Calculate probability for click-through rates

Question

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

Codemia · Accepted Answer

We need to evaluate the click-through rates (CTR) for a new feature in Roblox. Let's define our parameters:

- Let $ p_0 $ be the baseline click-through rate (CTR) for the current feature, say $ p_0 = 0.05 $ (5%). 
- Let $ p_1 $ be the click-through rate for the new feature after implementation. 
- We will perform a hypothesis test where:
  - Null Hypothesis ($ H_0 $): There is no difference in click-through rates, $ p_1 - p_0 = 0 $ 
  - Alternative Hypothesis ($ H_a $): The new feature increases the click-through rate, $ p_1 - p_0 > 0 $.
- We will calculate the p-value to determine the significance of our results.

1. **Data Collection**: Assume we collected data from 1000 users, with 50 users clicking the link for the current feature (baseline) and 80 users clicking for the new feature.
2. **Sample Proportions**: 
   - Baseline CTR: $ \hat{p_0} = \frac{50}{1000} = 0.05 $ 
   - New Feature CTR: $ \hat{p_1} = \frac{80}{1000} = 0.08 $ 
3. **Standard Error Calculation**: 
   - Standard error (SE) for the difference in proportions is given by:
     $$ SE = \sqrt{\hat{p}(1 - \hat{p}) \left( \frac{1}{n_0} + \frac{1}{n_1} \right) } $$
   - Where $ \hat{p} = \frac{50 + 80}{1000 + 1000} = \frac{130}{2000} = 0.065 $
   - Thus, $ SE = \sqrt{0.065(1 - 0.065) \left( \frac{1}{1000} + \frac{1}{1000} \right)} = \sqrt{0.065 \cdot 0.935 \cdot 0.002} \approx 0.0077 $
4. **Z-Score Calculation**:
   - Z = $ \frac{\hat{p_1} - \hat{p_0}}{SE} = \frac{0.08 - 0.05}{0.0077} \approx 3.90 $
5. **P-value Calculation**:
   - Using Z-tables or a normal distribution calculator, find the p-value for Z = 3.90. This is approximately 0.00005.

Given our calculated p-value of 0.00005, we can interpret this in the context of the problem:
- If we set a significance level ($ \alpha $) at 0.05, since 0.00005 < 0.05, we can reject the null hypothesis.
- This suggests that the new feature significantly increases the click-through rate compared to the old feature.
- Additionally, we should consider the power of our test, which is related to the sample size and effect size. A power analysis might indicate that our sample size is adequate to detect this effect.

The result indicates that the new feature has a statistically significant higher click-through rate than the baseline. However, we must be cautious:
- **Practical Significance**: A p-value does not indicate how meaningful the difference is in a real-world context. A shift from 5% to 8% could be significant but may not lead to substantial revenue increases depending on user behavior.
- **Confounding Variables**: Factors such as user demographics, time of day, or marketing efforts could influence click-through rates. We might need to control for these through stratification or regression analysis.

This problem connects to several broader concepts:
- **Power Analysis**: Understanding how sample size impacts our ability to detect true effects is crucial, especially in product testing.
- **Multiple Testing Correction**: If we were testing multiple features simultaneously, applying corrections such as Bonferroni or False Discovery Rate to adjust p-values would be necessary.
- **Bayesian Inference**: Instead of frequentist hypothesis testing, we could also consider Bayesian methods to incorporate prior beliefs about CTRs and update our beliefs with the new data.

Calculate probability for click-through rates

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