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Extra Credit

You sell two types of chocolate, dark and light. You know that nationally more people buy light chocolate than dark, but you are wondering if this preference is stronger in one part of the county than another. You gather information on how many tons of chocolate were sold in the western, middle, and eastern parts of the country from the CCCCC (Cross Country Commission on Chocolate Consumption). Conduct a full hypothesis test using a chi-square (χ2) test at the alpha (α) = .05 level.

Part of Country

West

Middle

East

Dark

65

72

61

Light

91

155

145

Ho: There is no relationship between chocolate preference and the part of the country where one lives.

H1: There is a relationship between chocolate preference and the part of the country where one lives.

Alpha Level: Alpha (α) = .05

Rejection Rule: Reject H0 if chi-square computed (χ2computed) > 5.99. The degrees of freedom equals the number of columns of data minus 1 (3 - 1 = 2, in this case) times the number of rows of data minus 1 (2 - 1 = 1, in this case). Multiplying these (2 × 1) gives us 2 degrees of freedom. The general formula is written as (R - 1) (C - 1) = df. We then find the critical value in Table G on page 409, replicated, in part, below.

df

5%

1%

1

3.84

6.64

2

5.99

9.21

3

7.82

11.34

4

9.49

13.28

5

11.07

15.09

6

12.59

16.81

Computation: First we total all rows and all columns.

Part of Country

West

Middle

East

Total

Dark

65

72

61

198

Light

91

155

145

391

Total

156

227

206

589

Then we do something that is really bizarre, we multiply the row total by the column total then divide this product by the grand total (589, in our case) to obtain an expected number for each cell.

Part of Country

West

Middle

East

Total


Dark

 

Expect

65

52.44143

72

76.309

61

69.24958

 

198


Light

 

Expect

91

103.5586

155

150.691

145

136.7504

 

391

Total

156

227

206

589

Once you obtain the expected value (E) for each cell, you subtract the expected value from the observed value (O). You then square that difference for each cell. You then divide that squared difference by the expected value. You sum this last result for all of the cells and you have computed the Chi-Square.

O

E

O - E

(O - E)2

(O - E)2/E

65

52.441

12.559

157.7285

3.007732

72

76.309

-4.309

18.56748

0.243320

61

69.25

-8.25

68.0625

0.982852

91

103.559

-12.559

157.7285

1.523078

155

150.691

4.309

18.56748

0.123216

145

136.75

8.25

68.0625

0.497715

Total = Chi Square Computed =

6.377439

Decision: Reject H0

Conclusion: There is a relationship between chocolate preference and the part of the country where one lives, χ2(2, N = 589) = 6.377, p < .05.

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