Title: Hypothesis testing Chapter 8
1Lecture 23
Outline
- Hypothesis testing (Chapter 8)
- Introduction setup
- test statistics
- p-values
2Hypothesis testing introduction
- Recall in the cloud seeding example, we found
that a 95 confidence interval for - Based on this interval, do you believe
is
some differences are negative!
3Introduction to hypothesis testing
- Example paint drying time. A consumer
protection agency wants to test a claim that the
average drying time of a new fast-drying paint
is 20 minutes. A member of its research staff
takes 36 boards and paints each with paint from
36 different cans of paint and measures the time
it takes each board to dry. - How do we determine whether the paint drying time
is 20 minutes or longer?
4The hypothesis testing setup
- The null-hypothesis is that the mean drying time
is 20 minutes - this hypothesis is what we are trying to refute
- to do so we begin by assuming that it is true!
- to refute the null, we determine how likely is
our data (36 drying times) under the assumption
that the null hypothesis is true
5Paint drying time.
paint drying time
normal probability plot
see lecture 22, slide 16
6Test statistics
- Hypothesis tests are based on test statistics,
since and, underit seems reasonable to base
a test on - how reasonable is 2.94 under
t-distribution introduced in lecture 20, on
slide 11
7p-values
- A p-value is the probability of observing a
sample with a test statistic as extreme or more
extreme than the one we observed, assuming
is true.
one sided alternative
T35 distribution
two sided alternative
observed t
8p-values
- the p-value in the paint drying example is
very unlikely outcome if the average drying
time is in fact 20 minutes
9Example paint drying test
- Summary
- There is convincing evidence against the claim
that the mean paint drying time is 20 minutes
(p-value 0.0031, one sample t-test). The
sample average paint drying time in our
experiment was 20.758 minutes a 95 lower
confidence limit for the mean paint drying time
is 20.319 minutes.
10Notes about p-values
- The p-value is a measure of the statistical
significance of a test. Small p-values are
evidence against - The words as extreme or more extreme are
relative to the direction of - Interpretation A small p-value means it is
unlikely that we get a sample that gives us more
evidence against the null than the one we have
seen.
11Hypothesis testing procedure
- State the null and the alternative hypotheses
- Decide on a test statistic
- known distribution under null hypothesis
- Calculate the p-value (observed significance
level) - Report the results, evidence against
12Other hypothesis tests
- The crux of the testing procedure is deciding on
the test statistic (needs to have known
distribution) here are some examples
13Other hypothesis tests
14Example
- Recall the cloud seeding experiment from lecture
21. we wish to testvswhere - use the test statistic
15Example
- recall
- Summary There is convincing statistical evidence
the average log rainfall is greater for seeded
clouds than unseeded clouds (p-value 0.0072).
A 95 lower confidence bound for this difference
is 0.3903
interpretation?
16Notes on hypothesis testing
- Never misinterpret a p-value
- The 0.05 level of significance
- Statistical significance differs from practical
significance - The results of a test do not give the full story
and thus are usually accompanied by a confidence
interval - Relationship between hypothesis tests and
confidence intervals
17Some code
the t-test
lSeed_log(CloudsSeeded) lUSeed_log(CloudsUnseede
d) t.test(lSeed,lUSeed,pairedF,var.equalF,alt"g
reater")