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Chi-squareProcedures
Observedis score we get
Expected
we would get if there were no differences.
Chi-square formula
Once we have found the Chi square value, then we need to go to a Chi-square table of numbers for:
At the Level of Significance we set.
Chi-square value
Critical (or cut-off) value we must exceed
Chi-square formula
Director of recreation program wonders if there would be enough takers among his members to
warrant aBungee Cord Jumping program
He can ask all his customers. But what if 35 / 100 say yes. How will he decide if that’s a significant number.He decides to set up his inquiry as a scientific study, and so has a hypothetical question –There is not a significant difference between the yes and the nos. This is a null hypothesis, he assumes nothing.
His study will look something like this.
The Bungee Cord Jumping Dilemma
Determine the Chi-square statistic
from the data
Next Step
Categories Observed Expected
YES 20 50
NO 80 50Total 100 100
Chi-square calculationsOur Chi-square value
O- E O-E squared YES -30 900 NO 30 900
YES -30 900 900/50 = 18NO 30 900 900/50 = 18
Categories
Observed
Expected
O - E (O – E) 2 (O – E) 2E
YES 20 50 -30 900 / 50 18
NO 80 50 30 900 / 50 18
100 100 Chi-sq value
36
Chi-square calculations
1. Significance level was 0.52. Critical value at 0.5 is 3.85
3. Our Chi-square value is 364. This exceeds the critical value at the 0.5
level
5. We reject the null that there is no difference between the YES and NOs
Our Calculations
Two types = One – way or Two –way
One –way = One Column or Row (df = C – 1)
Different Forms of Table
Category Observed Expected
Yes
No
Category 1 2 3 4 5
Observed
Expected
Category 1 2 3 4 5
Observed 7 8 10 10 15
Expected 10 10 10 10 10
One-way calculations
N= 50
7-10 8-10- 10-10 10-10 10-10 10 10 10 10 10
9 4 0 0 25 .9 + .4 + 0 + 0 +25 3.8 10 10 10 10 10
O – E E
df = 5 - 1 = 4 Chi critical value = 9.49 No significant differences
Two-way tableCategory
Natural Observed
Natural Expected
Artificial Observed
ArtificialExpected
Margin Total
Won 39 21 60
Lost 6 14 20
MarginTotal
45 35 Total=80
How do we find the expected?Col + Row Margin
Grand Total
Nat/won = (60x45) / 80 = 33.75Nat/lost = (20 x 45) / 80 = 11.25Art /won = (60 x 35) / 80 = 26.25Art/ lost = (20 x 35 ) / 80 = 8.75
Two-way contCategory
Natural Observed
Natural Expected
Artificial Observed
ArtificialExpected
Margin Total
Won 39 33.75 21 26.5 60
Lost 6 11.25 14 8.75 20
MarginTotal
45 35 Total=80Nat/won = (60x45) / 80 = 33.75Nat/lost = (20 x 45) / 80 = 11.25Art /won = (60 x 35) / 80 = 26.25Art/ lost = (20 x 35 ) / 80 = 8.75Category Observed Expected (O – E) / E
Nat/ won 39 33.75 0.82
Nat/ lost 6 11.25 2.45
Art / won 21 26.25 1.05
Art / lost 14 8.75 3.15
Totals 80 80 X = 7.47
Two-way cont
Category Observed Expected (O – E) / E
Nat/ won 39 33.75 0.82
Nat/ lost 6 11.25 2.45
Art / won 21 26.25 1.05
Art / lost 14 8.75 3.15
Totals 80 80 X = 7.47
How we get df?
(C-1)(R-1)(2 -1)(2 –
1) = 1
Critical value at 0.05, 1 df = 3.84Our Chi = 7.47There is a significant differences between surfaces
Category 1 2 3 4 5
Observed 15 25 57 66 37
Expected 40 40 40 40 40
Homework One-way calculations
N= 50
15- 40 25-40 57-40 66-40 37-40 40 40 40 40 40
625 225 289 676 9 40 40 40 40 40
(O – E) 2 E
df = 5 - 1 = 4 Chi critical value = 9.49Reject NullThere are significant differences
45.6
Two-way tableCategory
YesObserved
YesExpected
NoObserved
NoExpected
Margin Total
18-39 45 44.34 35 35.67 80
40 - 59 56 44.34 24 35.67 80
60 + 32 44.34 48 35.67 80
MarginTotal
133 107 240
Col + Row Margin Grand Total
YES - 80 x 133 / 240 = 44.34
NO - 80 x 107 / 240 = 35.67
Df = ? Df = (c-1)(r-1) = (2-1)(3-1)= 1 x 2 = 2
YES45 - 44.34 /44.34 = (O. 66)2/44.34 = 0.01
56 - 44.34 /44.34 = 3.07 32 - 44.34 /44.34 =3.43
NO 35 - 35.67 / 35.67 = -(0.67)2/35.67 =
0.013 24 - 35.67 / 35.67 = 3.8 48 – 35.67 / 35.67 = 4.26 Sum all= 14.6 Critical value = 5.99 Reject Null
Chi value = (O – E) 2 E