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, 395 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 95 of the 140die-hard fans attended Cubs games at least once a month, but only 49 of the 255 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.

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.

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 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=95/140=0.6786

sample proportion for less loyal=49/255=0.1922

pooled proportion=(95+49)/(140+255)=0.3646

Test statistic:

z=(0.6786-0.1922)/SQRT(0.3646*(1-0.3646)*((1/140)+(1/255)))

z=9.61

p-value=P(z>9.61)=0.0000

95% confidence interval for difference in proportions

=(0.6786-0.1922)+/-1.96*sqrt((0.6786*(1-0.6786)/140)+(0.1922*(1-0.1922)/255))

=0.4864+/-0.0912

=(0.3952, 0.5776)

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