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, 365 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 136 die-hard fans, 86.8% reported that they had watched or listened to Cubs games when they were children. Among the 229 less loyal fans, 71.6% said that they watched or listened as children. (Let D = p_die-hard − p_{less 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 α = 0.01. Round your z-value to two decimal places and your P-value to four decimal places.)

z	=
P-value	=

Conclusion

Fail to reject the null hypothesis, there is significant evidence that a higher proportion of die hard Cubs fans watched or listened to Cubs games as children.Fail to reject the null hypothesis, there is not significant evidence that a higher proportion of die hard Cubs fans watched or listened to Cubs games as children.    Reject the null hypothesis, there is not significant evidence that a higher proportion of die hard Cubs fans watched or listened to Cubs games as children.Reject the null hypothesis, there is significant evidence that a higher proportion of die hard Cubs fans watched or listened to Cubs games as children.

(c) Express the results with a 95% confidence interval for the difference in proportions. (Round your answers to three decimal places.)

( _____ , _____ )

Answer 1

a)

Die hard fans = 0.868 * 136 = 118

Less loyal fans = 0.716 * 229 = 164

b)

z = ( p1 -p2)/sqrt(p1 *(1-p1)/n1 + p2 *(1-p2)/n2)
= ( 0.868 - 0. 716)/sqrt(0.868 *(1-0.868)/136 + 0.716 *(1-0.716)/229)
= 3.65

p value = 0.0001

Reject the null hypothesis, there is significant evidence that a higher proportion of die hard Cubs fans watched or listened to Cubs games as children.

c)

z value at 95% = 1.96

CI = ( p1 -p2) +/- z * sqrt(p1 *(1-p1)/n1 + p2 *(1-p2)/n2)
= ( 0.868 - 0. 716) + /- 1.96 * sqrt(0.868 *(1-0.868)/136 + 0.716 *(1-0.716)/229)

= (0.070,0.234)

= 3.65