Can people really identify their favorite brand of cola? Volunteers tasted Coca-
ID: 2958780 • Letter: C
Question
Can people really identify their favorite brand of cola? Volunteers tasted Coca-Cola Classic, Pepsi, Diet Coke, and Diet Pepsi, with the results shown below. Research question: At = .05, is the correctness of the prediction different for the two types of cola drinkers? Could you identify your favorite brand in this kind of test? Since it is a 2 x 2 table, try also a two-tailed two-sample z test for 1 = 2 (see Chapter 10) and verify that z2 is the same as your chi-square statistic.Which test do you prefer? Why? (Data are from Consumer Reports 56, no. 8 [August 1991], p. 519.) Data to construct the contingency table are below
Yes, got it right No, got it wrong
Regular Cola: 7 Regular Cola: 12
Diet Cola: 20 Diet Cola: 7
Explanation / Answer
Chi-square Test:
Let H0 = The correctness of prediction is independent of the cola type
and H1 = The correctness of prediction depends upon the cola type
Recorded Values:
Regular Cola
Diet Cola
Total
Yes, got it right
7
7
14
No, got it wrong
12
20
32
Total
19
27
46
Expected values:
[Example: Expected value for the Yes, got it right – Regular cola category is calculated as
(Yes, got it right Total * Regular cola Total / Grand Total) = (14 * 19)/46 = 5.783]
Regular Cola
Diet Cola
Yes, got it right
5.78
8.22
No, got it wrong
13.22
18.78
Chi Squares Table:
Recorded Value (R)
Expected Value (E)
(R – E)^2
(R – E)^2 /E
7
5.78
1.488
0.258
12
13.22
1.488
0.112
7
8.22
1.488
0.181
20
18.78
1.488
0.079
Chi-square value
0.63
The p- value for chi square = 0.63 and dof = (2 – 1)(2 – 1) = 1 is 0.427
Since 0.427 > 0.05, we fail to reject the null hypothesis. There is no statistical evidence to suggest that the correctness of prediction is dependent on the cola type.
Two-tail z- Test:
H0: There is no difference between the abilities to identify the colas; that is, the correctness of prediction is independent of the cola type; that is, p1 – p2 = 0
Ha: There is a difference between the abilities to identify the colas; that is, the correctness of prediction depends upon the cola type; that is, p1 – p2 0
n1 = 19, n2 = 27, p1 = 7/19 = 0.368, p2 = 7/27 = 0.259
p’ = (n1p1 + n2p2)/(n1 + n2) = (7 + 7)/46 = 0.304
q’ = 1 – p’ = 0.696
SE = Ö(p’q’) * Ö [(n1 + n2)/(n1n2)] = Ö(0.304 * 0.696) * Ö[46/(19 * 27)] = 0.138
z = (p1 – p2)/SE = (0.368 – 0.259)/0.138 = 0.791
[Note that z^2 = 0.791^2 = 0.626 » The value of chi-square.]
The two-tail p- value for z = 0.791 is 0.457
Since 0.457 > 0.05, we fail to reject the null hypothesis. There is no statistical evidence to suggest that the correctness of prediction is dependent on the cola type.
The chi-square test appears to better since it is a general test for this type of questions.
(Also note that 0.427 < 0.457)
Regular Cola
Diet Cola
Total
Yes, got it right
7
7
14
No, got it wrong
12
20
32
Total
19
27
46