According to literature on brand loyalty, consumers who are loyal to a brand are
ID: 3254476 • Letter: A
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, 370 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 69.3% of the 137 die-hard fans attended Cubs games at least once a month, but only 19.7% of the 233 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 alpha = 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.)Explanation / Answer
given n1=137,n2=233
p1^= 69.3% = 0.693 p2^= 19.7 % 0.197
Test statistic. The test statistic is a z-score (z) defined by the following equation.
z = (p1 - p2) / SE
Standard error. Compute the standard error (SE) of the sampling distribution difference between two proportions.
SE = sqrt{ p * ( 1 - p ) * [ (1/n1) + (1/n2) ] }
where p is the pooled sample proportion, n1 is the size of sample 1, and n2 is the size of sample 2.
.Since we're assuming that the standard error is the same, we're going to have to pool the values.
.693 * 137 = 95, .197 * 233 = 46
. Now, I'll pool them.
Ppool = (95 + 46) / (137 + 233) = 0.3810
to find our standard error of the two proportions.
SE(p1 - p2) = sqrt[(0.3810 * 0.619) / 137 + (0.3810 * 0.619) / 233] = 0.0522
z = (0.693- 0.197) / 0.0522 = 9.5 which is high so we reject null hypothesis
pvalue =1
The z* for 95% confidence is 1.96. We simply take the difference of the proportions and add/subtract the z* times the standard error.
(0.693-0.197)+/- 1.96 * 0.0522
=(0.5983 , 0.3936)
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