According to literature on brand loyalty, consumers who are loyal to a brand are

Question

According to literature on brand loyalty, consumers who are loyal to a brand are likely to consistently select the same product. This type of consistency could come from a positive childhood association. To examine brand loyalty among fans of the Chicago Cubs, 369 Cubs fans among patrons of a restaurant located in Wrigleyville were surveyed prior to a game at Wrigley Field, the Cubs' home field. The respondents were classified as "die-hard fans" or "less loyal fans." The study found that 67.4% of the 138 die-hard fans attended Cubs games at least once a month, but only 18.2% of the 231 less loyal fans attended this often. Analyze these data using a significance test for the difference in proportions. (Let D = pdie-hard pless loyal. Use = 0.05. Round your value for z to two decimal places. Round your P-value to four decimal places.)

Explanation / Answer

pdie-hard = 0.674

Number of die-hard fans, n1 = 138

pless loyal = 0.182

Number of less loyal fans = 231

Pooled Sample proportion, p = (p1 * n1 + p2 * n2) / (n1 + n2)

= (0.674 * 138 + 0.182 * 231 ) / (138 + 231) = 0.366

Standard error (SE) of the sampling distribution difference between two proportions,

SE = sqrt{ p * ( 1 - p ) * [ (1/n1) + (1/n2) ] }

= sqrt{ 0.366 * ( 1 - 0.366 ) * [ (1/138) + (1/231) ] } = 0.0518

z test statistic is give as,

z = (pdie-hard pless loyal)/SE = (0.674 - 0.182)/ 0.0518 = 9.50

p-value for z = 9.50 is 0 which is less than 0.05, so we reject the null hypothesis and conclude that there is significant difference between the proportions of die-hard and less loyal fans.

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