Special Case: Paired Sample T-Test

Examples Paired-sample?A. Car Radial Belted

1 ** ** Radial, Belted tires

2 ** ** placed on each car.

3 ** **

4 ** **

B. Person Pre Post

1 ** ** Pre- and post-test

2 ** ** administered to each

3 ** ** person.

4 ** **

C. Student Test1 Test2

1 ** ** 5 scores from test 1,

2 ** ** 5 scores from test 2.

3 ** **

4 ** **

Example*Nine steel plate girders were subjected to two methods for predicting sheer strength. Partial data are as follows:

Girder Karlsruhe Lehighdifference, d

1 1.186 1.061

2 1.151 0.992

9 1.559 1.052

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

• Compare χ2calc to χ2

crit

• Make a decision about H0

• State your conclusion

n

i i

ii

E

EO

1

22 )(

Tests of Independence

• HypothesesH0: independence

H1: not independent

• ExampleChoice of pension plan.

1. Develop a Contingency Table

Worker Type

Pension Plan

Total#1 #2 #3

Salaried 160 140 40 340

Hourly 40 60 60 160

Total 200 200 100 500

Example

2. Calculate expected probabilities

P(#1 ∩ S) = _______________ E(#1 ∩ S) = _____________

P(#1 ∩ H) = _______________ E(#1 ∩ H) = _____________

(etc.)

Worker Type

Pension Plan

Total#1 #2 #3

Salaried 160 140 40 340

Hourly 40 60 60 160

Total 200 200 100 500

#1 #2 #3

S (exp.)

H (exp.)

Hypotheses

3. Define Hypotheses

H0: the categories (worker & plan) are independent

H1: the categories are not independent

4. Calculate the sample-based statistic

= ________________________________________

= ______

n

i i

ii

E

EO

1

22 )(

The Test5. Compare to the critical statistic, χ2

α, 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