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