Transcript H 0

Section 9.1(re-visited)
Making Sense of Statistical
Significance
Inference as Decision
Warm-up
 The one-sample t statistic for testing H0: μ = 0
and Ha: μ > 0 from a sample of n = 15
observations has the value t = 1.82
 What are the degrees of freedom for this statistic?
 Between what two values does the P-value of the test
fall?
 Is the value t = 1.82 significant at the 5% level? Is it
significant at the 1% level?
Practical Applications
 In practice, statistical tests are used for
marketing, research, and the
pharmaceutical industry.
 The decisions we make as statisticians
must have practical significance. This
means that it must be worthwhile to use
the information we find significant.
Points to Keep in Mind
 If you are going to make a decision based
on a statistical test, choose α in advance.
 When choosing α, ask these questions:
 Does H0 represent an assumption that people
have believed for years? If so, then strong
evidence (small α) is needed to persuade
them.
 What are the consequences of rejecting H0?
Costly changes will require strong evidence.
Statistical Significance
vs.
Practical Significance
 Statistical significance is based on the
hypothesis test.
 A large sample size will almost always
show that small deviations are significant.
 Why?
 Practical significance means the data isn’t
convincing enough to make a change.
Example of statistical significance
that is not practical
 Suppose we are testing a new antibacterial
cream, “Formulation NS” on a small cut made on
the inner forearm. We know from previous
research that with n medication, the mean
healing time (defined as the time for the scab to
fall off) is 7.6 days, with a standard deviation of
1.4 days. The claim we want to test here is that
Formulation NS speeds healing. We will use a
5% significance level.
 We cut 25 volunteer college students and apply
Formulation NS to the wound. The mean
healing time for these subjects is x-bar = 7.1
days. We will assume that σ = 1.4 days.
Solution
 We find that the data is statistically
significant.
 However, it does not appear that the effect
is all that great. Is it practical to use this
treatment if it only reduces the amount of
time you have a scab by about a day?
PROCEED
WITH…..
Cautions
 Watch out for badly designed surveys or
experiments!
 Statistical inference cannot correct for
basic flaws in design.
 Always plot the data (if it’s given to you)
and look for outliers or other deviations
from a consistent pattern.
Type I and Type II Errors
 Sometimes our decision (reject or fail to
reject H0) will be wrong.
 We could reject H0 when we shouldn’t
have.
 We could fail to reject H0 when we should
have.
Type I and Type II Errors
H0 is True
H0 is False
Reject H0
Type I Error

Fail to Reject H0

Type II Error
In words…
 Type I Error: Reject H0 when H0 is
actually true.
 Type II Error: Fail to reject H0 when H0 is
actually false.
Why do we care about errors?
 If a potato chip factory rejects bags of
chips that statistically fail to meet a salt
value, they lose money if the batch is
really ok.
 On the other hand, if they fail to reject a
batch that has too much salt, they will
have unhappy customers.
Probability of Type I error
 The probability of a Type I error occurring
is equal to alpha.
Find Probability of Type I error
 The mean salt content of a certain type of
potato chips is supposed to be 2.0mg. The
salt content of these chips varies normally
with standard deviation σ = 0.1mg. From
each batch produced, an inspector takes a
sample of 50 chips and measures the salt
content of each chip. The inspector rejects
the entire batch if the sample mean salt
content is significantly different from 2mg at
the 5% significance level.
I’ve Got the Power!
 Power is good!
 Power is the probability that a fixed α level
significance test will reject H0 if Ha is true.
 Power of a test = 1 – P(Type II Error)
 Power can increase by having a larger n.
More and more often, statisticians are
looking at the power of a study along with
confidence intervals and significance tests
Try this out for size
Test is
HIV +
Test is
HIV-
Have HIV Do not
have HIV
26
38
1
H0: A person
has HIV.
1235
 What is the alternative hypothesis?
 What is a Type I error? What is a Type II error?
 What is the probability of a Type I error? Type 2
error? Power?
 Error Probabilities
The potato-chip producer wonders whether the significance test of H0
: p = 0.08 versus Ha : p > 0.08 based on a random sample of 500
potatoes has enough power to detect a shipment with, say, 11%
blemished potatoes. In this case, a particular Type II error is to fail
to reject H0 : p = 0.08 when p = 0.11.
Earlier, we decided to reject
H0 at α = 0.05 if our sample
yielded a sample proportion
to the right of the green line.
( pˆ  0.0999)
Power and Type II Error
The power of a test against any alternative is
1 minus the probability of a Type II error
for that alternative; that is, power = 1 - β.

Significance Tests: The Basics
What if p = 0.11?
Since we reject H0 at α= 0.05
if our sample yields a
proportion > 0.0999, we’d
correctly reject the shipment
about 75% of the time.
Homework
Chapter 9
# 19 - 23