91Ó°ÊÓ

The concept of difference in proportions is crucial when comparing two separate groups and their outcomes in an experiment, like the one described in this exercise. In our particular case, the two groups are those who received a donation request email with a deadline and those who didn't. The proportion refers to the fraction or percentage of individuals in each group that made a donation after receiving the email.

To find the difference in proportions, you simply subtract one group's proportion from the other. Here, the donation proportion for the deadline group, denoted as \(p_1\), was 0.012, or 1.2%. The proportion for the group without a deadline, denoted as \(p_2\), was 0.008, or 0.8%. So, the point estimate of the difference is simply:

• \( p_1 - p_2 = 0.012 - 0.008 = 0.004 \)

This represents a preliminary estimate of how much more likely individuals were to donate when given a deadline, calculated in a simple and straightforward manner. It's important not to stop here, though, because without considering the variability of this estimate, you're only seeing part of the picture.

The standard error (SE) is a measure of the variability or uncertainty associated with our estimate, specifically, the difference in the proportions across two groups. It accounts for how much our sample-based difference might shift if we were to take many samples similarly.

The formula for standard error with respect to difference in proportions is: \[ SE = \sqrt{ \frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2} } \] Where:

• \(p_1\) is the proportion with a deadline
• \(p_2\) is the proportion without a deadline
• \(n_1\) and \(n_2\) are the sample sizes of the respective groups

Plugging in our numbers from the exercise, we get: \[ SE = \sqrt{ \frac{0.012 \times (1 - 0.012)}{2000} + \frac{0.008 \times (1 - 0.008)}{2000} } \] Which results in a standard error of approximately 0.0023.

This value gives us a sense of spread or dispersion around our estimated differences in proportions and tells us how precise our estimate is likely to be.

The margin of error expands our understanding of the point estimate by giving a range within which we expect the true population difference in proportions to lie, based on our sample statistics.

To compute the margin of error for a confidence interval, we multiply the standard error by the z-score corresponding to our desired level of confidence. In our scenario, to find the 90% confidence interval, a z-score of 1.645 is appropriate. The margin of error then becomes:

• Margin of Error = Z-score \( \times \) SE = 1.645 \( \times \) 0.0023 \( \approx 0.0038 \)

This margin indicates the degree of deviation we might expect from our point estimate (difference in proportions) when predicting the true difference in the wider population.

By adding and subtracting the margin of error from our point estimate, we derive a confidence interval. This tells us that with 90% confidence, the true difference between those two email groups' donation rates lies somewhere between 0.0002 and 0.0078. The interval does not contain zero, suggesting a legitimate effect of the deadline on donation behavior.