Topics for Today
Example
Interpreting a p-value
Introducing  (again)
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Fall 2011 – Week 8 Lecture 2
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Complete Example for
Hypothesis test of a Single Mean
The Degree of Reading Power (DRP) is a test
that measures how well a reader understands
the meaning of text. Scores range from 0
(can’t read) to 100 (perfect understanding of
complicated literature).
http://www.apsva.us/cms/lib2/VA01000586/Centricity/Domain/12/drp_scale_of_text_difficulty.pdf
Someone in a particular school district
believes the DRP in their district is higher than
the national average (32). She takes a random
sample of 44 students from the district and has
them perform the test. She now wants to know
if the data supports her hypothesis.
Research Hypothesis:
Individual:
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Fall 2011 – Week 8 Lecture 2
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Population:
Variable of Interest:
Parameter of Interest:
Statistical Hypotheses:
Test Statistic:
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Determining the p-value:
First, sketch the picture.
Next, for a one-sided hypothesis test, look in
Table C (pg 520 … tho there is no page
number, and it’s the 2nd half of Table C).
Now, find the row closest to the degrees of
freedom we have (43), and look across until
you identify the two columns which bracket our
test statistic.
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Fall 2011 – Week 8 Lecture 2
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In this problem, the two values bracketing the
test statistic are from the row with df = 40, and
they are
…
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So … the p-value is somewhere between the
two probabilites associated with these values:
0.05 and 0.025
Conclusions:
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How small, the p-value
How _____ does a p-value have to consider
the null hypothesis ________?
We usually define our threshold for ‘unlikely’
similarly to the way we defined the confidence
level for our confidence interval.
By defining something called  … and usually
setting  to ____.
So, we say that if our p-value is smaller than
, the null hypothesis is ________ to be ____.
This is called ________________________.
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Fall 2011 – Week 8 Lecture 2
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From the previous example, on DRP, for a
 = 0.05, is the mean of students in the district
significantly higher than the national average?
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Interpret Another P-value
Let’s go back to the in-class age experiment,
calculate the p-value, compare it to  = 0.05
and make a conclusion.
test statistic:
p-value:
Statistically significant?
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Fall 2011 – Week 8 Lecture 2
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Decision Rules
These are synonymous
Here are the conclusions we draw based on
our p-value and the  level.
p-value < 
p-value > 
statistical
____________
no statistical
____________
evidence
supports the
___________
evidence
supports the
____
______ Null
Hypothesis
______________
the Null
Hypothesis
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What is  anyway?
First off, to reiterate,  is a pre-defined,
_________ value. Typically chosen to be __.
What does  mean?
 is the ___________ that we make a Type I
error.
Remember … just because we have a small pvalue, doesn’t mean we’ve definitely made the
right decision.
Null is True
Accept Null
Reject Null
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Fall 2011 – Week 8 Lecture 2
Valid, negative
result!
Type I Error
Null is False
Type II Error
Valid positive!
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So … how is  the probability of a Type I
Error?
Consider the following t-distribution. Shade the
region of the lowest 5%.
This region defines values of the test statistic
for which we’d reject the null hypothesis.
But … is it possible to get a test-statistic in this
region if the null is true?
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___! Which would mean we’d be rejecting the
null … but the null is ____.
Hence,  is the probability of making this type
of error, _______________.
So, setting  = 0.05 = 5% =
_______________.
Let’s do another example
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Example 4 Cont’d: Cable company
installation is not 45 minutes.
Statistical hypotheses:
H0: µ = 45
Ha: µ  45
Sample Statistics:
n = 20
s=
X=
Test-Statistic:
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p-value: (sketch the region first – but note, this
time we have a two-sided hypothesis)
Conclusion:
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Today’s Topics
Interpreting p-values
- represents how ‘consistent’ the sample is
with the null hypothesis
- p-value is the probability of obtaining the
observed test statistic if the null
hypothesis is true
- calculated differently for one-sided and
two-sided hypotheses
Decision Rules
- p-value < , reject null hypothesis
- p-value > , fail to reject null hypothesis
- reject the null is the same as “statistically
significant” difference

- defines the ‘evidence threshold’ for
significance
- is the P(Type I Error)
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Reading
Chapter 7 up to page 235
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