First USA, a major credit card company, is planning a new...

First USA, a major credit card company, is planning a new offer for their current cardholders. The offer will give double airline miles on purchases for the next 6 months if the cardholder goes online and registers for the offer. To test the effectiveness of the campaign, First USA recently sent out offers to a random sample of 50,000 cardholders. Of those, 1184 registered. a) Give a 95% confidence interval for the true proportion of those cardholders who will register for the offer. b) If the acceptance rate is only 2% or less, the campaign won’t be worth the expense. Given the confidence interval you found, what would you say?

First USA, a major credit card company, is planning a new offer for their current cardholders. The offer will give double airline miles on purchases for the next 6 months if the cardholder goes online and registers for the offer. To test the effectiveness of the campaign, First USA recently sent out offers to a random sample of 50,000 cardholders. Of those, 1184 registered.
a) Give a 95% confidence interval for the true proportion of those cardholders who will register for the offer.
b) If the acceptance rate is only 2% or less, the campaign won’t be worth the expense. Given the confidence interval you found, what would you say?

Key Concepts

Decision Making Based on Confidence Intervals

Using the confidence interval in decision making involves comparing the estimated range to a pre-determined threshold or benchmark of interest. In contexts such as campaign evaluations, if the entire confidence interval falls above or below a target level, it can provide strong evidence for or against the campaign's effectiveness and cost efficiency.

Standard Error of a Proportion

The standard error of a proportion quantifies the variability of the sample proportion from the true population proportion. It is calculated using the formula ?[p(1-p)/n], where p is the sample proportion and n is the sample size. This measure is critical in assessing the precision of the estimated proportion and determining the width of the confidence interval.

Confidence Interval for a Proportion

A confidence interval for a proportion is a range of values, derived from sample data, that is likely to contain the true population proportion. It gives an estimate of the proportion with an associated margin of error, providing a level of confidence (such as 95%) that the true parameter falls within this range.

Normal Approximation and the Central Limit Theorem

When the sample size is large, the sampling distribution of the sample proportion can be approximated by a normal distribution thanks to the Central Limit Theorem. This approximation allows for the use of z-scores to calculate confidence intervals for the population proportion, even when the underlying variable is binary.

Transcript

00:01 In this question, the scenario is that a credit card company is planning to offer a new offer to its card holders.

00:09 And to test the offer, they first send the offer out to a random sample of 50 ,000 of their clients.

00:17 So we have a sample size of 50 ,000.

00:25 And of those 50 ,000, 1 ,184 accepted the offer.

00:31 So the sample proportion for accepting the offer is 1 ,000.

00:38 184 divided by 50 ,000, which is about 0 .024.

00:52 And for part a, we were asked to give a 95 % confidence interval for the true proportion of cardholders who would accept the offer.

01:04 So to make inference for the overall population of credit card holders that this company has, in order to do a 1 proportion z confidence interval, we'd want to check the conditions for inference.

01:17 The independence assumption, we want to satisfy the randomization condition, and we are told from the question that it is a random survey, and we would want to check the 10 % condition, which is the sample of 50 ,000 at most 10 % of the total population of card holders.

01:37 So assuming those are met, the other thing to check is the success failure condition, where we want the expected number of successes to be at least 10, and the expected number of failures to be at least 10.

01:51 We know from the way the question was set up that the expected number of successes is 1 ,184.

01:59 If you multiply 50 ,000 times 0 .024, you get 1 ,184...

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