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Chapter 14
Introduction to Inference
Part B
1
Hypothesis Tests of
Statistical
“Significance”
• Test a claim about a parameter
• Often misunderstood
• Has an elaborate vocabulary
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4-step Process
Hypothesis Testing
Introduction to Inference
3
Step 1: Statement
• The illustrative example will
address whether people in a
population are gaining weight
• We collect an SRS of n = 10
individuals from the population
and determine the mean weight
change in the sample (x-bar)
• At what point do we declare that
an observed increase “statistically
significant” and applies to the
entire population?
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Step 2: Plan
1. Identify the parameter (in
this chapter we try to infer
population mean µ)
2. State the null and alternative
hypotheses (next slide)
3. Determine what test is
appropriate (in this chapter we
use the one-sample z test)
Introduction to Inference
5
Null and alt hypotheses H0 & Ha
• H0 = a claim of “no difference” or “no change”
• Ha = a claim of “difference” or “change”
• Ha can be one-sided or two-sided
• One-sided Ha specifies the direction of
the change (e.g., weight GAIN)
• Two-sided Ha does not specify the
direction of change (e.g., weight
CHANGE, either increase or decrease)
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Step 3: “Solve” has 3 sub-steps
1. Simple conditions (see prior lecture)
a) SRS
b) Normality
c) σknown before collecting data
2. Calculate test statistics
In this chapter  “z Statistic”
3. Find P-value
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Conditions
• Data were collected via SRS
• We know population standard
deviation σ= 1 for weight
change in current population
• If H0 is true, then the sampling
distribution of the mean
based on n = 10 will be Normal with
µ = 0 and

1
x 

 0.316
n
10
Introduction to Inference
8
What
• In practice we rely on
is the
?
and
to shed light on population
shape
may suggests a population is Normal
• If individual data points are available,
by making stemplot
• The shape of the data (sample) should parallel the
shape of the population.
• Beware that small samples have a lot of chance
variation.  It is is difficult to judge “Normality” in small
samples (but you can still check only for clear
departures)
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Test Statistic
Standardize the sample mean
zstat 
x  0

n
Suppose: x-bar = 1.02, n = 10, σ= 1
x  μ0
zstat 
σ
n
1.02  0

1
10
 3.23
 X-bar is ~3 standard deviations greater than expected under H0
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P-Value from Z Table
If Ha: μ > μ0
P-value = Pr(Z > zstat)
= right-tail beyond zstat
If Ha: μ < μ0
P-value = Pr(Z < zstat)
= left tail beyond zstat
If Ha: μ μ0
P-value = 2×one-tailed
P-value
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P-value from Z Table
• Draw
• One-sided P-value
= Pr(Z > 3.23)
= 1 − .9994
= .0006
• Two-sided P-value
= 2 × one-sided P
= 2 × .0006
= .0012
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P-value: Interpretation
• P-value ≡ probability data would take a value as extreme
or more extreme than observed data when H0 is true
• Measure of evidence: Smaller-and-smaller Pvalues → stronger-and-stronger evidence
against H0
• Conventions
.10 < P < 1.0  insignificant evidence against H0
.05 < P ≤ .10  marginally significant evidence vs. H0
.01 < P ≤ .05  significant evidence against H0
0 < P ≤ .01  highly significant evidence against H0
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“Significance Level”
• α (alpha) ≡ threshold for “significance”
• If we choose α = 0.05, we require evidence so
strong that a false rejection would occur no
more than 5% of the time when H0 is true
• Decision rule
P-value ≤ α  evidence is significant
P-value > α  evidence not significant
• For example, the two-sided P-value of 0.0012
is significant at α = .002 but not at α = .001
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Step 4: Conclusion
• The P-value of .0012 provides “highly
significant” evidence against H0: µ = 0
• Conclude: The sample demonstrates a
significant increase in weight (mean weight gain
= 1.02 pounds, P = .0012).
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Basics of Significance Testing
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Take out pencil and calculator
•
•
•
•
•
Reconsider the weight change example
Recall σ= 1
Now take a different sample of n = 10
This new sample has a mean of 0.3 lbs
Carry out the four-step solution to test
whether the population is gaining weight
based on this new data
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