Application of sampling distribution Joe Greene, a new manager at Pilgrim Bank wants to better understand profitability data for bank’s customers. Joe is able to obtain a random sample of 31,634 customers on the following variables – Profitability (in $, for the most recent completed year, i. e. 2006), whether or not the customer uses the online banking channel, customer tenure, age and income where available, as well as the customer’s residential area. Descriptive statistics for Profits indicates that the average profit per customer is 1.

50 with a standard deviation of $272. 4. a. Is Joe justified in assuming that this is a “large” sample? (see slide 7-14) YES, BECAUSE 31,634 IS A LOT MORE THAN 30 OBVERSATIONS. GENERALLY, THE LARGER THE SAMPLE, THE MORE RELIABLE ARE THE ESTIMATES. THE KEY IS TO HAVE A RANDOMLY SELECTED SAMPLE TO REDUCE THE RISK OF BIASED ESTIMATES b. Joe has been informed that the bank serves approximately 5 million customers nationwide – should he worry about using finite population correction factor (slide 7-9).

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FINITE POPULATION CORRECTION (FPC) IS NOT NECESSARY, SINCE THE POPULATION OF APPROXIMATELY 5 MILLION IS QUITE LARGE.

FPC IS ONLY RECOMMENDED FOR SMALL POPULATIONS. c. Joe wants to estimate average profit for the entire population based on his sample. He knows that the “point estimate” for average profit would be $111. 50, but, he will need to calculate the margin of error. The first step for this is to calculate the standard error (see slide 7-6); provide the value below. 272. 84 / SQRT(31,634) = $1. 53 d. The second step in calculating the margin or error (often simply called error) is to multiply the standard error by 1. 96; provide the value below. 1. 6*$1. 53=$3. THUS THE MARGIN OF ERROR FOR OUR ESTIMATE OF AVERAGE PROFIT PER CUSTOMER, FOR THE ENTIRE CUSTOMER BASE WILL BE $3 e. Joe can combine responses from c and d to report the estimated average profit customer for all the bank’s customers as the point estimate of average profit plus/minus margin of error. Thus, in Joe’s case this will be: JOE’S ESTIMATE FOR AVERAGE PROFIT PER CUSTOMER, FOR THE ENTIRE CUSTOMER BASE IS: $111. 50+/-$3 = $108. 50 TO $114. 50 Note: The interval reported above is often called a 95% confidence level estimate. . Of the 31,634 customers, 3853 use bank’s online services. Proportion of online service users in the sample can thus be calculated as the ratio of users divided by the sample size. This is called the estimated percentage of online service users. On slide 7-20, the box shown on top left is used to ensure that we can use sampling theory – can you confirm that: n*p>=5 and n*(1-p)>=5 YES, BOTH RELATIONSHIPS ARE SATISFIED (n*p=3853, WHICH IS GREATER THAN 5, AND n*(1-p)=31,634-3853, WHICH IS ALSO GREATER THAN 5) . Does Joe have to worry about using finite population correction (slide 7-21)? NO, FOR THE SAME REASONS GIVEN IN b. h. Calculate the standard error for the percentage of online service users for Joe (see the box at the bottom, right, on slide 7-20). USING THE SUGGESTED FORMULA, STANDARD ERROR =0. 00184 i. Joe can now calculate the margin of error (or simply error) as 1. 96*standard error. Calculate the margin of error. 1. 96*0. 00184=0. 00364 BECAUSE THE SAMPLE SIZE IS SO LARGE, THE MARGIN OF ERROR IS 0. 36% j.

Joe can combine responses from above to report the estimated percentage of online users among all the bank’s customers as the estimated percentage of online service users plus/minus margin of error. Thus, in Joe’s case this will be: 0. 1218+/-0. 00364 = 11. 82% TO 12. 84% OF BANK’S ENTIRE CUSTOMER BASE IS EXPECTED TO USE ITS ONLINE SERVICES Based on the work done on pages 1 and 2, Joe can now conclude that: Average profit among all of the bank’s customers ranges from $108. 50 to $114. 50 Also, the percentage of online service users among all of the bank’s customers ranges from 11. 2% to 12. 84% How would these estimates be affected if the sample size were 1000 instead of 31,634 (assume all others numbers, such as average and standard deviation of profit, as well as the percentage of online service users stays the same). THE STANDARD ERROR IS NOW $8. 63; MARGIN OF ERROR IS $16. 91 (COMPARED TO $3 BEFORE). OUR ESTIMATE OF PROFIT PER CUSTOMER NOW RANGES FROM $94. 59 TO $128. 41 (WHICH IS A LOT WIDER THAN BEFORE) FOR THE ESTIMATED PROPOTION OF ONLINE SERVICE USERS, THE MARGIN OF ERROR GOES UP FROM . 36% TO 2. 02%; THE RANGE NOW IS 10. 15% TO 14. 21%