Transcript Test of Significance

Carrying Out
Significance Tests
Review of a Significance Test
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A test of significance is intended to assess the evidence
provided by data against a null hypothesis H0 in favor of
an alternate hypothesis Ha.
The statement being tested in a test of significance is
called the null hypothesis. Usually the null hypothesis is
a statement of “no effect” or “no difference.”
A one-sided alternate hypothesis exists when we are
interested only in deviations from the null hypothesis in
one direction
H0 : =0
Ha : >0 (or <0)
If the problem does not specify the direction of the
difference, the alternate hypothesis is two-sided
H0: =0
Ha: ≠0
CONDITIONS
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These should look the same as in the last
chapter (for confidence intervals)
– SRS
– Normality
 For means—population distribution is Normal or
you have a large sample size (n≥30)
 For proportions--np≥10 and n(1-p)≥10
– Independence
CAUTION
 Be
sure to check that the conditions
for running a significance test for
the population mean are satisfied
before you perform any
calculations.
INFERENCE TOOLBOX (p 705)
DO YOU REMEMBER WHAT THE STEPS ARE???
Steps for completing a SIGNIFICANCE TEST:
 1—PARAMETER—Identify the population of interest
and the parameter you want to draw a conclusion
about. STATE YOUR HYPOTHESES!
 2—CONDITIONS—Choose the appropriate inference
procedure. VERIFY conditions (SRS, Normality,
Independence) before using it.
 3—CALCULATIONS—If the conditions are met, carry
out the inference procedure.
 4—INTERPRETATION—Interpret your results in the
context of the problem. CONCLUSION, CONNECTION,
CONTEXT(meaning that our conclusion about the
parameter connects to our work in part 3 and includes
appropriate context)
Example 1-sided Test
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The diastolic blood pressure for American women
aged 18-44 has approximately the Normal
distribution with mean =75 milliliters of mercury
(mL Hg) and standard deviation σ=10 mL Hg.
We suspect that regular exercise will lower
blood pressure. A random sample of 25 women
who jog at least five miles a week gives sample
mean blood pressure x =71 mL Hg. Is this good
evidence that the mean diastolic blood pressure
for the population of regular exercisers is lower
than 75 mL Hg?
Step 1
The parameter of interest is the mean diastolic
blood pressure .
 Our null hypothesis is that the blood pressure is
no different for those that exercise.
 Our alternative hypothesis is one-sided because
we suspect that exercisers have lower blood
pressure.
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H0:  = 75
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Ha:  < 75
Step 2
Since we know the population standard deviation we
will be performing a z-test of significance.
 We were told that the sample is random, but we do
not know if it is an SRS from the population of
interest. This may limit our ability to generalize.
 Since the population distribution is approximately
Normal, we know that the sampling distribution of x
will also be approximately Normal. So we are safe
using the z procedures.
 The blood pressure measurements for the 25 joggers
should be independent. Note that the population of
interest is at least 10 times as large as the sample.
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Step 3
A curve should be drawn, labeled, and shaded.
 You can use the formula to calculate your z test
statistic for this problem
x
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
0
 z
In this case z = -2.00
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n
 Mark this on your sketch.
 Based on our calculations the P-value is 0.0228.
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x  71 , σ=10, n=25
Step 4
Since there is no predetermined level of significance
if we are seeking to make a decision, this could be
argued either way. If exercisers are no different, we
would get results this small or smaller about 2.28%
of the time by chance.
 This result is significant at the 5% level, but is not
signficant at the 1% level.
 We would likely reject H0.
 There is not much chance of obtaining a sample like
we did if there is no difference, so we would reject
the idea that there is no difference and conclude that
the mean diastolic blood pressure of American
women aged 18-44 that exercise regularly is probably
less than 75 mL Hg.
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DUALITY
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A level α two-sided significance test rejects a
hypothesis H0 : = 0 exactly when 0 falls outside
a level 1- α confidence interval for .
This relationship is EXACT for a TWO-SIDED
hypothesis test FOR A MEAN, but IS NOT EXACT FOR
tests involving PROPORTIONS.
 Essentially, if the parameter value given in the null
hypothesis falls inside the confidence interval, then
that value is plausible. If the parameter value lands
outside the confidence interval, then we have good
reason to doubt H0.
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