Question

In the middle of the 2016 football season the Seattle Seahawks got a new surface on their football field at Century Link Stadium. The Seahawks started the season playing on real grass and ended the season playing on artificial turf. The table below shows the win-lose-tie point record on the two types of grass.

	Win	Loss	Tie
Real Grass	1	3	1
Artificial Grass	9	2	0

Does it seem that the type of grass made the team have a better winning record? Or, do you think the season was a just a fluke? Explain your thinking.

Answer 1

Solution

Solution is based on Chi-square Test of Independence or also known as Contingency Chi-square Test.

Back-up Theory and Working

Suppose we have a contingency table with r rows representing r levels/grades of one attribute and c columns representing c levels/grades of another attribute.

The Chi-square Test of Independence is designed to test if the two attributes are associated.

In the given case, the two attributes are: Type of grass and Game status.

Hypotheses

Null H₀: The two attributes are independent Vs

Alternative H₁: The two attributes are not independent

Test Statistic

χ² = ∑(i = 1 to r, j = 1 to c){(Oij - Eij)²/Eij}, where Oij and Eij are respectively, the observed and the expected frequencies of the ijth cell of the contingency table.

Distribution

Under H₀, χ² ~ χ²_n, where n = {(r - 1)(s - 1)}

Critical Value

Given level of significance as α, critical value is the upper α% point of χ²_n.

p-value

P(χ²_n > χ²_cal)

Rejection Region

χ²_cal > χ²_crit or equivalently, p-value < α.

Calculations

Oij
Grass	Win	Loss	Tie	Oi.
Real	1	3	1	5
Artificial	9	2	0	11
Total	10	5	1	16
Eij
Grass	Win	Loss	Tie	Total
Real	3.125	1.5625	0.3125	5
Artificial	6.875	3.4375	0.6875	11
Total	10	5	1	16
Chi-square
Grass	Win	Loss	Tie	Total
Real	1.445	1.3225	1.5125	4.28
Artificial	0.656818	0.601136	0.6875	1.945455
Total	2.101818	1.923636	2.2	6.225455
α	0.05	DF	2
χ2cal	6.225455
χ2crit	5.991465
p-value	0.044479

Decision

Since χ²_cal > χ²_crit, H₀is rejectedor equivalently, p-value < α, H₀is rejected.

Conclusion

There is not enough evidence to suggest that The two attributes, namely, degree of interest in political election and level of education are independent. i.e.,

Game result is dependent on the type of grass. Answer

DONE