11-2 Goodness-of-Fit

In this section, we consider sample data consisting of observed frequency counts arranged in a single row or column (called a one-way frequency table). We will use a hypothesis test for the claim that the observed frequency counts agree with some claimed distribution, so that there is a good fit of the observed data with the claimed distribution.

Definition

A goodness-of-fit test is used to test the hypothesis that an observed frequency distribution fits (or conforms to) some claimed distribution.

Goodness-of-Fit Test

Notation

k n O E

represents the observed frequency from the sample data.

of an outcome, found represents the expected frequency of an outcome, found by assuming that the distribution is as claimed.

represents the number of different categories or cells.

represents the total number of trials .

Goodness-of-Fit

Hypotheses and Test Statistic

H

0 : The frequency counts agree with the claimed distribution.

H A

: The frequency counts do not agree with the claimed distribution.

x

2   (

O



E

) 2

E

Critical Values 1. Found in Table A-4 using

k –

freedom, where

k

1 degrees of = number of categories.

2. Goodness-of-fit hypothesis tests are always

right tailed

.

Finding Expected Frequencies

If all expected frequencies are assumed equal:

E



n k

If all expected frequencies are assumed not equal:

E



np

for each individual category

Goodness-of-Fit Test

A close agreement between observed and expected values will lead to a small value of

χ

2 and a large

P

-value. (Do Not Reject Ho.) A large disagreement between observed and expected values will lead to a large value of

χ

2 and a small

P

-value.

A significantly large value of

χ

2 will cause a rejection of the null hypothesis of no difference between the observed and the expected. (Reject Ho)

Goodness Of-Fit Tests

.

Example

A random sample of 100 weights of Californians is obtained, and the last digit of those weights are summarized on the next slide.

When obtaining weights, it is extremely important to actually measure the weights rather than ask people to self-report them.

By analyzing the last digit, we can verify the weights were actually measured since reported weights tend to be rounded to something ending with a 0 or a 5.

Test the claim that the sample is from a population of weights in which the last digits do not occur with the same frequency.

.

Example - Continued

.

.

Example - Continued

The hypotheses can be written as:

H

0 :

p

0 

p

1 

p

2 

p

3 

p

4 

p

5 

p

6 

p

7 

p

8 

p

9

H

1 : At least one of the probabilities is different. No significance level was specified, so we select

α

0.05.

=

.

.

Example - Continued

The calculation of the test statistic is given:

.

.

Example - Continued

The test statistic is

χ

2 = 212.800 and the critical value is

χ

2 = 16.919 (Table A-4).

.

.

Example - Continued

Since the = 212.8 > =16.919 we have SE to

TS

 2

CV

sufficient evidence to support the claim that the last digits do not occur with the same relative frequency.

In other words, we have evidence that the weights were self-reported by the subjects, and the subjects were not actually weighed.

.

.