Calculate expected value for click-through rates

Question

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

Codemia · Accepted Answer

To calculate the expected value for click-through rates (CTR) and subsequently the p-value in this context, we first need to define our variables clearly. Let's denote:

- **CTR**: Click-through rate, which is the probability of a user clicking on a link.
- **Revenue per session**: The average revenue generated from a session when a user clicks on a link.

Assuming we want to test the effect of a marketing campaign on CTR, we set up the following hypotheses:
- **H0** (null hypothesis): The marketing campaign has no effect on CTR (i.e., the mean CTR before and after the campaign is the same).
- **H1** (alternative hypothesis): The marketing campaign has a significant effect on CTR (i.e., the mean CTR after the campaign is different from before).

We will conduct a hypothesis test to determine if we can reject H0 in favor of H1 based on the observed data.

1. **Collect Data**: Gather CTR data from sessions before and after the marketing campaign. Let’s denote the pre-campaign CTR as 'p1' and the post-campaign CTR as 'p2'.

2. **Calculate Sample Means and Standard Deviations**: For both pre- and post-campaign data, calculate the sample means ($\bar{p_1}, \bar{p_2}$) and standard deviations ($s_1, s_2$).

3. **Formulate the Test Statistic**: Use a two-proportion z-test as the test statistic, which is given by:
   $$ z = \frac{\bar{p_1} - \bar{p_2}}{\sqrt{\hat{p}(1 - \hat{p}) \left(\frac{1}{n_1} + \frac{1}{n_2}\right)}} $$
   where $\hat{p} = \frac{x_1 + x_2}{n_1 + n_2}$ with $x_1, x_2$ as the number of successes (clicks) and $n_1, n_2$ as the sample sizes.

4. **Calculate P-value**: The p-value is determined using the standard normal distribution, where you find the probability of observing a test statistic as extreme as the calculated z under the null hypothesis.

Let's assume:
- Pre-campaign data: 200 sessions with 40 clicks (CTR = 0.20)
- Post-campaign data: 180 sessions with 54 clicks (CTR = 0.30)

1. Calculate sample proportions:
   - $\bar{p_1} = 0.20$ (40/200)
   - $\bar{p_2} = 0.30$ (54/180)

2. Calculate pooled proportion:
   $$ \hat{p} = \frac{40 + 54}{200 + 180} = \frac{94}{380} \approx 0.2474 $$

3. Calculate standard error:
   $$ SE = \sqrt{0.2474(1 - 0.2474)\left(\frac{1}{200} + \frac{1}{180}\right)} \approx 0.045 $$

4. Compute z:
   $$ z = \frac{0.20 - 0.30}{0.045} \approx -2.22 $$

5. Find p-value:
   Using a standard normal distribution table, the p-value corresponding to z = -2.22 is approximately 0.0132. Since this is less than the typical threshold of 0.05, we reject H0.

The calculated p-value of 0.0132 indicates that there is strong evidence against the null hypothesis. This suggests that the marketing campaign has had a statistically significant effect on the click-through rate, increasing it from 20% to 30%. Practically, this means that the campaign was successful in engaging users, leading to higher revenue potential per session due to increased clicks.

This analysis can extend into several related concepts:
- **Causal Inference**: Further investigations could involve establishing a causal relationship between the marketing campaign and CTR using techniques like A/B testing or propensity score matching.
- **Bayesian Methods**: Employing Bayesian statistics could provide a different perspective on the data, allowing for the updating of beliefs about CTR as new data becomes available.
- **Simpson's Paradox**: Be cautious about potential confounding variables (e.g., seasonality, user demographics) that might affect CTR, leading to misleading interpretations if not controlled for.
- **Regression Analysis**: Implementing regression models could help in understanding the relationship between multiple factors influencing CTR and isolate the effect of the marketing campaign.

Calculate expected value for click-through rates

Redfin

What the Interviewer Expects

Key Topics to Cover

How to Approach This

Possible Follow-up Questions

Sharpen Your Skills on Codemia

Sample Answer

Problem Formulation

Solution Approach

Submit Your Answer

Redfin Data Scientist Interview Guide

Related Questions

Interpret mixed results across segments from an experiment

Explain instrumental variables with an example

Calculate conditional probability for coin flips

Explain difference-in-differences with an example

Explain p-value with an example