Question

The number of credit card holders of a bank in two different cities (city - X and city - Y) settling their excess withdrawal amounts in time without attracting interest follows binomial distribution.

The manager (collections) of the bank feels that the proportion of the number of such credit card holders in the city - X is not different from the proportion of the number of such credit card holders in the city - Y.

To test his intuition, a sample of 200 credit card holders is taken from the city - X and it is found that 160 of them are settling their excess withdrawal amount in - time without attracting interest.

Similarly, a sample of 180 credit card holders is taken from the city - Y and it is found that 50 of them are settling their excess withdrawal amount in - time without attracting interest,

Question: Check the intuition of the sales manager at a significance level of 0.05.

Answer 1

Ans. We need to check whether the proportion of card holders settling their excess withdrawal amounts in time without attracting interest in city X and city Y are no different.

Therefore we need to perform a 2 sample discrete test of difference in proportions.

Our Test hypothesis is :

H0: There is no significant difference between City X and City Y i.e Px is equal to Py

H1: there is a significant difference between City X and City Y i.e Px is not equal to Py

Test-Statistic:

Z= (X-Y) - (P1-P2)/sqrt( (X(1-X)/Nx) + (Y(1-Y)/Ny)) ~ N(0,1)

under H0 : Z = (X-Y) /sqrt( (X(1-X)/Nx) + (Y(1-Y)/Ny)) ~ N(0,1)

Z= 11.935

We know that the two tailed Z value at 5% level of significance is 1.96 .

Since our calculated Z value is Greater than our tabulated Z value , We have sufficient evidence to reject our Null Hypothesis at 5% level of significanece.

Therefore the sales manager is completely wrong in saying that there is no difference in proportion of card holders with excess withdrawal amounts in time without attracting interest between city X and City Y .