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.
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 ~ χ2n, where n = {(r - 1)(s - 1)}
Critical Value
Given level of significance as α, critical value is the upper α% point of χ2n.
p-value
P(χ2n > χ2cal)
Rejection Region
χ2cal > χ2crit 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 χ2cal > χ2crit, 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
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