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, 369 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 137 die-hard fans, 88.3%reported that they had watched or listened to Cubs games when they were children. Among the 232 less loyal fans, 68.1% 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.)

(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.)

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

  ,

Explanation / Answer

a.
sample one, x1 =120.971, n1 =137, p1= x1/n1=0.883
sample two, x2 =157.992, n2 =232, p2= x2/n2=0.681
b.
Given that,
null, Ho: p1 = p2
alternate, H1: p1 != p2
level of significance, = 0.01
from standard normal table, two tailed z /2 =
since our test is two-tailed
reject Ho, if zo < -2.576 OR if zo > 2.576
we use test statistic (z) = (p1-p2)/(p^q^(1/n1+1/n2))
zo =(0.883-0.681)/sqrt((0.756*0.244(1/137+1/232))
zo =4.365
| zo | =4.365
critical value
the value of |z | at los 0.01% is 2.576
we got |zo| =4.365 & | z | =2.576
make decision
hence value of | zo | > | z | and here we reject Ho
p-value: two tailed ( double the one tail ) - Ha : ( p != 4.365 ) = 0
hence value of p0.01 > 0,here we reject Ho
ANSWERS
---------------
null, Ho: p1 = p2
alternate, H1: p1 != p2
test statistic: 4.365
critical value: -2.576 , 2.576
decision: reject Ho
p-value: 0

c.
Confidence Interval for Diffrence of Proportion
CI = (p1 - p2) ± Z a/2 Sqrt(p1(1-p1)/n1 + p2(1-p2)/n2 )
Proportion 1
No.Of Observed (n1)=137
P1= X1/n1=0.883
Proportion 2
No.Of Observed (n2)=232
P2= X2/n2=0.681
C.I = (0.883-0.681) ±Z a/2 * Sqrt( (0.883*0.117/137) + (0.681*0.319/232) )
=(0.883-0.681) ± 1.96* Sqrt(0.002)
=0.202-0.081,0.202+0.081
=[0.121,0.283]

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