Week 9
October 27-31
Four Mini-Lectures
QMM 510
Fall 2014
Chapter 10
Two-Sample Hypothesis Tests
Chapter Contents
10.1 Two-Sample Tests
10.2 Comparing Two Means: Independent Samples
10.3 Confidence Interval for the Difference of Two Means, 1  2
10.4 Comparing Two Means: Paired Samples
10.5 Comparing Two Proportions
So many topics,
so little time …
10.6 Confidence Interval for the Difference of Two Proportions, 1  2
10.7 Comparing Two Variances
10-2
Chapter 10
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
nonsample benchmark.
Basis of Two-Sample Tests
•
Two-sample tests are especially useful because they possess
a built-in point of comparison.
•
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.
10-3
Chapter 10
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.
10-4
Chapter 10
Comparing Two Means:
Independent Samples
ML 9.1
Format of Hypotheses
•
The hypotheses for comparing two independent population means
µ1 and µ2 are:
10-5
Chapter 10
Comparing Two Means: Independent Samples
Case 1: Known Variances
•
When the population variances 12 and 22 are known, use the
normal distribution for the test (assuming a normal population).
•
The test statistic is:
10-6
Chapter 10
Comparing Two Means: Independent Samples
Case 2: Unknown Variances, Assumed Equal
•
If the variances are unknown, they must be estimated and the
Student’s 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 sp2.
10-7
Case 3: Unknown Variances, Assumed Unqual
•
If the population variances cannot be assumed equal, the distribution
of the random variable x1  x2 is uncertain (Behrens-Fisher problem)..
•
The Welch-Satterthwaite test addresses this difficulty by estimating
each variance separately and then adjusting the degrees of freedom.
10-8
A quick rule for degrees of freedom is to use min(n1 – 1, n2 – 1).
You will get smaller d.f. but avoid the tedious formula above.
Chapter 10
Comparing Two Means: Independent Samples
Test Statistic
•
•
•
If the population variances 12 and 22 are known, then use the normal
distribution. Of course, we rarely know 12 and 22 .
If population variances are unknown and estimated using s12 and s22,
then use the Student’s t distribution (Case 2 or Case 3)
If you are testing for zero difference of means (H0: µ1−µ2 = 0) the
formulas are simplified to:
10-9
Chapter 10
Comparing Two Means: Independent Samples
Chapter 10
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 and therefore the
p-values 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.
10-10
Large Samples
•
If both samples are large (n1  30 and n2  30) and the population is
not badly skewed, it is reasonable to assume normality for the
difference in sample means and use Appendix C.
• Assuming normality makes the test easier. However, it is not
conservative to replace t with z.
• Excel does the calculations, so we should use t whenever population
variances are unknown (i.e., almost always).
10-11
Chapter 10
Comparing Two Means: Independent Samples
Three Caveats:
• Are the populations severely skewed? Are there outliers? Check using
•
histograms and/or dot plots of each sample. t tests are OK if
moderately skewed, while outliers are more serious.
In small samples, the mean may not be a reliable indicator of
central tendency and the t-test will lack power.
• In large samples, a small difference in means could be “significant”
but may lack practical importance.
10-12
Chapter 10
Comparing Two Means: Independent Samples
Are the means equal?
Test the hypotheses:
Example: Order Size
H0: μ1 = μ2
H0: μ1 ≠ μ2
Summary statistics in 8
spreadsheet cells and use
MegaStat:
Assuming either Case 2 or
Case 3, we would not reject
H0 at α = .05 (because the pvalue exceeds .05)
Friday Saturday
22.32
25.56
4.35
6.16
13
18
Hypothesis Test: Independent Groups (t-test, pooled variance)
Friday Saturday
22.32
25.56 mean
4.35
6.16 std. dev.
13
18 n
29
-3.24000
30.07397
5.48397
1.99604
0
df
difference (Friday - Saturday)
pooled variance
pooled std. dev.
standard error of difference
hypothesized difference
Hypothesis Test: Independent Groups (t-test, unequal variance)
Friday Saturday
22.32
25.56 mean
4.35
6.16 std. dev.
13
18 n
28
-3.24000
1.88777
0
df
difference (Friday - Saturday)
standard error of difference
hypothesized difference
-1.716 t
.0972 p-value (two-tailed)
-1.623 t
.1154 p-value (two-tailed)
10-13
Chapter 10
Comparing Two Means: Independent Samples
ML 9.2
Paired Data
•
Data occur in matched pairs when the same item is observed
twice but under different circumstances.
•
For example, blood pressure is taken before and after a
treatment is given.
•
Paired data are typically displayed in columns.
10-14
Chapter 10
Comparing Two Means:
Paired Samples
Paired t Test
•
Paired data typically come from a before/after experiment.
•
In the paired t test, the difference between x1 and x2 is
measured as d = x1 – x2
•
The mean and standard deviation for the differences d are:
•
The test statistic becomes just a one-sample t-test.
10-15
Chapter 10
Comparing Two Means: Paired Samples
Chapter 10
Comparing Two Means: Paired Samples
Steps in Testing Paired Data
•
Step 1: State the hypotheses. For example:
H0: µd = 0
H1: µd ≠ 0
•
Step 2: Specify the decision rule. Choose  (the level of significance)
and determine the critical values from Appendix D or with use of Excel.
•
Step 3: Calculate the test statistic t.
•
Step 4: Make the decision. Reject H0 if the test statistic falls in the
rejection region(s) as defined by the critical values.
10-16
Chapter 10
Comparing Two Means: Paired Samples
Analogy to Confidence Interval
A two-tailed test for a zero difference is equivalent to asking whether
the confidence interval for the true mean difference µd includes zero.
10-17
Chapter 10
Comparing Two Means: Paired Samples
Example: Exam Scores
Using MegaStat:
Right-tailed test to see if mean scores improved
Name
Cecil
David
Edward
Fred
Gary
Henry
Post-Test
85
97
81
77
96
68
Pre-Test
79
87
78
82
96
69
Diff
6
10
3
-5
0
-1
Mean difference
St dev of differences
2.1667
5.3448
H 0: μ d = 0 (no change in mean)
t calc
0.9930
H 1: μ d > 0 (improved mean score)
t .05
2.015
0.1832
p -value
Hypothesis Test: Paired Observations
0.000 hypothesized value
84.000 mean Post-Test
Do not reject H 0 because t calc does not exceed t .05 (p > .05).
tcalc 
d
2.1667

sd / n (5.3448) / 6
81.833 mean Pre-Test
=T.DIST.RT(0.9930,5)
confidence interval includes
zero
2.167
5.345
2.182
6
5
mean difference (Post-Test - Pre-Test)
std. dev.
std. error
n
df
0.993 t
.1832 p-value (one-tailed, upper)
-3.442 confidence interval 95.% lower
7.776 confidence interval 95.% upper
5.609 margin of error
10-18
Chapter 10
Comparing Two Proportions
ML 9.3
Testing for Zero Difference: 1  2 = 0
To test for equality of two population proportions, 1, 2, use
the following hypotheses:
10-19
Chapter 10
Comparing Two Proportions
Testing for Zero Difference: 1  2 = 0
Sample Proportions
The sample proportion p1 is a point estimate of 1 and
p2 is a point estimate of 2:
10-20
Chapter 10
Comparing Two Proportions
Testing for Zero Difference: 1  2 = 0
Pooled Proportion
If H0 is true, there is no difference between 1 and 2, so the
samples are pooled (or averaged) in order to estimate the
common population proportion.
10-21
Chapter 10
Comparing Two Proportions
Testing for Zero Difference: 1  2 = 0
Test Statistic
•
•
If the samples are large, p1 – p2 may be assumed normally
distributed.
The test statistic is the difference of the sample proportions divided
by the standard error of the difference.
The standard error is calculated by using the pooled proportion.
•
The test statistic for the hypothesis 1  2 = 0 is:
•
10-22
Chapter 10
Comparing Two Proportions
Example: Hurricanes
p1 
p
x1 19
x
45

=.4130 p2  2 
 .6429
n1 46
n2 70
 2.435
x1  x2 19  45 64


 .5517
n1  n2 46  70 116
Hypothesis test for two independent proportions
… or using MegaStat:
p1
p2
0.413
0.6429
19/46
pc
0.5517 p (as decimal)
45/70 64/116 p (as fraction)
19.
45.
46
70
-0.2298
0.
0.0944
-2.435
.01491
64. X
116 n
difference
hypothesized difference
std. error
z
p-value (two-tailed)
-0.468 confidence interval 99.% lower
0.0084 confidence interval 99.% upper
0.2382 margin of error
=2*NORM.S.DIST(-2.435,1)
10-23
Chapter 10
Comparing Two Proportions
Testing for Zero Difference: 1  2 = 0
Checking for Normality
• We have assumed a normal distribution for the statistic p1 – p2.
• This assumption can be checked.
• For a test of two proportions, the criterion for normality is n  10 and
n(1 − )  10 for each sample, using each sample proportion in place of
.
• If either sample proportion is not normal, their difference cannot safely
be assumed normal.
• The sample size rule of thumb is equivalent to requiring that each
sample contains at least 10 “successes” and at least 10 “failures.”
10-24
Chapter 10
Comparing Two Proportions
Testing for Nonzero Difference
10-25
Chapter 10
Comparing Two Variances
ML 9.4
Format of Hypotheses
We may need to test whether two population variances are equal.
10-26
Chapter 10
Comparing Two Variances
The F Test
•
The test statistic is the ratio of the sample variances:
•
If the variances are equal, this ratio should be near unity: F = 1.
10-27
Chapter 10
Comparing Two Variances
The F Test
•
If the test statistic is far below 1 or above 1, we would reject the
hypothesis of equal population variances.
•
The numerator s12 has degrees of freedom df1 = n1 – 1 and the
denominator s22 has degrees of freedom df2 = n2 – 1.
•
The F distribution is skewed with mean > 1 and mode < 1.
Example: 5% right-tailed area for F11,8
10-28
Chapter 10
Comparing Two Variances
F Test: Critical Values
•
For a two-tailed test, critical values for the F test are denoted FL (left
tail) and FR (right tail).
•
A right-tail critical value FR may be found from Appendix F using df1 and
df2 degrees of freedom.
FR = Fdf1, df2
•
Excel function is:
=F.INV.RT(α, df1, df2)
A left-tail critical value FL may be found by reversing the numerator and
denominator degrees of freedom, finding the critical value from
Appendix F and taking its reciprocal:
FL = 1/Fdf2, df1
Excel function is:
=F.INV(α, df1, df2)
10-29
Chapter 10
Comparing Two Variances
Two-Tailed F-Test:
•
Step 1: State the hypotheses:
H0: 12 = 22
H1: 12 ≠ 22
•
Step 2: Specify the decision rule.
Degrees of freedom are:
Numerator: df1 = n1 – 1
Denominator: df2 = n2 – 1
Choose α and find the left-tail and right-tail critical values from
Appendix F or from Excel.
•
Step 3: Calculate the test statistic.
•
Step 4: Make the decision. Reject H0 if the test statistic falls in the
rejection regions as defined by the critical values.
10-30
One -Tailed F-Test
Example: 5% left-tailed area for F11,8
•
Step 1: State the hypotheses. For example:
H0: 12  22
H1: 12 < 22
•
Step 2: State the decision rule. Degrees of freedom are:
Numerator: df1 = n1 – 1
=F.INV(0.05,11,8)
Denominator: df2 = n2 – 1
Choose α and find the critical value from Appendix F or Excel.
•
Step 3: Calculate the test statistic.
•
Step 4: Make the decision. Reject H0 if the test statistic falls in the
rejection region as defined by the critical value.
10-31
Chapter 10
Comparing Two Variances
Chapter 10
Comparing Two Variances
EXCEL’s F Test
Note: Excel uses a left-tailed
test if s12 < s22
So, if you want a two-tailed
test, you must double Excel’s
one-tailed p-value.
Conversely, Excel uses a
right-tailed test if s12 > s22
10-32
Assumptions of the F Test
•
The F test assumes that the populations being sampled are
normal. It is sensitive to nonnormality of the sampled
populations.
•
MINITAB reports both the F test and a robust alternative called
Levene’s test along with its p-values.
10-33
Chapter 10
Comparing Two Variances