Question

The manager of a brokerage firm with 500 customers asked them to rate their brokers. The results have been tabulated below. The columns describe the customers’ incomes and the rows describe their rating of the brokers. Under $20,000 $20,000 to 50,000 Over $50,000 Excellent 50 60 40 Average 100 120 50 Poor 30 35 15


.If we consider a customer is satisfied if he/she gave a rating of excellent or average, otherwise unsatisfied. Use a 5% significant level to test the claim that the percentage of customers with an income of at most $50,000 satisfied with their broker is higher than the percentage of those with an income over $50,000.

Answer 1

For at most 50000:

n1 = 395, x1 = 330

p̂1 = x1/n1 = 0.8354

For over 50000:

n2 = 105, x2 = 90

p̂2 = x2/n2 = 0.8571

Null and Alternative hypothesis:

Ho : p1 = p2

H1 : p1 > p2

Pooled proportion:

p̄ = (x1+x2)/(n1+n2) = (330+90)/(395+105) = 0.84

Test statistic:

z = (p̂1 - p̂2)/√ [p̄*(1-p̄)*(1/n1+1/n2)] = (0.8354 - 0.8571)/√[0.84*0.16*(1/395+1/105)] = -0.539

p-value = 1- NORM.S.DIST(-0.5391, 1) = 0.7051

Decision:

p-value > α, Do not reject the null hypothesis

Conclusion:

There is not enough evidence to conclude that the percentage of customers with an income of at most $50,000 satisfied with their broker is higher than the percentage of those with an income over $50,000.