In Exercise 24 of Chapter 8 , you learned that First USA, a...

In Exercise 24 of Chapter 8 , you learned that First USA, a major credit card company, is planning a new offer for their current cardholders. First USA will give double airline miles on purchases for the next 6 months if the cardholder goes online and registers for this offer. To test the effectiveness of this campaign, the company recently sent out offers to a random sample of 50,000 cardholders. Of those, 1184 registered. A staff member suspects that the success rate for the full campaign will be comparable to the standard $2 \%$ rate that they are used to seeing in similar campaigns. What do you predict? a) What are the hypotheses? b) Are the assumptions and conditions for inference met? c) Do you think the rate would change if they use this fundraising campaign? Explain.

In Exercise 24 of Chapter 8 , you learned that First USA, a major credit card company, is planning a new offer for their current cardholders. First USA will give double airline miles on purchases for the next 6 months if the cardholder goes online and registers for this offer. To test the effectiveness of this campaign, the company recently sent out offers to a random sample of 50,000 cardholders. Of those, 1184 registered. A staff member suspects that the success rate for the full campaign will be comparable to the standard $2 \%$ rate that they are used to seeing in similar campaigns. What do you predict?
a) What are the hypotheses?
b) Are the assumptions and conditions for inference met?
c) Do you think the rate would change if they use this fundraising campaign? Explain.

Key Concepts

Campaign Effect Analysis

In studying the effect of an intervention, such as an altered campaign offer, the goal is to determine whether the new strategy has led to a significant change in a key metric relative to the standard or previous rate. This involves comparing the observed success rate from the intervention with the established baseline rate and assessing whether any differences are statistically significant — which informs decisions about the effectiveness and potential scalability of the campaign.

Sampling Distribution of Proportions

The sampling distribution of the sample proportion refers to the probability distribution of the sample proportion over many independent samples from the population. Under the null hypothesis and with a sufficiently large sample size, this distribution is approximately normal with a mean equal to the hypothesized proportion and a standard error calculated as the square root of [p(1-p)/n]. This concept is fundamental when estimating population parameters or testing hypotheses about proportions.

Assumptions and Conditions for Inference

Before conducting an inference on a proportion, certain key assumptions and conditions must be satisfied: the sample should be a random sample to ensure independence, and the sample size must be large enough so that the expected number of successes and failures (np and n(1-p)) are both sufficiently large (commonly at least 10) for the normal approximation to be valid. This ensures that the sampling distribution of the proportion can be approximated by a normal distribution, which is critical for many inferential procedures.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about population parameters based on sample data. In the context of proportions, this involves setting up a null hypothesis (typically stating that the population proportion is equal to a known value) and an alternative hypothesis (indicating that the proportion is different, either in one direction or both). It provides a framework to determine if observed deviations from the hypothesized value are significant or if they can be attributed to sampling variability.

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