Chapter 10 - two sample hypothesis test
Transcript of Chapter 10 - two sample hypothesis test
Two-Sample Hypothesis Test
Vo Duc Hoang Vu
University of Economics Ho Chi Minh City
April 16, 2014
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Chapter Contents
Two-Sample Tests
Comparing Two Means: Independent Samples
Confidence Interval for the Difference of Two Means, µ1 − µ2
Comparing Two Means: Paired Samples
Comparing Two Proportions
Confidence Interval for the Difference of Two Proportions, π1 − π2
Comparing Two Variances
Vo Duc Hoang Vu (ISB) Two-Sample Hypothesis Test April 16, 2014 2 / 17
Chapter Contents
Two-Sample Tests
Comparing Two Means: Independent Samples
Confidence Interval for the Difference of Two Means, µ1 − µ2
Comparing Two Means: Paired Samples
Comparing Two Proportions
Confidence Interval for the Difference of Two Proportions, π1 − π2
Comparing Two Variances
Vo Duc Hoang Vu (ISB) Two-Sample Hypothesis Test April 16, 2014 2 / 17
Chapter Contents
Two-Sample Tests
Comparing Two Means: Independent Samples
Confidence Interval for the Difference of Two Means, µ1 − µ2
Comparing Two Means: Paired Samples
Comparing Two Proportions
Confidence Interval for the Difference of Two Proportions, π1 − π2
Comparing Two Variances
Vo Duc Hoang Vu (ISB) Two-Sample Hypothesis Test April 16, 2014 2 / 17
Chapter Contents
Two-Sample Tests
Comparing Two Means: Independent Samples
Confidence Interval for the Difference of Two Means, µ1 − µ2
Comparing Two Means: Paired Samples
Comparing Two Proportions
Confidence Interval for the Difference of Two Proportions, π1 − π2
Comparing Two Variances
Vo Duc Hoang Vu (ISB) Two-Sample Hypothesis Test April 16, 2014 2 / 17
Chapter Contents
Two-Sample Tests
Comparing Two Means: Independent Samples
Confidence Interval for the Difference of Two Means, µ1 − µ2
Comparing Two Means: Paired Samples
Comparing Two Proportions
Confidence Interval for the Difference of Two Proportions, π1 − π2
Comparing Two Variances
Vo Duc Hoang Vu (ISB) Two-Sample Hypothesis Test April 16, 2014 2 / 17
Chapter Contents
Two-Sample Tests
Comparing Two Means: Independent Samples
Confidence Interval for the Difference of Two Means, µ1 − µ2
Comparing Two Means: Paired Samples
Comparing Two Proportions
Confidence Interval for the Difference of Two Proportions, π1 − π2
Comparing Two Variances
Vo Duc Hoang Vu (ISB) Two-Sample Hypothesis Test April 16, 2014 2 / 17
Chapter Contents
Two-Sample Tests
Comparing Two Means: Independent Samples
Confidence Interval for the Difference of Two Means, µ1 − µ2
Comparing Two Means: Paired Samples
Comparing Two Proportions
Confidence Interval for the Difference of Two Proportions, π1 − π2
Comparing Two Variances
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Learning Objectives - LO
1 Recognize and perform a test for two means with known σ1 and σ2
2 Recognize and perform a test for two means with unknown σ1 and σ2
3 Recognize paired data and be able to perform a paired t test
4 Explain the assumptions underlying the two-sample test of means
5 Perform a test to compare two proportions using z
6 Check whether normality may be assumed for tow proportions
7 Use Excel to find p-values for two-sample tests using z or t
8 Carry out a test of two variances using the F distribution
9 Construct a confidence interval for µ1 − µ2 or π1 − π2
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Two-sample Tests
What is a Two - Sample Test
A Two-sample test compares two sample estimates with each other.
A one-sample test compares a sample estimate to a non-sample
benchmark
Basis of Two-Sample Tests
The logic of two-sample tests is based on the fact that two samples
drawn from the same population may yield different estimates of a
parameter due to chance.
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Two-sample Tests
What is a Two - Sample Test
A Two-sample test compares two sample estimates with each other.
A one-sample test compares a sample estimate to a non-sample
benchmark
Basis of Two-Sample Tests
The logic of two-sample tests is based on the fact that two samples
drawn from the same population may yield different estimates of a
parameter due to chance.
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Two-Sample Tests
What is a Two-Sample Test
If the two sample statistics differ by more than the amount
attributable to chance, then we conclude that the samples came from
populations with different parameter values.
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Test Procedure
State the hypotheses
Set up the decision rule
Insert the sample statistics
Make a decision based on the critical values or using p − values
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Comparing Two Means: Independent Samples
Recognize and perform a test for two means with known σ1 and σ2
The hypotheses for comparing two independent population means
µ1 and µ2are :
Left-Tailed Test Two-Tailed Test Right-Tailed Test
H0 : µ1 − µ2 ≥ 0 H0 : µ1 − µ2 = 0 H0 : µ1 − µ2 ≤ 0
H1 : µ1 − µ2 < 0 H1 : µ1 − µ2 6= 0 H1 : µ1 − µ2 > 0
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Comparing Two Means: Independent Samples
LO4: Explain the assumptions underlying the two-sample test of
means:
Case 1: Known Variances
When the variances are known, use the normal distribution for the
test (assuming a normal population).
The test statistic is:
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Recognize and perform a test for two means with unknownσ1 and σ2
Case 2: Unknown Variances, assumed equal
Since the variances are unknown, they must be estimated and the
Students t distribution used to test the means.
Assuming the population variances are equal, s12 and s22 can be used
to estimate a common pooled variance s2p .
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Recognize and perform a test for two means with unknownσ1 and σ2
Case 3: Unknown Variances, assumed unequal
If the unknown variances are assumed to be unequal, they are not
pooled together.
In this case, the distribution of the random variable x̄1x̄2 is not certain
(Behrens-Fisher problem).
Use the Welch-Satterthwaite test which replaces
σ21 and σ22 with s21 and s22 in the known variance z formula, then use a
Students t test with adjusted degrees of freedom.
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Comparing Two Means: Independent Samples
Case 3: Unknown Variances, assumed unequal
Welch-Satterthwaite test
A Quick Rule for degrees of freedom is to use min(n1 − 1, n2 − 1).
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Comparing Two Means: Independent Samples
Summary for the Test Statistic
If the population variances σ21 and σ22 are known, then use the normal
distribution.
If population variances are unknown and estimated using s21 and s22 ,
then use the Students t distribution.
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Steps in Testing Two Means:
1 Step 1: State the hypotheses
2 Step 2: Specify the decision rule. Choose α (the level of significance)
and determine the critical value(s).
3 Step 3: Calculate the Test Statistic
4 Step 4: Make the decision Reject H0 if the test statistic falls in the
rejection region(s) as defined by the critical value(s).
5 Step 5: Take action based on the decision.
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Comparing Two Means: Independent Samples
Which Assumption is Best:
If the sample sizes are equal, the Case 2 and Case 3 test statistics will
be identical, although the degrees of freedom may differ.
If the variances are similar, the two tests will usually agree.
If no information about the population variances is available, then the
best choice is Case 3.
The fewer assumptions, the better.
Must Sample Sizes Be Equal?
Unequal sample sizes are common and the formulas still apply.
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Comparing Two Means: Independent Sample
Large Samples?
For unknown variances, if both samples are large
(n1 ≥ 30 and n2 ≥ 30) and the population is not badly skewed, use
the following formula with appendix C.
zcalc =x̄1 − x̄2√s21n1
+s22n2
(large samples, symmetric populations)
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Comparing Two Means: Independent Sample
Caution: Three Issues
1 Are the populations skewed? Are there outliers?
Check using histograms and/or dot plots of each sample. t-tests are
OK if moderately skewed, especially if samples are large. Outliers are
more serious.
2 Are the sample sizes large (n ≥ 30)?
If samples are small, the mean is not a reliable indicator of central
tendency and the test may lack power.
3 Is the difference important as well as significant?
A small difference in means or proportions could be significant if the
sample size is large.
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LO9: Confidence Interval for the Difference of Two Means:µ1 − µ2
Construct a confidence interval for µ1 − µ2 or π1 − π2Assuming equal variances:
(1 − x̄2)± tα/2
√(n1 − 1)s21 + (n2 − 1)s22
n1 + n2 − 2
√1
n1+
1
n2
Assuming unequal variances:
(1 − x̄2)±
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