare |Z| and Zα/2
If |Z| > Zα/2, reject H0
If |Z| <= Zα/2, accept H0
Computation
•
•
•
•
•
•
•
|Z|=|(μ-μ0)/ (σ0/√n)|
=|(72-69)/(5/√45)|
=3/(5/6.71)
=3/0.745
=4.027
|Z| > Zα/2=0.025=1.96
Reject H0
Another Case
• An experiment on reading comprehension was done
on two groups of students: an advanced class with 63
students and a normal class with 67 students. The
investigator obtained the following data:
• Advanced class
M: 2.33929
s: 0.69483
n: 56
• Normal class
M: 1.864407
s: 0.797791
n: 59
Question
• Is there any statistically significant
difference between the two groups at the
95% confidence level?
Analysis
• The advanced class is taken as a sample from a
population represented by the normal class. The
advanced students are supposed to possess a
higher level of reading comprehension which can
make systematic difference from average students.
• Therefore
• null hypothesis
H0: μ=μ0
• alternative hypothesis
H1: μ≠μ0
Computation
• |Z|=|(μ-μ0)/ (σ0/√n)|
=|(2.33929-1.864407)/( 0.797791/√56)|
=3/(5/6.71)
=3/0.745
=4.027
• |Z| > Zα/2=0.025=1.96
• Reject H0
One Tailed vs. Two Tailed
• When we ask: Is A different from B, we
need a two tailed test.
One Tailed vs. Two Tailed
• When we ask: is A better or worse than B,
we need a one tailed test.