EGR 252 S09 Ch.10 Part 3 Slide 1

Statistical Hypothesis Testing - Part 3

A statistical hypothesis is an assertion concerning one or more populations.

In statistics, a hypothesis test is conducted on a set of two mutually exclusive statements:

H0 : null hypothesis

H1 : alternate hypothesis New test statistic of interest:

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EGR 252 S09 Ch.10 Part 3 Slide 2

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.

Example:

H0: The data come from a normal distribution.

H1: The data do not come from a normal distribution.

EGR 252 S09 Ch.10 Part 3 Slide 3

Goodness of Fit Tests (cont.)Test statistic is χ2

Draw the picture Determine the critical value for goodness of fit test

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.Discussion: Look at Table 10.4 in text.

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EGR 252 S09 Ch.10 Part 3 Slide 4

Tests of Independence Example: Choice of pension plan. Hypotheses

H0: Pension Plan Choice and Worker Type are independent

H1: Pension Plan Choice and Worker Type are not independent

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

EGR 252 S09 Ch.10 Part 3 Slide 5

Worker vs. Pension Plan 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.)

EGR 252 S09 Ch.10 Part 3 Slide 6

Hypotheses

3. Define Hypotheses

H0: the categories (worker & plan) are independent

H1: the categories are not independent

4. Calculate the sample-based statistic

(160-136)^2/136 + (140-136)^2/136 + (40-68)^2/68 + (40-64)^2/64 + (60-64)^2/64 + (60-32)^2/32

= 49.63

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EGR 252 S09 Ch.10 Part 3 Slide 7

The Chi-Squared Test of Independence

5. Compare to the critical statistic for a test of independence, χ2α, r

where r = (a – 1)(b – 1) a = # of columnsb = # of rows

For our example, let’s use α = 0.01 _

χ20.01,2_ = 9.210 (from Table A.5, pp 756)

Comparison: χ2 calc> χ2 crit

Decision: Reject the null hypothesis

Conclusion: Worker and plan are not independent.

EGR 252 S09 Ch.10 Part 3 Slide 8