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Statistical Consulting
Program
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Hypothesis Testing
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Department of Science and Mathematics
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Montclair State University
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- About
Hypothesis Testing
Hypothesis Testing is a process
of evaluating validity of some claim made about a parameter or parameters
in a population. This claim, called the Null Hypothesis, is an assertion
about a particular value. Very often we test validity of claims made in
previous testing. It can also be a guess which an analyst chooses to validate.
The Null Hypothesis can be then an old estimation or someone's claim.
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Question 1: How does someone test a
claim?
An analyst collects a sample of individuals
from a population, calculates a particular sample parameter (mostly mean
or variance) to further determine the probability of the sample parameter
coming from a population with the claimed parameter. In other words, if
we are told that the average income in our town is $40,000, that statement
is considered a claim. How can we validate it? We can select a random group
of people in our town and ask them for their incomes. Then, after calculating
the average income of our sample, we can compare it to the given claim.
The procedure for testing a null
hypothesis or a claim often boils down to knowing how to interpret P-values,
especially if we have access to statistical software such as graphing calculators,
one of them being a TI-83 Plus, JMP and SAS. Small P-values provide evidence
again the null hypothesis because they indicate that it is unlikely for
the observed result (sample result) to come from a population with the
claimed mean or variance. Large P-values fail to give evidence necessary
to reject the null hypothesis.
*It is very important to remember
that the same P-value may yield a significant result in one case and a
non-significant one in another case, depending on the level of significance.
We choose the significance level according to our needs, based on how reliable
we want our results to be. The smaller the significance level is, however,
the more reliable are the results. And when we say "more reliable," we
mean that the likelihood of us making a mistake by rejecting the null hypothesis
is considerably small. The generally agreed upon traditional minimum standard
significance level for not accepting the hypothesized claim is a P-value
less than 5%. Statisticians also frequently use 10% and 1%, depending on
what the case might be.
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Questions 2 & 3: What does it mean
for 5% to be the significance level? When does the significance level
enable us to reject the hypothesized claim as valid?
Whenever the P-value is less
than the significance level, and we always determine the significance level
prior to testing, there exist sufficient evidence to claim that the null
hypothesis is wrong. Another words, we reject the null hypothesis as true.
The significance level is nothing else but the measure of risk that we
are willing to take when we reject a claim. The risk comes from the fact
that we might in fact be rejecting a true hypothesis. Some mistakes have
more severe consequences than others. Medicine is one of those fields that
rely on an incredible accuracy and precision of statistical results. The
significance level of our findings depends on the kind of risk we can afford
to take. In the light of our study's importance to us, we can decide upon
the level that best fits our case.
If we want to be 95% confident
that the null hypothesis is false, we risk 5%, and the significance level
becomes 0.05. However, if the nature of our analysis is to directly affect
people's lives, we might want to be at least 99% positive and risk only
1%. As mentioned above, the choice of a significance level depends on the
circumstances.
Example
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Suppose
it is early June and you just heard on TV that last year's average summer
noon temperature for your state was 85° F. In addition, the same weather
forecast stated that this year's summer should be the same as last year's.
People should then expect the average temperature at noon time to be 85°
F. You quickly make a comparison of last year's weather with this year's
and you realize that this year's winter and spring temperatures were much
lower than last year's. You begin to doubt the accuracy of this forecast,
expecting the summer temperatures to be lower than 85° F. You decide
to measure temperature throughout the whole summer to test 85° F. You will measure temperature every other day.
After
the summer is over.......Here are your average weekly temperature readings:
for the months of June (75, 76, 74, 80), July (80, 83, 85, 92), and August
(90, 92, 88, 85). How do we test whether the average temperature is 85°F?
Let's use some technological aids such as a TI-83 Plus graphing calculator
and JMP to generate our results. If you are interested in learning how
to do the analysis by yourself, scroll down for details.
Here
is what we were able to find using a TI-83 Plus graphing calculator:
  
The
averages were put into a list, which was later used in a t-test.
On
the right, there are similar results, obtained using the JMP software.
Look
in the Test Mean=value box of the output (last box on bottom right)
under "Prob < t."
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mean ~83.333
standard deviation
~6.43
t test statistic
= -0.898
Prob < t = 0.1942
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Analysis:
Null
Hypothesis: Average summer temperature = 85°F
Alternative
Hypothesis: Average summer temperature < 85°F
Let the significance level be 5% or 0.05.
Was
the average summer temperature this year less than 85°F ?
Observation:
P-value = .1942 > 0.05 ---> Failure to reject the Null Hypothesis
Conclusion:
At
the 5% significance level, there is not enough evidence to claim that the
average summer temperature this year was less than 85°F. |
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Go
to STANDARD TESTING above to learn about a t-test
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Click on P-VALUES above to learn
more about the P-values and the significance level.
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