Calculate variance for A/B test results

Question

Given the following scenario about revenue per session, calculate the the p-value.

Codemia · Accepted Answer

In this scenario, we are tasked with calculating the p-value to assess the significance of the difference in revenue per session between two groups in an A/B test. We will set up the null hypothesis (H0) and the alternative hypothesis (H1):

- **H0**: There is no difference in revenue per session between Group A (control) and Group B (treatment).
- **H1**: There is a difference in revenue per session between Group A and Group B.

Let $ \mu_A $ be the mean revenue per session for Group A and $ \mu_B $ for Group B. We will use the sample means, variances, and sizes of both groups to calculate the test statistic and subsequently the p-value.

To calculate the p-value, we will:
1. Gather the sample means ($ ar{x}_A $ and $ ar{x}_B $), sample sizes ($ n_A $ and $ n_B $), and sample variances ($ s_A^2 $ and $ s_B^2 $) for both groups.
2. Calculate the test statistic using the formula for the t-test for independent samples:
   
   $$ t = \frac{\bar{x}_A - \bar{x}_B}{\sqrt{\frac{s_A^2}{n_A} + \frac{s_B^2}{n_B}}} $$

3. Determine the degrees of freedom using the formula:
   
   $$ df = n_A + n_B - 2 $$

4. Use the t-distribution to find the p-value associated with the calculated t-statistic.

Assume we have the following data:
- Group A (Control): $ n_A = 100, \bar{x}_A = 20, s_A^2 = 4 $
- Group B (Treatment): $ n_B = 100, \bar{x}_B = 22, s_B^2 = 5 $

1. Calculate the test statistic:
   $$ t = \frac{20 - 22}{\sqrt{\frac{4}{100} + \frac{5}{100}}} = \frac{-2}{\sqrt{0.04 + 0.05}} = \frac{-2}{\sqrt{0.09}} = \frac{-2}{0.3} = -6.67 $$

2. Calculate the degrees of freedom:
   $$ df = 100 + 100 - 2 = 198 $$

3. Use a t-table or software to find the p-value corresponding to $ t = -6.67 $ with 198 degrees of freedom. This will yield a very small p-value (e.g., < 0.0001), indicating strong evidence against the null hypothesis.

The calculated p-value suggests that there is a statistically significant difference in revenue per session between the two groups. In practical terms, this means that the treatment applied in Group B has a significant positive impact on revenue compared to the control group. This finding can prompt the company to consider implementing the changes tested in Group B to enhance revenue generation.

This exercise connects to related concepts such as:
- **Power Analysis**: Understanding the power of the test helps in determining the sample size needed to detect an effect.
- **Confidence Intervals**: We could also construct confidence intervals for the difference in means to provide a range of plausible values.
- **Type I and Type II Errors**: It's crucial to understand the implications of rejecting or failing to reject the null hypothesis, as these errors can significantly affect business decisions.
- **Bayesian Inference**: An alternative approach could involve Bayesian methods, which may offer additional insights into the results by considering prior beliefs about the parameters.

Calculate variance for A/B test results

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