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, 368 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." Of the 137die-hard fans, 89.1% reported that they had watched or listened to Cubs games when they were children. Among the 231 less loyal fans, 69.7% said that they watched or listened as children. (Let D = pdie-hard pless loyal.)

(a) Find the numbers of die-hard Cubs fans who watched or listened to games when they were children. Do the same for the less loyal fans. (Round your answers to the nearest whole number.)

die-hard fans

less loyal fans

(b) Use a one sided significance test to compare the die-hard fans with the less loyal fans with respect to their childhood experiences relative to the team. (Use your rounded values from part (a). Use = 0.01. Round your z-value to two decimal places and your P-value to four decimal places.)

Explanation / Answer

a. Numbers of die-hard fans, who had watched or listened to Cubs games is: 0.891*137=122.067~122

Similarly, numbers of less-loyal fans, who had watched or listened to Cubs games is: 0.697*231=161.007~161

b. Hypotheses:

H0: pdh-pll=0 (there is no difference in population proportin of die hard and less loyal fans, with respect to their childhood experinence)

H1:pdh-pll>0 (proportion of die hard fans is higher than proportion of less loyal fans, with respect to their childhood experience)

Assumptions:

Independent group assumption: This is an experiment, and randomization give independent groups.

Independence assumption: Childhood experience is independent from die-hard fans to less loyal fans.

Randomizatin condition: The result is a suitable random sample.

10% condition: the samples include far fewer than 10% of all children who are die hard and less loyal fans and who had watched or listened cubs games.

Success/failure condition: For die-hard fans, x=122, and n-x=137-122=15, and for less-loyal group, x=161, and n-x=231-161=70, which are atleast 10. Note, x denote number of events, and n denote number of success.

Test statistic:

Phatpooled=(event1+event2)/n1+n2=(122+161)/(137+231)=0.7690

Z=(p1hat-p2hat)-0/sqrt[phatpooled(1-phatpooled)/n1+phatpooled(1-phatpooled)/n2]

=(122/137-161/231)-0/sqrt[0.7690(1-0.7690)/137+0.7690(1-0.7690)/231]

=4.26

p value:0.0000

Conclusion: Rule states, reject H0, if p value is less than 0.05. Here, p value is less than 0.05, reject H0 and conclude that proportion of die hard fans is significantly higher than proportion of less loyal fans, with respect to their childhood experience.

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