Download STA 291 Summer 2010

Lecture 16
Dustin Lueker

Charlie claims that the average commute of his
coworkers is 15 miles. Stu believes it is greater
than that so he decides to ask some of his
coworkers what their commute is. He asks 36 of
them and finds that their average commute is
16.88 miles with a standard deviation of 6 miles.
◦ Does this prove that Stu is correct and the average
commute is greater than 15 miles?
 If not how could you explain the sample mean being greater
than 15 if the true, population mean (all the coworkers)
isn’t?
◦ Can we use anything we have already learned to
investigate this further?
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
A way of statistically testing a hypothesis by
comparing the data to values predicted by the
hypothesis
◦ Data that fall far from the predicted values provide
evidence against the hypothesis
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1.
2.
State a hypothesis that you would like to
find evidence against
Get data and calculate a statistic
1. Sample mean
2. Sample proportion
3.
4.
Hypothesis determines the sampling
distribution of our statistic
If the sample value is very unreasonable
given our initial hypothesis, then we
conclude that the hypothesis is wrong
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
H0: μ=μ0
◦ μ0 is the value we are testing against

H1: μ≠μ0
◦ Most common alternative hypothesis
 This is called a two-sided hypothesis since it includes
values falling on two sides of the null hypothesis
(above and below)
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
The research hypothesis is usually the
alternative hypothesis
◦ The alternative is the hypothesis that we want to
prove by rejecting the null hypothesis

Assume that we want to prove that μ is larger
than a particular number μ0
◦ We need a one-sided test with hypotheses
H 0 :   0
H 0 :   0
H1 :    0
H1 :    0
 Null hypothesis can also be written with an equal sign
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
Assumptions

Hypotheses
◦ Type of data, population distribution, sample size
◦ Null hypothesis
 H0
◦ Alternative hypothesis
 H1

Test Statistic

P-value

Conclusion
◦ Compares point estimate to parameter value under the null hypothesis
◦ Uses the sampling distribution to quantify evidence against null
hypothesis
◦ Small p-value is more contradictory
◦ Report p-value
◦ Make formal rejection decision (optional)
 Useful for those that are not familiar with hypothesis testing
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
The z-score has a standard normal
x  0
distribution
z
s
n
◦ The z-score measures how many estimated
standard errors the sample mean falls from the
hypothesized population mean

The farther the sample mean falls from  0 the
larger the absolute value of the z test
statistic, and the stronger the evidence
against the null hypothesis
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
How unusual is the observed test statistic
when the null hypothesis is assumed true?
◦ The p-value is the probability, assuming that the
null hypothesis is true, that the test statistic takes
values at least as contradictory to the null
hypothesis as the value actually observed
 The smaller the p-value, the more strongly the data
contradicts the null hypothesis
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
Has the advantage that different test results
from different tests can be compared
◦ Always a number between 0 and 1, no matter what
type of data is being examined


Probability that a standard normal
distribution takes values more extreme than
the observed z-score
The smaller the p-value, the stronger the
evidence against the null hypothesis and in
favor of the alternative hypothesis
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
In addition to reporting the p-value,
sometimes a formal decision is made about
rejecting or not rejecting the null hypothesis
◦ Most studies require small p-values like p<.05 or
p<.01 as significant evidence against the null
hypothesis
 “The results are significant at the 5% level”
 α=.05
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
Charlie claims that the average commute of
his coworkers is 15 miles. Stu believes it is
greater than that so he decides to ask some
of his coworkers what their commute is. He
asks 36 of them and finds that their average
commute is 16.88 miles with a standard
deviation of 6 miles.
◦ Construct a hypothesis test to see if Stu is correct
using the P-Value method with a 5% level of
significance
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
p-value<.01
◦ Highly significant
 “Overwhelming evidence”

.01<p-value<.05
◦ Significant
 “Strong evidence”

.05<p-value<.1
◦ Not Significant
 “Weak evidence

p-value>.1
◦ Not Significant
 “No evidence”
 Whether or not a p-value is considered significant typically depends
on the discipline that is conducting the study
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
Significance level
◦ Alpha level
 α
 Number such that one rejects the null hypothesis if the
p-values is less than it
 Most common are .05 and .01
◦ Needs to be chosen before analyzing the data
 Why?
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Decision
True
Condition
of H0
False
Reject H0
Do Not
Reject H0
Type I
Error
Correct
Correct
Type II
Error
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

α=probability of Type I error
β=probability of Type II error
Power=1-β
◦ The smaller the probability of Type I error, the
larger the probability of Type II error and the
smaller the power
 If you ask for very strong evidence to reject the null
hypothesis (very small α), it is more likely that you fail
to detect a real difference

In reality, α is specified, and the probability
of Type II error could be calculated, but the
calculations are often difficult
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
In a criminal trial someone is assumed innocent
until proven guilty
◦ What type of error (in terms of hypothesis testing) would
be made if an innocent person is found guilty?
◦ What type of error would be made if a guilty person is
found not guilty?
◦ What does the Power represent (1-β)?
 Also, the reason we only do not reject H0 instead of saying
that we accept H0 is because of the way our hypothesis tests
are set up
 Just like in a criminal trial someone is found not guilty
instead of innocent
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
If the consequences of a Type I error are very
serious, then α should be small
◦ Criminal trial example


In exploratory research, often a larger
probability of Type I error is acceptable
If the sample size increases, both error
probabilities decrease
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
Which area of study would be most likely to
use a very small level of significance?
◦ Social Sciences
◦ Medical
◦ Physical Sciences
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