Practical Statistics

Chi-Square Statistics

There are six statistics that willanswer 90% of all questions!1. Descriptive2. Chi-square3. Z-tests4. Comparison of Means5. Correlation6. Regression

Chi-square:

Chi-square is a simple test for counts…..

Which means: nominal dataand… if some cases…

Ordinal data

Chi-square:

There are three types:

1. Test for population variance2. Test of “goodness-of-fit”3. Contingency table analysis

Which is essentially a measure of association!

Chi-square:

There are three types:

1. Test for population variance

2

2

2

1

n S

Chi-square:

There are three types:

1. Test for population variance2. Test of “goodness-of-fit”

22

1

( )o e

ei i

ii

k

Where o = frequency of actual observation, and e = frequency you expected to find

22

1

( )o e

ei i

ii

k

Coin thrown 100 times:Expect (e): heads = 50, tails = 50Observed (o):

heads = 40, tails = 60

Is this a “fair” coin?

According to marketing research, the clienteleof a Monkey Shine Restaurant is made up of 30% Western businessmen, 30% women who stop in while shopping, 30% Chinese businessmen, and 10% tourists.

A random sample of 600 customers at the Kowloon Monkey Shine found 150 Western businessmen, 190 Chinese businessmen, 100 tourists, and

65 women who were shopping.

Is the clientele at this establishment different than the norm for this company?

Type Percent Expected

600

Observed

600

Western

Business

30%

180 150

Chinese

Business

30%

180 190

Women

Shoppers

30%

180 160

Tourist 10%

60

100

22

1

( )o e

ei i

ii

k

( ) ( ) ( ) ( )180 150

180

180 190

180

180 160

180

60 100

60

2 2 2 2

= 5.00 + 0.56 + 2.22 + 26.67 = 34.45

With (4-1) degrees of freedom

The chi-square distribution is highly skewedand dependent upon how many degrees offreedom (df) a problems has.

The chi-square for the restaurant problem was:

Chi-square = 34.45, df = 3

By looking in a table, the critical value of Chi-square with df = 3 is 7.82. The probability that the researched frequency equals the frequency found in the MR project was p < .001.

http://www.fourmilab.ch/rpkp/experiments/analysis/chiCalc.html

( ) ( ) ( ) ( )180 150

180

180 190

180

180 160

180

60 100

60

2 2 2 2

= 5.00 + 0.56 + 2.22 + 26.67 = 34.45 df = 3

By looking at the analysis, it is obvious thatthe largest contribution to chi-square came fromthe tourists.

Hence, the Kowloon property is attracting moretourist than what would be expected at the MonkeyShine.

Chi-square:

There are three types: 1. Test for population variance2. Test of “goodness-of-fit”3. Contingency table analysis

22

1

( )o e

ei i

ii

k

Where o = frequency of actual observation, and e = frequency you expected to find

A contingency table is a table with numbers grouped by frequency.

A contingency table is a table with numbers grouped by frequency.

Consider a study:

There are three groups: brand loyal customers,regular buyers, and occasional buyers.

Each is asked if they like the taste of new product over the old. They answer with a “yes” or a “no.”

A contingency table would look like this:

YES NO Totals

Loyal 50 40 90

Regular 60 40 100

Occasional 40 40 80

Total 150 120 270

A contingency table is a table with numbers grouped by frequency.

All the numbers in the table are “observed”frequencies (o).

So, what are the expected values?

YES NO Totals

Loyal 50 40 90

Regular 60 40 100

Occasional 40 40 80

Total 150 120 270

The expected values (e) would be a randomdistribution of frequencies.

YES NO Totals

Loyal 50 40 90Regular 60 40 100Occasional 40 40 80

Total 150 120 270

The expected values (e) would be a randomdistribution of frequencies. These can be calculatedby multiplying the row frequency by the column frequency and dividing by the total number of observations.

YES NO Totals

Loyal 50 40 90Regular 60 40 100Occasional 40 40 80

Total 150 120 270

For example, the expected values (e) of “loyal”and “yes” would be (150 X 90)/270 = 50

YES NO Totals

Loyal 50 40 90Regular 60 40 100Occasional 40 40 80

Total 150 120 270

For example, the expected values (e) of “regular”And “no” would be (120 X 100)/270 = 44.4

The expected values (e) for the entire tablewould be:

YES NO Totals

Loyal 50.0 40.0 90

Regular 55.6 44.4 100

Occasional 44.4 35.6 80

Total 150 120 270

The chi-square value is calculated for every cell,and then summed over all the cells.

YES NO Totals

Loyal 50.0 40.0 90

Regular 55.6 44.4 100

Occasional 44.4 35.6 80

Total 150 120 270

The chi-square value is calculated for every cell:For Cell A: (50-50)^2/50 = 0

For Cell D: (40-44.4)^2/44.4 = 0.44

YES NO Totals

Loyal A 50.0 40.0 90

Regular 55.6 D 44.4 100

Occasional 44.4 35.6 80

Total 150 120 270

The chi-square value is calculated for every cell:

YES NO Totals

Loyal 0 0

Regular .36 .44

Occasional .44 .55

Total

The chi-square value is calculated for every cell:Chi-square = 0 + 0 + .35 + .44 + .44 + .54 = 1.77The df = (r-1)(c-1) = 1 X 2 = 2

YES NO Totals

Loyal 0 0

Regular .35 .44

Occasional .44 .54

Total

A chi-square with a df = 2 has a critical valueof 5.99, this chi-square = 1.77, so the resultsare nonsignificant.

The probability = 0.4127. This means that the distribution is random, and there is no association between customer typeand taste preference.

http://www.fourmilab.ch/rpkp/experiments/analysis/chiCalc.html

A chi-square with a df = 2 has a critical valueof 5.99, this chi-square = 1.77, so the resultsare nonsignificant.

This means that the distribution is random, and there is no association between customer typeand taste preference.

Note: This type of chi-square is a test ofassociation using nothing but

counts (frequency);VERY useful in business research.

Service Encounter and Personality

Normally, 60% of our shoppers are women. Is our sample correct?0.6 X 271 = 163 women

.4 X 271 = 109 men

Service Encounter and Personality

Do men and women shop at different times?

Service Encounter and Personality

Do men and women shop at different times?