The wise CPA expands beyond compliance and adds value by using tools such as net present value, marginal analysis, and cost and revenue analysis. In a data-driven environment, the CPA will grow his or her worth by updating their analytical skill set with tools from the school of probability and statistics.
by Bruce S. Weitzman, CPA, and Leslie Marlo, FCAS, MAAA Dec 3, 2018, 16:24 PM
You need to determine whether the two mailings are substantially equal (with any difference being random noise: that the mean response rate of offer A is equal to the mean of offer B – or the null hypothesis) or if there is a true statistical difference in the results.
One way to determine this is to use the common statistical technique of the z-test. A z-test is a hypothesis test. In this example, the z-statistic measures how many standard deviations above or below the true population mean the marketing data is. In this case, the test compares the calculated z-statistic to the z-value at 2.5 percent (a probability level of 95 percent since we are measuring deviation both above and below the true mean). If the calculated z-statistic is less than the z-value at 2.5 percent, then they would be substantially equivalent since it will show the result is not an outlier. But if the calculated z-statistic is greater than the z-value at 2.5 percent, then it is an outlier and the difference in results is significant.
The calculation of the z-statistic follows this formula:
z = (MeanA – MeanB) - (μA - μB)
[ (σA2/nA) + (σB2/nB) ]0.5
The mean and variance are calculated from the samples, and n is equal to the number of observations in each sample. We do not know the true μA or μB but since we are testing to see if there is no significance between the two (the null hypothesis) then μA = μB, so μA - μB equals 0.
Our calculated means and variances (σ = response rates) are 2.1 percent for offer A and 1.9 percent for offer B. (Note: here, the calculated means and variances are equal, but that is coincidental.)
Therefore, the numerator would be:
(0.021 - 0.019) = 0.002
The denominator would be:
[ (0.021/5000) + (0.019/3500) ]0.5 = 0.0031
So, the calculated z-statistic is:
0.002/0.0031 = 0.645
Now, compare this to the z-value at 2.5 percent. If you go to www.calculators.org/math/z-critical-value.php, the calculator returns a z-value of 1.96.
Since the calculated z-statistic of 0.645 is less than the z-value at 2.5 percent of 1.96, the mail campaigns are statistically similar.