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Tests of Significance:Stating Hypothesis; Testing Population Mean
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Tests of SignificanceUse a confidence interval to…
estimate a population parameter (µ)Use a Test of Significance to…
Assess the evidence provided by data about some claim concerning a population
Tim claims he can run a mile in under 6 minutes 80% of the time. You make Tim run a mile while you watch and it takes
him 20 minutes. You then call him a liar!!!You would use a significance test to determine the probability that
he would run a 20 minute mile if his claim of 6 minutes was actually true which would either prove or disprove his statement.
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Breaking Down a Significance Test
The question a significance test asks is:Does the sample result reflect
something that is a true result of the experiment OR
Is the result simply a result of CHANCE?We will investigate this question by
investigating two Hypotheses:
The Null Hypothesis (Ho) The Alternative Hypothesis (Ha)
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Quick Glance at the Hypotheses
Null Hypothesis (Ho)Says there is NO effect or NO change
in the populationAlternative Hypothesis (Ha)
The Alternative to no effect or no change
Often stated as “greater than” or “less than” the population
We are set out to decide between one of these Hypotheses…
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P-ValueThis value is the basis for our
conclusions about the Null Hypothesis
This value is the probability of being more extreme than the observed x-bar valueThe p-value is the probability of a result at least as far out as
the result we got.
The smaller the p-value, the stronger the evidence
AGAINST the Ho.
Why? The less likely your sample is to occur given
Ho
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More About PSmall p-values : p<.05
Give evidence against Ho because they say the observed result is unlikely to occur by chance (accept Ha)
Large p-values : p>.05Fail to give evidence (or reject)
against the Ho(we never actually accept Ho, we just fail
to reject it)Want to be Statistically Significant?
Have a p-value LESS than 0.05
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The Outline of a Significance Test
1. Describe the desired effect in terms of a parameter (i.e. - )
2. Write the Hypotheses3. Calculate a statistic (x-bar) that
estimates the parameter4. Find the p-value for that statistic
and decide if you should reject or fail to reject the null
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Stating HypothesesThe Null Hypothesis (Ho)
The statement being tested in a test of significance
Statement of “no effect” or “no difference”
We are testing the strength of the evidence AGAINST the Ho
Census Bureau data show that the mean household income in the area served by Turfland Mall is $42,500 per year. A
market research firm suspect the mean household income of mall shoppers is higher that of the general population.
Ho: = $42,500 (The population mean is $42,500 per year)
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Stating HypothesesThe Alternative Hypothesis (Ha)
The part of the claim you are investigating
Statement varies depending on problem
Either 1-sided (≥,≤,<,>) or 2-sided (≠)Census Bureau data show that the mean household income in
the area served by Turfland Mall is $42,500 per year. A market research firm suspect the mean household income of
mall shoppers is higher that of the general population.
Ha: > $42,500 (The population mean greater than $42,500 per year)
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Duracell BatteriesDuracell claims the average lifespan of
their Ultra Coppertop C Batteries is 22.3 hours with a standard deviation of 23 minutes. You take a SRS of 200 batteries and find the mean to be 21.4 hours. At a 5% significance level, is there enough evidence to refute Duracell’s claim? Ho: The mean lifetime of a Duracell Ultra Coppertop
C Battery is 22.3 hours. µ = 22.3hrs Ha: The mean lifetime of a Duracell Ultra Coppertop C Battery is less than 22.3 hours. µ < 22.3hrs
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Deciding Between 1 or 2 Sided Testing
One SidedExamination (stated in problem) is
looking for greater than, less than, or some specific side of the Ho
Two SidedExamination (stated in problem) is
vague; just looking at not equal to the mean, but not mentioning a specific side of the Ho
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Investigating the P-ValueP value is the probability that a
value as extreme or worse than what was observed could actually happenSmaller the P-value, the stronger
evidence AGAINST the HoFind the Sample Mean, find the z-score, and then find the appropriate area to represent p.
≠< or ≤> or ≥
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Extreme and SignificantSignificance Level (a)
Predetermined p-value that becomes the decisive value that determines your rejection of the Ho
Often a = .05
Statistical SignificanceP-value ≤ a
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Inference for population mean
Identify the population of interest & the parameter you want to draw conclusions about. State the null and alternative hypothesis
Choose the appropriate inference procedure and verify the conditions.
If the conditions are met, carry out the procedure Calculate the test statistic (z-value) Find the p-value and compare to a
Interpret your results in context
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Inference for population mean
(z-test for population mean Conditions needed for this procedure:
Testing H0: µ = µ0 Known σ and SRS from sample of size n
Now, let’s look at the components of the procedure:
Test Statistic (z-score)
nxbarz
/0
Let’s look at how to find the p-value…
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Finding the right pThe p-value is based on the Ha
Ha: µ > µ0 Ha: µ < µ0
Ha: µ ≠ µ0
One-sample tests
2-sided test
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Duracell BatteriesDuracell claims the average lifespan of
their Ultra Coppertop C Batteries is 22.3 hours with a standard deviation of 23 minutes. You take a SRS of 200 batteries and find the mean to be 21.4 hours. At a 5% significance level, is there enough evidence to refute Duracell’s claim? Ho: µ = 22.3hrs
Ha: µ ≤ 22.3hrs 2003833.96.14.21
Does their mean fit in our
CI?
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Z Tests for µ with Fixed Significance Level (alpha)
With these tests you are given an alpha level against which you test your p-valuep ≤ a – Reject the null; accept the Hap > a – Fail to reject the null
Instead of comparing to the standard a =. 05, we use the significance level the problem requires
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Confidence Interval & 2-Sided Tests
A two sided test is just looking to see if your value fits into a parallel confidence
interval For a 2 sided z-test…a = .05 for 2 sided
has .025 on each side
But… So does a 95% Confidence
Interval
So for a 2 sided z test...
You could construct a CI
to test for significance
And test to see if µ0 fits in your CI… If not, reject H0
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Duracell BatteriesDuracell claims the average lifespan of
their Ultra Coppertop C Batteries is 22.3 hours with a standard deviation of 23 minutes. You take a SRS of 200 batteries and find the mean to be 22.22 hours. At a 5% significance level, is there enough evidence to show the average is less than Duracell’s claim? Ho: µ = 22.3hrs
Ha: µ ≤ 22.3hrs
2003833.
3.2222.22z
z = -2.95 p =
What does this mean?
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Homework#43,44,46-49,54,55Next class – testing on calculator
and decisions