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, 390 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 92 of the 132 die-hard fans attended Cubs games at least once a month, but only 40 of the 258 less loyal fans attended this often. Analyze these data using a significance test for the difference in proportions. (Let D = p_die-hard − p_{less loyal}. Use α = 0.05. Round your value for z to two decimal places. Round your P-value to four decimal places.)

z	=
P-value	=

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.

Answer 1

Ans:

sample proportion for die hard fans=92/132=0.6970

sample proportion for less loyal fans=40/258=0.1550

pooled proportion=(92+40)/(132+258)=0.3385

Test statistic:

z=(0.697-0.155)/SQRT(0.3385*(1-0.3385)*((1/132)+(1/258)))

z=10.70

p-value=P(z>10.70)=0.0000

95% confidence interval for difference in proportions

=(0.6970-0.1550)+/-1.96*sqrt(0.3385*(1-0.3385)*((1/132)+(1/258)))

=0.542+/-0.099

=(0.443, 0.641)

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