Download Topic 10: Two-Sample Hypothesis Testing

ECON 4630
ECON 5630
TOPIC #10: TWO-SAMPLE HYPOTHESIS TESTING
I.
Introduction
A.
Motivation
Sometimes we are interested in comparing two separate samples. What we are
really interested in is the following question: are these two samples from the same
underlying population, or are they from two separate populations?
B.
II.
General Method
Tests When σ is Known
A.
Test statistic:
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B.
Example
A financial analyst is interested in comparing the turnover rates, in percent, for
shares of oil-related stocks versus other (non-oil) stocks. She selects 32 oilrelated stocks and 49 other stocks, and discovers that the mean turnover rate of
oil-related stocks is 31.4 and the mean sample turnover rate for the non-oil stocks
is 34.9. Suppose we know that the standard deviation of oil-related stocks is 5.1
and the standard deviation for non oil-related stocks 6.7. Is there a significant
difference in the turnover rates of the two kinds of stocks?
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III.
Tests When σ is Unknown
A.
Test statistic:
B.
Examples
Example 1:
Nurses (1)
541
590
521
471
550
559
525
529
X 1  535.75
Teachers (2)
545
526
527
575
484
509
502
520
529
530
542
532
X 2  526.75
Is the weekly salary of nurses higher? Test at the 90% confidence level.
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Student’s t-Distribution
df
0.100
1
2
3
4
5
0.020
3.078
1.886
1.638
1.533
1.476
Confidence Intervals
90%
95%
98%
99%
Level of Significance for One-Tailed Test
0.050
0.025
0.010
0.005
Level of Significance for Two-Tailed Test
0.10
0.05
0.02
0.01
6.314
12.706
31.821
63.657
2.920
4.303
6.965
9.925
2.353
3.182
4.541
5.841
2.132
2.776
3.747
4.604
2.015
2.571
3.365
4.032
6
7
8
9
10
1.440
1.415
1.397
1.383
1.372
1.943
1.895
1.860
1.833
1.812
2.447
2.365
2.306
2.262
2.228
3.143
2.998
2.896
2.821
2.764
3.707
3.499
3.355
3.250
3.169
5.959
5.408
5.041
4.781
4.587
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12
13
14
15
1.363
1.356
1.350
1.345
1.341
1.796
1.782
1.771
1.761
1.753
2.201
2.179
2.160
2.145
2.131
2.718
2.681
2.650
2.624
2.602
3.106
3.055
3.012
2.977
2.947
4.437
4.318
4.221
4.140
4.073
16
17
18
19
20
1.337
1.333
1.330
1.328
1.325
1.746
1.740
1.734
1.729
1.725
2.120
2.110
2.101
2.093
2.086
2.853
2.567
2.552
2.539
2.528
2.921
2.898
2.878
2.861
2.845
4.015
3.965
3.922
3.883
3.850
21
22
23
24
25
1.323
1.321
1.319
1.318
1.316
1.721
1.717
1.714
1.711
1.708
2.080
2.074
2.069
2.064
2.060
2.518
2.508
2.500
2.492
2.485
2.831
2.819
2.807
2.797
2.787
3.819
3.792
3.768
3.745
3.725
26
27
28
29
30
1.315
1.314
1.313
1.311
1.310
1.706
1.703
1.701
1.699
1.697
2.056
2.052
2.048
2.045
2.042
2.479
2.473
2.467
2.462
2.457
2.779
2.771
2.763
2.756
2.750
3.707
3.690
3.674
3.659
3.646
40
60
120

1.303
1.296
1.289
1.282
1.684
1.671
1.658
1.645
2.021
2.000
1.980
1.960
2.423
2.390
2.358
2.326
2.704
2.660
2.617
2.576
3.551
3.460
3.373
3.291
80%
99.9%
0.0005
0.001
636.619
31.599
12.924
8.610
6.869
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Example 2:
A sample of 15 male UNT Economics grad students has a mean of 659.3 and a
standard deviation of 78.5. A sample of 11 female UNT Economics grad students
has a mean of 685.5 and a standard deviation of 99.3. Test (at the 95% level) the
hypothesis that female graduate students have higher GRE quantitative scores
than males.
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IV.
Large Sample Tests Involving Proportions
A.
Test Statistic
B.
Example
The so-called “hot hand” is generally assumed to exist. That is, streak shoot is
typically believed to occur in which a player, having made three shots in a row is
more likely to make his or her fourth shot than to miss it. Is there statistical
evidence of this?
A sample from the 2002-2003 season reveals the following information about
Dallas Mavericks star Dirk Nowitzki:
Shooting percentage after 3 misses: 48.2% (sample size = 245)
Shooting percentage after 3 hits: 53.3% (sample size = 214)
Test the “hot hand” hypothesis at the 90% level.
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NOTES:
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