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, 376 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 66.4% of the 134 die-hard fans attended Cubs games at least once a month, but only 18.2% of the 242 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.)

Analyze these data using a 95% confidence interval for the difference in proportions. (Round your answers to three decimal places.)

  ,

Write a short summary of your findings.

Reject the null hypothesis, there is significant evidence that a higher proportion of die hard Cubs fans attend games at least once a month.Reject the null hypothesis, there is not significant evidence that a higher proportion of die hard Cubs fans attend games at least once a month.    Fail to reject the null hypothesis, there is not significant evidence that a higher proportion of die hard Cubs fans attend games at least once a month.Fail to reject the null hypothesis, there is significant evidence that a higher proportion of die hard Cubs fans attend games at least once a month.

Explanation / Answer

pdie-hard = 0.664, pless loyal = 0.182

H0: pdie-hard pless loyal = 0
HA: pdie-hard pless loyal 0

p = (p1 * n1 + p2 * n2) / (n1 + n2) = (0.664*134 + 0.182*242)/(134+242) = 0.35377659574

SE = sqrt{ p * ( 1 - p ) * [ (1/n1) + (1/n2) ] } = sqrt{0.353777*(1-0.353777)*[(1/134)+(1/242)]} = 0.0514861

z = (p1 - p2) / SE = (0.664 - 0.182)/0.0514861 = 9.36

p = 0

lower bound = (p1 - p2) - zcrit*SE = (0.664-0.182) - 1.96*0.0514861 = 0.381
upper bound = (p1 - p2) + zcrit*SE = (0.664-0.182) + 1.96*0.0514861 = 0.583

Reject the null hypothesis, there is significant evidence that a higher proportion of die hard Cubs fans attend games at least once a month.

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