A P S T A T I S T I C S

C H A P T E R 2 5

Comparing Counts:Chi-Square Goodness-of-Fit Test

1

Most real life statistical problems have one or more nonstandard features. There are no routine statistical questions; only questionable statistical routines.

David R. Cox (1924 - )

Down and Dirty2

In the television series, “Mythbusters”, the cast conducts studies to test urban myths and oldwives-tales; based on their findings, the cast declares the myth Busted, Plausible or Confirmed. A 2013 episode, entitled “Down and Dirty” tested the theory that in a public restroom, the bathroom stall closest to the entrance is used the least often and contains the least amount of bacteria. The cast observed 119 people enter a public restroom with 4 stalls and recorded the stall number (#1 closest to entrance and #4 farthest from entrance). The tally of the users of each stall showed that the stalls were used by 23, 38, 34 and 24 people, respectively. To ascertain if there is evidence to support the myth, test the hypothesis that the stalls are selected randomly.

Down and Dirty3

Here are the observed counts (total of 119 participants):

Stall 1 2 3 4

Obs 23 38 34 24

Chi-Square Distribution4

We want to compare our observed data to what we would expect to see if there was nothing unusual (i.e. our null hypothesis).

Of course, small differences between the observed counts and the expected counts could just be natural sampling variation.

So we need a way to quantify how much variation there is and when is that variation large enough for us to be surprised.

Chi-Square Distribution5

When we worked with a single proportion, we created a test statistic by standardizing our sample proportion to the Standard Normal Model.

But to match several proportions to a hypothetical distribution we need a new sampling model. It is called the Chi-Square Distribution (Model).

With this model we will look at the differences between what is observed in the data and what we would expect under the null hypothesis.

Chi-Square Distribution6

All chi-square distributions are skewed to the right and described by one number: the degrees of freedom

Chi-Square Goodness-of-Fit Test (GOF) 7

Purpose of GOF

Used to compare the distribution of a single categorical variable to a hypothesized distribution.

Model Conditions

1. Randomness2. Independence3. Sample large enough to use appropriate model

(most expected counts > 5)

Test Statistic

2

2Obs Exp

Exp

Degrees of Freedom

are # categories - 1

Chi-Square Goodness-of-Fit Test (GOF) 8

The difference between each observed count and the corresponding expected count contributes to the total of the chi-square statistic:

2

2

2 2 2

1 1 2 22

1 2

n n

n

Obs Exp

Exp

Obs Exp Obs Exp Obs Exp

Exp Exp Exp

Chi-Square Goodness-of-Fit Test (GOF) 9

We will state the hypotheses verbally:

H0: proportions in the population are what is expected

Ha: one or more proportions differ from expected

Because of the squaring, the chi-square test always produces a positive value.

So, the larger the chi-square statistic, the stronger the evidence to reject the null hypothesis.

Down and Dirty10

To ascertain if there is evidence to support the myth, test the hypothesis that the stalls are selected randomly.

Stall 1 2 3 4

Obs 23 38 34 24

Exp

Down and Dirty11

Observed Expected Residuals Standardized

residuals

23 29.75 -6.75 -1.237543

38 29.75 8.25 1.5125525

34 29.75 4.25 0.77919372

24 29.75 -5.75 -1.0542033

N DF Chi-Square P-value

119 3 5.5378151 0.1364

Chi-Square goodness-of-fit

results:

Observed: Obs

Expected: Exp

The Common Core12

The vast majority of states and the District of Columbia

have adopted the Common Core State Standards

(CCSS) for math and English language arts. Do teachers

support the CCSS? In March 2003, The American

Federation of Teachers (AFT) asked AFT member

teachers “Based on what you know about the Common

Core State Standards and the expectations they set for

children, do you approve or disapprove of your state’s

decision to adopt them?”

The Common Core13

The national results were:

• Strongly approve 27%

• Somewhat approve 48%

• Somewhat disapprove 14%

• Strongly disapprove 8%

• Not sure 3%

The Common Core14

A district superintendent asked the same question to 230 teachers in her district to assess the level of teacher support for the CCSS within the district. She obtained the following results.

Response Stronglyapprove

Somewhatapprove

Somewhat disapprove

Stronglydisapprove

Not sure

Observed 55 106 28 32 9

Is there evidence that teachers in this district match the national approval distribution?

Assignment15

Read Chap 25

Chap 25 #3-13 odd

John Landers (www.landers.co.uk)