Transcript Special Case: Paired Sample T-Test

Special Case: Paired Sample T-Test
Examples
A.
B.
C.
Car
Radial
1
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2
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3
**
4
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Person Pre
1
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2
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3
**
4
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Student Test1
1
**
2
**
3
**
4
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Paired-sample?
Belted
**
**
**
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Post
**
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**
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Test2
**
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Radial, Belted tires
placed on each car.
Pre- and post-test
administered to each
person.
5 scores from test 1,
5 scores from test 2.
Example*
Nine steel plate girders were subjected to two methods
for predicting sheer strength. Partial data are as follows:
Girder
1
2
Karlsruhe
1.186
1.151
Lehigh
1.061
0.992
9
1.559
1.052
difference, d
Conduct a paired-sample t-test at the 0.05 significance
level to determine if there is a difference between the
two methods.
* adapted from Montgomery & Runger, Applied Statistics and Probability for
Engineers.
Example (cont.)
Hypotheses:
H0: μD = 0
H1: μD ≠ 0
t__________
= ______
Calculate difference scores (d), mean and
standard deviation, and tcalc …
d = 0.2736
sd = 0.1356
tcalc = ______________________________
What does this mean?
• Draw the picture:
• Decision:
• Conclusion:
Goodness-of-Fit Tests
• Procedures for confirming or refuting hypotheses
about the distributions of random variables.
• Hypotheses:
H0: The population follows a particular distribution.
H1: The population does not follow the distribution.
Examples:
H0: The data come from a normal distribution.
H1: The data do not come from a normal distribution.
Goodness of Fit Tests (cont.)
• Test statistic is χ2
– Draw the picture
– Determine the critical value
χ2 with parameters α, ν = k – 1
• Calculate χ2 from the sample
2
(
O

E
)
i
2   i
Ei
i 1
n
• Compare χ2calc to χ2crit
• Make a decision about H0
• State your conclusion
Tests of Independence
• Hypotheses
H0: independence
H1: not independent
• Example
Choice of pension plan.
1. Develop a Contingency Table
Worker Type
Salaried
Hourly
Total
Pension Plan
#1
#2
160
140
40
200
60
200
#3
40
60
100
Total
340
160
500
Example
Worker Type
Salaried
Hourly
Total
Pension Plan
#1
#2
160
40
200
140
60
200
#3
40
60
100
Total
340
160
500
2. Calculate expected probabilities
P(#1 ∩ S) = _______________
E(#1 ∩ S) = _____________
P(#1 ∩ H) = _______________
(etc.)
E(#1 ∩ H) = _____________
#1
S (exp.)
H (exp.)
#2
#3
Hypotheses
3. Define Hypotheses
H0: the categories (worker & plan) are independent
H1: the categories are not independent
4. Calculate the sample-based statistic
2
(
O

E
)
i
2   i
Ei
i 1
n
= ________________________________________
= ______
The Test
5. Compare to the critical statistic, χ2α, r
where r = (a – 1)(b – 1)
for our example, say α = 0.01
χ2_____ = ___________
Decision:
Conclusion:
Homework for Wednesday, Nov. 10
• pp. 319-323: 25, 27
• Pp. 345-346: 12, 13
Homework