6/19/2012
Hypothesis Tests
The Null Hypothesis, H0
• Objective decision-making tools
• A hypothesis test focuses on a single question
• Are used to answer very specific questions.
• That question is formulated in terms of a “TRUE or
FALSE” statement called the Null Hypothesis, also
denoted by H0.
For example,
“Are the two gum groups equivalent in terms of mean
change in DMFS?”
• Decision rules for possible answers to your question are
based on how consistent the possible answers are with
the observed data.
• The evidence from a hypothesis test is in the form of how
sure we are that H0 is not true.
• Thus, the null hypothesis is formulated in such a way that
it is the statement that we are expecting to disprove.
Example: Chewing Gum Study
Creating the Decision Rule
• We are hoping to find evidence that the gums are
different in terms of DMFS progression.
We will base our decision about H0 on a statistic
that:
1. is a function of the observed data.
• Thus, we would phrase the null hypothesis:
H0: The mean change in DMFS in both
groups is the same
• We hope to prove the gums are different by
disproving H0
• Disproving H0 = “rejecting” the null hypothesis
2. gives information about H0, the question of
interest.
3. will be predictable (we know it’s probability
distribution) when H0 is true
Example: change in facial height
Example: change in facial height
• Study to determine whether facial height of
people aged 21-26 years changes over a ten
year period.
• Facial heights were measured from a sample
of graduate students.
• After ten years, the measurements were taken
again on 84 of the same students.
The average change in facial height was
with s= 6.8 mm.
1.7 mm
We define our null hypothesis to be:
H0: µ = 0,
where μ = mean change in facial height.
Evidence against H0 will mean evidence that facial
heights change.
1
6/19/2012
The t statistic
The t statistic
To test H0: µ = 0 we use the t statistic
To test H0: µ = 0 we use the t statistic
0
0
• If H0 is true, then we would expect the sample mean,
, to be close to µ = 0.
• If H0 is not true, then we would expect the to be
far from µ = 0.
• Thus, values of the t statistic far from zero will
indicate evidence against H0.
Furthermore, If the null hypothesis is true, then we
know that T has t distribution with n-1 degrees of
freedom.
0
Making the Decision
Example: change in facial height
• Compute the statistic from the observed data
• Calculate the probability of seeing the observed
statistic if the null hypothesis were true
The average change in facial height for the 84 former
grad students was X̅ = -1.7 mm with s = 6.8 mm.
The t statistic is:
1.7
• If the observed statistic would have low
probability of occurring if the null hypothesis
were true this will be considered evidence
against the null hypothesis
0
6.8⁄ 84
2.29
Example: change in facial height
Example: change in facial height
The probability of observing T = -2.29 or something
even more unlikely when H0 is true would be
Because the probability p = 0.024 is fairly low (less than
1 in 40), we consider this good evidence against H0.
P(t83 < -2.29) + P(t83 > 2.29) = 2 × 0.012 = 0.024*
This probability, p = 0.024, is an example of a p-value
t83distribution
t83distribution
prob = 0.012
prob = 0.012
prob = 0.012
prob = 0.012
-2.29
2.29
-2.29
2.29
* Computed using Excel function “TDIST”
2
6/19/2012
P-value
• The final decision on whether or not we should believe the null
hypothesis is based upon the p-value.
• The p-value is defined as the probability that if one performed an
experiment in which it was known that H0 was true, that we would still
see data producing a statistic as “extreme” (or more extreme) as the
one observed.
• By an “extreme” statistic we mean one that would be unlikely if H0
were true. Or a statistic that appears to show evidence that H0 is not
true.
Decision Rule
•How small of a p-value do we require to decide that the null
hypothesis is just too unlikely?
•We decide upon a threshold value a priori, before looking at the data.
•This threshold value is called the significance level and is usually
denoted by α.
Decision Rule
Reject H0 if the p-value < α
• The smaller the p-value, the less likely our observed statistic would
be if H0 were true.
•If the p-value is less than α and we reject H0 this is often called
“statistically significant”
• Since we know that the observed statistic is real, this means that
small p-values are actually evidence against H0 .
•Standard choices for α are .05 or .01
Example: change in facial height
Test the null hypothesis H0: µ = 0 at significance level α = 0.05.
• The p-value was p = 0.024. Since 0.024 is less than .05, we
REJECT H0.
“There is good evidence at the α = 0.05 significance level to that
H0 is not true.”
• If the significance level was α = 0.01 instead, then we would
NOT REJECT H0.
“There is not good evidence at the α = 0.01 significance level
that H0 is not true.”
Using Table 4 to find out whether the pvalue is above or below α
• Look up the 1-α/2th percentile of the tn-1 distribution
in Table 4.
• Compare T to tn-1, 1-α/2
• If |T| > tn-1, 1-α/2 , then p-value < α. REJECT H0
Computing the p-value
• If you have access to Excel, you can calculate the
probabilities from any t distribution
– http://courses.washington.edu/dphs568/course/excel-t.htm
• If not, you can use Table 4 in the course notes to tell
whether the p-value is below certain values
• Specifically, it will be good to know whether the pvalue is above or below α.
How does that work?
For the facial height data test
H0: µ = 0 at significance level α = 0.05.
t83,0.975 = 1.99, so
prob = 0.05
P(t83 < -1.99) + P(t83 > 1.99) = .05
-1.99
P(t83 < -2.29) + P(t83 > 2.29)
< P(t83 < -1.99) + P(t83 > 1.99)
= .05
1.99
t83distribution
Since T = 2.29 > 1.99, this means
• If |T| < tn-1, 1-α/2 , then p-value > α. DO NOT REJECT H0
t83distribution
p-value
-2.29
2.29
3
6/19/2012
Example: chewing gum data
Do the data indicate at the α=0.01 significance level
that the mean DMFS has changed in Group A?
• Let µ = mean change in DMFS
• Test hypothesis H0: µ = 0
• Note: “µ = 0” would indicate “no change in DMFS”
• For Group A: n = 25, X̅ = -0.72, and s = 5.37.
• The t statistic is:
0.72 0
0.67
5.37⁄ 25
Example: chewing gum data
0.72
0
5.37⁄ 25
0.67
• Since α=0.01, we look up (1- α/2) = 99.5th percentile
of a t24 distribution.
• From Table 4, t24,0.995 = 2.80
• Since |T| = 0.67 < 2.80, DO NOT REJECT H0
t24distribution
• p-value = P(|t24| > .67) = 0.51
p-value
-0.67
Properties of a hypothesis test
Properties of a hypothesis test
Scenario #1
• The decision of a hypothesis test is based on the
random sample.
0.67
H0 is true
Scenario #2
H0 is not true
test
rejects H0
type I
error
test
rejects H0
OK
test does
not reject
H0
OK
test does
not reject
H0
type II
error
• The decision of the test can be incorrect.
• There are two possible scenarios, and two
possible test outcomes in each scenario:
•The probability of making a type I error is equal to the
significance level, α (which is decided upon by us)
•The probability of not making a type II error is called the
power of the test.
“Yes”: Reject H0
•If we reject H0 with, say, α=.05, the implication is clear.
•Because we set up our hypothesis test with a specified
α=.05, we know there is only a 5% chance that our
decision is incorrect (when H0 is true).
•Note that if we had rejected when α=.01, then we would
be even more sure that H0 is not true. There would only
be a 1% chance that we were incorrect.
Properties of a hypothesis test
Scenario #1
H0 is true
Scenario #2
H0 is not true
test
rejects H0
type I
error
test
rejects H0
OK
test does
not reject
H0
OK
test does
not reject
H0
type II
error
•The probability of making a type I error is called the
significance level, α.
•Hypothesis testing limits the probability making a type I error
4
6/19/2012
“No”: Do not reject H0
• Failure to reject H0 indicates that the null hypothesis
assumption is not clearly contradicted by the data.
Properties of a hypothesis test
Scenario #1
H0 is true
Scenario #2
H0 is not true
•This could be because H0 is true.
test
rejects H0
type I
error
test
rejects H0
OK
•It could also be that H0 is not true, but we just do not have
enough evidence to clearly indicate this. (In other words,
we made a type II error.)
test does
not reject
H0
OK
test does
not reject
H0
type II
error
•The probability of not making a type II error is called the
power of the test.
•Hypothesis testing does not control the probability of
making a type II error
Recap
• A hypothesis test allows you to make a decision
• The null hypothesis is always formulated in such a way
that it is the statement that we want to disprove.
• Reject H0 if the p-value < α
(“statistically significant”)
• A hypothesis test only controls the type I error
– If we reject H0 our evidence is clear
– If we fail to reject H0 then it is not as clear
5