2) A bank wonders whether omitting the annual credit card fee for customers who
ID: 3356402 • Letter: 2
Question
2) A bank wonders whether omitting the annual credit card fee for customers who charge at least $2400 in a year will increase the amount charged on its credit cards. The bank makes this offer to an SRS of 200 of its credit card customers. It then compares how much these customers charge this year with the amount that they charged last year. The mean increase in the sample is $332, and the standard deviation is $108 (a) Is there significant evidence at the 1% level that the mean amount charged increases under the no-fee offer? State Ho and Ha and carry out a t test. (b) Give a 99% confidence interval for the mean amount of the increase. (c) The distribution of the amount charged is skewed to the right, but outliers are prevented by the credit limit that the bank enforces on each card. Use of the t procedures is justified in this case even though the population distribution is not Normal. Explain why.Explanation / Answer
Here sample size n = 200
THe mean increase in the sample xd = $ 332
standard deviation of the increase in the amount sd = $ 108
Standard error of the difference se0 = sd /sqrt(n) = 108/ sqrt(200) = 7.6367
(a) H0 : There is no increase in charge customers paid in no fee offer as compared to charge customers paird last year. d = 0
Ha : THere is significant difference in charge customers paid in no fee offer as compared to charge customers paird last yeasr. d > 0
Test statistic
t = xd / (se0) = 332/ 7.6367 = 43.47
Here dF = 199 and a = 0.01
tcritical = t199, 0.01 = 2.6
so t > tcritical so we shall reject the null hypothesis.
(b) 99% confidence interval for the mean amount of increase = xd +- t0.01,199 se0
= 332 + 2.6 * 7.6367
= (312.14, 381.86)
(c) Here in this case t procedure is justified as it is applicable for sample mean distribution. If we use z procedure it is applicable for population, but as said in question the distribution is skewed right. It is not suitable. As central limit theorem also entails that as we increase sample size, distribution of means tend to be normal and t - distribution is perfect explaintation to approximation of normal distribution.
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