Today’s Agenda
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Review of ANOVA
Module 9
Review for Exam 2
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your participation will be graded by the
number of questions you answer (no
matter whether correct or not)
Second Midterm Exam
• 27th Mar. 6pm-7pm30 1324 EH
• Material:
Ch. 9-13: Inference for the big 5 parameters
HW 4-9 and all lab material
• Bring: PICTURE ID, Pen/pencils, erasers, and a
calculator. Formula cards with tables will be provided
with the exam.
• Alternate, randomized seating will be used in the exam rooms.
• No cell phones, beepers, palm pilots allowed in the exam rooms.
• All bags/backpacks will be placed against the exam room wall, so
the less extra things you bring, the better.
Comments on p-value
When using tables from yellow card…
• Z-statistic: the probability from z-table (of yellow card) corresponds
to the left direction
• T-statistic: the probability from t-table (of yellow card) corresponds to
the tail direction, or
to the left direction when t<0
to the right direction when t>0
• Then, one-minus or divided by two when needed.
When to tell whether there is a difference of
two population means
• Using hypothesis test with
H0: μ1=μ2 Ha: μ1≠μ2
and check whether p-value<α
• Constructing CI
and check whether 0 is included in CI
• When the significant level α and confidence level of CI are
consistent, e.g., 5% and 95%, two methods always give the same
conclusion
• When Ha is μ1>μ2 or μ1<μ2, the two methods do not always give
the same conclusion.
• In other words, when the question asks whether the mean of
population 1 is larger (smaller) than the mean of population 2, only
hypothesis test can be used.
ANOVA
• ANOVA (Analysis Of Variance): basically an extension of
two independent samples pooled t test
• Hypothesis:
H0: all the population means are equal, or
μ0 = μ1 = … = μk
Ha: at least one population mean is different
• Assumptions:
1) Normality of each population
2) The k groups have equal population variances
3) k independent random samples
ANOVA Table
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MSE is a good (unbiased) estimator of the common population
variance σ2
MSG is a good estimator for the variance σ2 only when the null
hypothesis for ANOVA F- Test is true
F-statistic is a ratio of two estimators
ANOVA
• F statistic has F(k-1,N-k) distribution
• When F is large (p-value is smaller than α), reject H0
(equal population means)
• P-value is the area to the right of the observed F-test
statistic value
• When H0 is rejected, use Tukey’s procedure to tell which
population mean(s) appear to be different
• Tukey’s procedure produces confidence intervals for the
differences in each pair of population means. We need
to check whether 0 is in the interval or not
Module 9: ANOVA
What is the appropriate
null hypothesis?
Ho: μ1-μ2-μ3 = 0
Ho: μ1=μ2=μ3 = 0
Ho: μ1=μ2=μ3
Any of the above would be appropriate.
What is the appropriate
alternative hypothesis?
H a : μ1 ≠ μ 2 ≠ μ 3
Ha: at least one is different
Ha: at least one μi is different
Any of the above would be appropriate.
Which provides an estimate of the
common population standard deviation
for the response?
Mean square groups
Mean square error
None of the above
What is the distribution of the test statistic if
there really is no difference between the popul.
average GPA for the three social classes?
N(0,1)
t(n-1)
t(n1+n2-2)
F(k-1,N-k)
Yes or No
Based on our decision, it would be
appropriate to use Tukey’s multiple
comparison procedure.
Select the pair(s) for which the popul.
means are significantly different.
Choose all that apply...
A) Lower class, Middle class
B) Lower class, Upper class
C) Middle class, Upper class
Exam 2 Review
Inference
• We’ve talked about two broad inference
procedures?
– Do you know what “inference” means?
– Do you know what the two general
procedures are?
– Do you know when it is appropriate to use
each one?
Suppose we want to estimate the
magnitude of an effect.
Which procedure would be best to use?
Confidence interval
Hypothesis test
Confidence Intervals (CI)
• Can you interpret the confidence interval?
• Can you interpret the confidence level?
• Do you know how to use a CI to make a
decision?
Hypothesis Tests (HT)
• What are the 5 steps to conducting a HT?
• Can you interpret the test statistic?
• Can you interpret the p-value?
Suppose we want to make a “maybe
yes” or “maybe no” type decision.
Which procedure would be best to use?
Confidence interval
Hypothesis test
The “Big 5”
• Do you know what this refers to?
• Do you know the distinction between the
following symbols? (Hint: 2 are parameters, 2 are statistics)
p̂
p
μ
x
• What other 3 parameters complete the “Big 5”?
Order the steps for conducting a
hypothesis test...
1)
2)
3)
4)
5)
Assumptions and test statistic
Conclusion
Find the p-value assuming Ho true
State the hypotheses
Decision
Which of the following tools could be
used to make a decision for a
two-sided hypothesis test?
A) Confidence interval
B) p-value
C) Either of the above
A 95% CI for the popul. mean is (35, 51). Which
of the following are correct interpretation(s) of
the confidence level? Choose all that apply...
A) The popul. mean will be in the interval
(35, 51) 95% of the time.
B) We are 95% confident that the popul.
mean is in the interval (35,51).
C) If the procedure were repeated many
times, we would expect 95% of the
resulting CIs to contain the popul. mean.
Which of the following are correct
interpretation(s) of the p-value?
Choose all that apply...
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The probability of observing a test
statistic as extreme or more extreme (in
the direction of Ha) than what we found,
assuming Ho is true.
The probability the null hypothesis is
true.
Which of the following inference
scenarios will NOT be on Exam 2?
Choose all that apply...
Single population proportion p
Single population mean 
Population mean difference d
Difference in 2 population means 1 - 2
Difference in 2 popul. proportions p1 - p2
ANOVA 1,…, k