Advanced High-School Mathematics

412 CHAPTER 6 Inferential Statistics

number in this category was found to be 69, then the contribution to
theχ^2 statistic would be

(69− 300 pq)^2

300 pq

.

Likewise, there would be three other contributions to theχ^2 statistic,
one for each “cell” in the above table.

However, it’s unlikely that we know the parameters of eitherX or
Y, so we use the data in the table to estimate these quantities. Clearly,
the most reasonable estimate ofp is ˆp =^147300 and the most reasonable
estimate forqis ˆq=^155300. This says that the estimated expected count of
those in the Male/Algebra category becomesE(n 11 ) = 300×^147300 ×^155300 =
147 · 155
300. This makes the corresponding to theχ

(^2) statistic
(n 11 −E(n 11 ))^2
E(n 11 )

=

(69(−^147300 ·^155 )^2

147 · 155

300

).

The fullχ^2 statistic in this example is

χ^2 = (n^11 −E(n^11 ))

2

E(n 11 ) +

(n 12 −E(n 12 ))^2

E(n 12 ) +

(n 21 −E(n 21 ))^2

E(n 21 ) +

(n 22 −E(n 22 ))^2

E(n 22 )

= (69−

147 · 155

Ä^300 )^2

147 · 155

300

ä +(86−

153 · 155

Ä^300 )^2

153 · 155

300

ä +(78−

147 · 145

Ä^300 )^2

147 · 145

300

ä +(67−

153 · 145

Ä^300 )^2

153 · 145

300

ä

≈ 2. 58.

We mention finally, that the aboveχ^2 has only 1 degree of freedom: this
is the number of rows minus 1 times the number of columns minus 1.
TheP-value associated with the above result isP(χ^21 ≥ 2 .58) = 0.108.
Note this this result puts us in somewhat murky waters, it’s small
(significant) but perhaps not small enough to reject the null hypothesis
of independence. Maybe another survey is called for!

In general, given a two-way contingency table, we wish to assess
whether the random variables defined by the rows and the columns