Transcript Inference: Hypothesis Test

Hypothesis Testing
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An understanding of the method of
hypothesis testing is essential for
understanding how both the natural and
social sciences advance.
In science one begins with a theory, then
collects data (hopefully under carefully
controlled conditions) and asks the central
question:
Does the data fit the theory?
Does the data fit the Theory/Model?
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This question is not as easily answered as
you might think.
As we know, samples vary, measurements
almost always contain small errors, so it is
unreasonable to expect exact agreement with
a theory/model based upon actual
observations.
When can we say that the sample we have
carefully collected does or does not fit the
theory/model?
Today we focus on the population mean
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We believe we know the true population
mean
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We collect a sample and the sample average
differs from what we believe the true
population mean to be.
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Does this mean we have the wrong
population mean, or is there a difference just
because samples vary?
When can we say that the sample we have
carefully collected does not fit the model?
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There is no one answer:
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Rather, we calculate the probability that a random sample
would vary from that predicted by the theory/model by as
much or more than the value we obtained. (This value is
called the p-value.)
For example:
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If we believe that the mean height of students at CSUMB is
5’6” we can collect a sample and calculate how likely it is
that the average height of a sample of this size would vary
from 5’6” by as much or more than that of our sample.
The average height we calculate from the sample is called
the test statistic.
Ho determines the model. Small p-values
indicate the sample data does not fit the
model
What Model do we use?
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We have seen that the average of almost all large
samples (of size n) is modeled by a normal
distribution with mean equal to the population mean
and standard deviation:

x
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x n
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So, if we know the population standard deviation,
we have the two parameters needed for our model
and we can ask if the data fits the model.
If we do not know the population standard deviation,
we can use the sample standard deviation as long
as the sample is large. (Generally this means > 25.)
Hypothesis Testing about The Mean
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The model is defined by the parameters
mean µ and standard deviation σ.
Since we can use the sample standard
deviation in place of σ, we really only have
one assumption: That we know the mean of
the population.
This assumption µ = µ0 (a known value) is
called the null hypothesis and is designated
Ho
Hypothesis Testing about The Mean
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The model is defined by the null hypothesis
Ho :µ = µ0
(in our example of student heights µ0 =5’ 6")
If the null hypothesis is not true then one of the following
alternative hypotheses (HA) must be true:
µ < µ0 , or µ > µ0
If we have no idea which of these to expect, we can
state the alternative hypothesis as HA:µ ≠ µ0
although this is rarely used and I discourage you
from ever using it in practice.
Inference: Null Hypothesis
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The null hypothesis (H0) is the hypothesis/theory
that is being tested.
H0 can never be proved, only disproved!
This is how the sciences advance, by disproving a theory
with data and suggesting an alternative theory that seems
to agree with the data.
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It is always a statement of the value of a population
parameter. E.g., H0:  = 0 signifies the population
mean has the value 0.
H0 is presumed TRUE until there is sufficient
evidence to reject it.
The Big Idea
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The null hypothesis provides us with a model for the
population from which the sample is selected.
A sample is collected, and the sample average (test
statistic) is compared to the population parameter.
In other words, we place the average of the sample
on the model and ask how reasonable is it that we
obtain a test statistic that varies from 0 by this
much or more.
Generally values that are within 2 standard
deviations of the (assumed) mean are considered
reasonable.
The Model is determined by Ho
Quantifying the improbable: p-value
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p-value: The probability of observing, when
the null hypothesis is true, a value of the test
statistic that is as extreme or more extreme
than the value observed. (memorize this!)
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In the preceding example, the value 0.139 is
the p-value of the test.
Inference: Statistical significance
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Traditionally, the decision to reject H0 was based
upon selection of a level of significance () used to
derive a critical value for the test statistic. The
critical value set a gating value beyond/beneath
which a test statistic must fall in order that H0 may
be rejected.
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Most technology tools produce p-values directly.
The p-values carry more information about the test
statistic, since they enable reporting the smallest
possible significance level for which the results are
statistically significant.
Inference: Conducting a Hypothesis Test (5 steps)
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Identify the parameter to be tested and state the two
hypotheses in symbolic terms.
Restate the hypotheses in context of the problem.
Analyze the sample data and report the p-value of
your test.
Interpret the results: Does the data provide evidence
against Ho? At which of the standard confidence
levels should you reject Ho?
State the conclusion in the context of the problem.
Example 1
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Standards set by government agencies indicate
that Americans should not exceed an average
daily sodium intake of 3300 milligrams (mg). To
find out whether Americans are exceeding this
limit, a sample of 100 Americans is selected.
The mean and standard deviation of daily
sodium intake are found to be 3400 mg and
1100 mg, respectively.
Inference: Conducting a Hypothesis
Test (State H0, HA)
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H0:  = 3300 mg
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Americans’ average daily sodium intake is 3300
mg.
HA:  > 3300 mg
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Americans’ average daily sodium intake exceeds
3300 mg.
The Test Statistic
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The test statistic is the value produced from
the sample. We place this value on our model
(a normal distribution with mean 3300 and
standard deviation: 1100/ 100  110 )
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The p-value is the probability of getting a
value of 3400 or larger on this normal curve.
The Test Statistic
1100/ 100  110
Conclusion
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The p-value represents the chance of getting a
value as high or higher than 3400 when the true
average is 3300.
The p-value of .1814 means there is an 18.14%
chance that whenever we conduct a similar
experiment we would find a sample average of 3400
or higher.
We conclude that there is not enough evidence to
show that Americans’ average daily sodium intake
exceeds 3300 mg.
(1-Confidence level)= significance level
Inference: Guidelines & Language of
Statistical Significance
Range of p-value
.01 > p-value
.05 > p-value ≥ .01
.10 > p-value ≥ .05
p-value ≥ .10
Level of significance
Results are highly
significant.
Results are statistically
significant.
Results tend toward
statistical significance.
(H0 usually not rejected)
Results are not statistically
significant.
(H0 not rejected)
Confidence Levels
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If p-value is less than 0.1, reject Ho at 90%
confidence level, otherwise keep Ho.
If p-value is less than 0.05, reject Ho at both
90% and 95% level, otherwise keep Ho.
If p-value is less than 0.01, reject Ho at 90%,
95%, and 99% levels, otherwise keep Ho.
Example2: Water Quality
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An environmentalist group knows that historically a certain
stream has had a dissolved oxygen content of 5 mg per liter, with
 = 0.92 mg. The group collects a liter of water from each of 45
random locations along a stream and measures the amount of
dissolved oxygen in each specimen. The sample mean is 4.801
milligrams (mg) per liter.
Is this strong evidence that the stream has a mean dissolved
oxygen content of less than 5 mg per liter?
1. State the hypotheses
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0 = 5 mg/liter
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H0:  = 0
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The historical mean of the population
The null hypothesis claim is that the true dissolved oxygen
level will be exactly 5 mg/liter
HA:  < 0
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The alternative hypothesis states that the true dissolved
oxygen level will be less than 5 mg/liter
2. Calculate the test statistic
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Use the values…
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Sample mean is 4.801 mg/liter
Historical mean is 5 mg/liter
Population standard deviation is 0.92 mg/liter
Sample size is 45
The standard error for the sample
0.92 / 45
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Since the sample average is below the mean,
4.801 is an upper limit.
3. Find the p-value
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The p-value is the probability of getting a value of
4.801 or smaller on the normal curve with mean=5
and standard deviation=0.1371.
Using the normal worksheet you can find the pvalue is about 0.073.
. Interpret the Results
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p-value= 0.073
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A mean sample level as low as 4.801
mg/liter would occur 7.3% of the time if the
true population mean were still 5 mg/liter.
This is modest evidence that the true
mean dissolved oxygen level is less than 5
mg/liter.
5. State your conclusion in the context of the
problem
p-value= 0.0735
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Significance level  = 0.10
 0.0735 < 0.10
 Reject the null hypothesis and accept the alternative
hypothesis
Significance level  = 0.05
 0.0735 > 0.05
 Fail to reject the null hypothesis