Advanced High-School Mathematics

SECTION 6.6 χ^2 and Goodness of Fit 405

Active Ingredient

(in milligrams)

Pharmacy 1 2 3 4 5 6 7 8 9 10

Name brand 245 244 240 250 243 246 246 246 247 250

Generic brand 246 240 235 237 243 239 241 238 238 234

Based on the above data, what should be the consumer group’s

laboratory report about the difference in the active ingredient in

the two brands of pills? Give appropriate statistical evidence to

support your response.

6.6 χ^2 and Goodness of Fit

Perhaps an oversimplification, the χ^2 statistic gives us a means for
measuring the discrepancy between how we feel something ought to
behave versus how it actually appears to behave. In order to flesh
out this very cryptic characterization, suppose we have a die which we
believe to be fair, and roll it 200 times, with the outcomes as follows:

Outcome 1 2 3 4 5 6

No. of occurrences 33 40 39 28 36 24

Does this appear to be the way a fair die should behave? Is there a
statistic appropriate for us to measure whether its descrepancy from
“fairness” is significant?
Notice that the underlying null hypothesis would be that the die is
fair, expressed in the form

H 0 : p 1 =

1

6

, p 2 =

1

6

, p 3 =

1

6

, p 4 =

1

6

, p 5 =

1

6

, p 6 =

1

6

,

(where the probabilities have the obvious definitions) versus the alter-
native

Ha: at least one of the proportions exceeds

1

6

.

The appropriate test statistic, sometimes called theχ^2 statistic, is