A bank is testing a new method for getting delinquent customers to pay their past-due credit card bills. The standard way was to send a letter (costing about $0.45) asking the customer to pay. That worked 30% of the time. The bank wants to test a new method that involves sending a DVD to customers encouraging them to contact the bank and setup a payment plan. Developing and sending the video costs about $10 per customer.
The bank sets up the experiment to test the effectiveness of the DVD. They mail the DVD to a randomly selected sample of delinquent customers and keep track of how many actually do contact the bank to arrange payments.
Using the information from the experiment, the bank statistician calculated the 90% confidence interval for proportion of delinquent customers who make payments. The interval is:
(0.29, 0.45)
1. Since the old send-a-letter method worked 30% of the time, would this confidence interval lead the bank to decide that this new strategy is effective?
A) Since the interval includes 30%, the bank can assume the new strategy which utilizes the DVD is extremely effective.
B) Since the interval extends above and below 30%, the bank can assume the new strategy which utilizes the DVD is extremely effective.
C) Since the interval extends above and below 30%, the bank cannot assume the new strategy which utilizes the DVD is extremely effective.
D) The confidence interval is too wide for any conclusions.
2. Given the confidence interval calculated by the bank statistician, what would you recommend to the bank president? Should they move forward with the DVD strategy or scrap it?
A) The results are very clear; the DVD is very effective and the bank should switch immediately.
B) Since the confidence interval is so wide, the DVD strategy is clearly better.
C) Since the confidence interval extends below 30%, we are not convinced that the DVD is effective. It would be helpful to conduct the experiment again with a larger sample size.
D) Since the confidence interval extends below 30%, we are not convinced that the DVD is effective. It would be helpful to conduct the experiment again with a smaller sample size.
E) The confidence interval clearly shows that the DVD is not effective. The bank should scrap this idea.
3. The bank president thinks that the confidence interval is too wide. Which of the following would result in a narrower confidence interval?
I. Increase the confidence level
II. Decrease the confidence level
III. Increase the sample size
IV. Decrease the sample size
A) I only
B) II only
C) III only
D) IV only
E) I and III
F) II and III
G) II and IV
A,E,G
Correction ** 1 - C, 2 - C, 3 - IDK
1. The correct answer is C) Since the interval extends above and below 30%, the bank cannot assume the new strategy which utilizes the DVD is extremely effective.
To reach this conclusion, we need to consider that the confidence interval calculated by the bank statistician is (0.29, 0.45), which means that the true proportion of delinquent customers who make payments could be anywhere within this interval with 90% confidence. Since the interval includes values both below and above 30%, it suggests that there is uncertainty about the effectiveness of the DVD strategy. Therefore, the bank cannot make the assumption that the new strategy is extremely effective.
2. The correct answer is C) Since the confidence interval extends below 30%, we are not convinced that the DVD is effective. It would be helpful to conduct the experiment again with a larger sample size.
Based on the confidence interval (0.29, 0.45), we cannot conclude with certainty that the DVD strategy is effective. The lower bound of the confidence interval extends below 30%, indicating that there is a possibility that the true proportion of delinquent customers who make payments using the DVD strategy is lower than the 30% success rate of the old send-a-letter method. To gain more confidence in the results, it would be recommended to conduct the experiment again with a larger sample size, which could reduce the variability and potentially provide a narrower confidence interval.
3. The correct answer is E) I and III.
To obtain a narrower confidence interval, we can increase the sample size (III) and increase the confidence level (I).
Increasing the confidence level provides a higher level of confidence but will also widen the interval. Therefore, reducing the confidence level would not result in a narrower confidence interval.
Increasing the sample size reduces the sampling variability, resulting in a narrower confidence interval. Decreasing the sample size would increase the variability and widen the interval.
Therefore, the combination of increasing the confidence level (I) and increasing the sample size (III) would result in a narrower confidence interval.