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, 368 Cubs fans among patrons of a restaurant located in Wrigley Ville 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 62.3% of the 138 die-hard fans attended Cubs games at least once a month, but only 20.4% of the 230 less loyal fans attended this often. Analyze these data using a significance test for the difference in proportions. (Let D = pdie-hard − pless loyal. Use α = 0.05. Round your value for z to two decimal places. Round your P-value to four decimal places.)
z = __________
To analyze the data using a significance test for the difference in proportions, we need to calculate the test statistic z. The formula for the test statistic in this case is:
z = (p1 - p2) / sqrt(p̂(1-p̂) * ((1/n1) + (1/n2)))
Where:
p1 = proportion of die-hard fans attending Cubs games at least once a month
p2 = proportion of less loyal fans attending Cubs games at least once a month
p̂ = (x1 + x2) / (n1 + n2)
In this case:
n1 = number of die-hard fans surveyed (138)
x1 = number of die-hard fans attending Cubs games at least once a month (138 * 0.623)
n2 = number of less loyal fans surveyed (230)
x2 = number of less loyal fans attending Cubs games at least once a month (230 * 0.204)
Let's calculate the values:
p1 = (138 * 0.623) / 138 = 0.623
p2 = (230 * 0.204) / 230 = 0.204
p̂ = (138 * 0.623 + 230 * 0.204) / (138 + 230) = 0.369
Now, we can calculate the test statistic z:
z = (0.623 - 0.204) / sqrt(0.369 * (1-0.369) * ((1/138) + (1/230)))
After calculating this, we get the value of z. Round this value to two decimal places and record it in the answer.