Practice: Interpreting Statistical Results

Okay, let's put the concepts from this chapter into practice. Statistical inference helps us move from observing a sample to making educated statements about the larger population. The key is not just getting a number (like an average or a p-value) but understanding what that number tells us (and what it doesn't).

In this section, we'll look at a few common scenarios where you might encounter statistical results and practice interpreting them correctly. Remember, the goal is to make sense of point estimates, confidence intervals, and p-values in context.

Scenario 1: Estimating Website Conversion Rate

Imagine you're working on an e-commerce website. You run an A/B test where 1000 visitors see a new checkout page design (Group B), while another 1000 see the old design (Group A). You want to estimate the conversion rate (percentage of visitors who make a purchase) for the new design.

After the test, you find that 55 out of the 1000 visitors who saw the new design made a purchase. Your analysis provides the following:

• Point Estimate for Conversion Rate (Group B): 0.055 (or 5.5%)
• 95% Confidence Interval for Conversion Rate (Group B): [0.042, 0.068] (or 4.2% to 6.8%)

Interpretation:

1. Point Estimate: The value 0.055 is our single best guess for the true conversion rate of the entire population of visitors if they were all shown the new design. This guess is based directly on our sample data (55 conversions / 1000 visitors).
2. Confidence Interval: The interval [0.042, 0.068] provides a range of plausible values for the true conversion rate of the new design. We are "95% confident" that the true conversion rate for all potential visitors lies between 4.2% and 6.8%.
   • What does "95% confident" mean? It refers to the method used to create the interval. If we were to repeat this experiment many times, constructing a 95% confidence interval each time, we'd expect about 95% of those intervals to successfully capture the true population conversion rate. It doesn't mean there's a 95% probability the true value is in this specific interval; rather, it reflects our confidence in the procedure used to generate the interval.
   • This interval gives us a sense of the uncertainty around our point estimate. A wider interval would indicate more uncertainty (perhaps due to a smaller sample size or more variability), while a narrower interval suggests a more precise estimate.

Scenario 2: Testing the Impact of a Feature Change

Let's say your team implemented a new recommendation algorithm on a streaming service. You want to know if this change significantly increased the average number of videos watched per user per week compared to the old algorithm.

You collect data from two groups of users over a month and perform a hypothesis test.

• Null Hypothesis ($H_0$): The new algorithm does not increase the average number of videos watched per user per week. (Mathematically, often written as $\mu_{new} \le \mu_{old}$ or $\mu_{new} - \mu_{old} \le 0$, where $\mu$ represents the population average).
• Alternative Hypothesis ($H_1$): The new algorithm does increase the average number of videos watched per user per week. ($\mu_{new} > \mu_{old}$ or $\mu_{new} - \mu_{old} > 0$).
• Significance Level ($\alpha$): You decide on a threshold of $\alpha = 0.05$. This means you're willing to accept a 5% chance of incorrectly concluding the new algorithm is better when it actually isn't (a Type I error).

The statistical software outputs the following results based on the sample data:

• Sample Average Videos (Old Algorithm): 8.2 videos/week
• Sample Average Videos (New Algorithm): 8.9 videos/week
• P-value: 0.028

Here's a simple visualization of the sample averages:

Sample averages show higher engagement with the new algorithm, but the p-value tells us if this difference is statistically significant.

Interpretation:

1. Compare p-value to $\alpha$: The calculated p-value (0.028) is less than our chosen significance level ($\alpha = 0.05$).
2. Decision: Because $p < \alpha$, we reject the null hypothesis ($H_0$).
3. Conclusion: We conclude that there is statistically significant evidence to suggest that the new recommendation algorithm increases the average number of videos watched per user per week compared to the old one. The observed difference (8.9 vs 8.2 videos in the samples) is unlikely to be due to random chance alone if the new algorithm was truly no better than the old one.

What if the p-value was 0.15?

If the p-value had been 0.15 (which is greater than $\alpha = 0.05$), our interpretation would change:

1. Compare p-value to $\alpha$: $0.15 > 0.05$.
2. Decision: We fail to reject the null hypothesis ($H_0$).
3. Conclusion: We conclude that there is not enough statistically significant evidence to say the new algorithm increases the average number of videos watched. Even though the sample average was higher (8.9 vs 8.2), a p-value of 0.15 suggests that such a difference could reasonably occur due to random sampling variation even if the true averages were the same (or if the new one was even worse). We haven't proven the algorithms are equally effective, only that we lack strong evidence to claim the new one is better.

Scenario 3: Comparing Machine Learning Model Performance

You've trained two different classification models (Model A and Model B) to predict customer churn. You test both models on the same hold-out test dataset and obtain their accuracy scores. You want to know if the difference in accuracy is statistically significant.

• Null Hypothesis ($H_0$): Both models have the same true accuracy on the underlying data distribution. (Accuracy$_A$ = Accuracy$_B$)
• Alternative Hypothesis ($H_1$): The models have different true accuracies. (Accuracy$_A \ne$ Accuracy$_B$)
• Significance Level ($\alpha$): You choose $\alpha = 0.05$.

You use an appropriate statistical test (like McNemar's test, which is suitable for comparing classifiers on the same dataset) and get:

• Model A Accuracy (on test set): 88%
• Model B Accuracy (on test set): 90%
• P-value: 0.21

Interpretation:

1. Compare p-value to $\alpha$: The p-value (0.21) is greater than the significance level ($\alpha = 0.05$).
2. Decision: We fail to reject the null hypothesis ($H_0$).
3. Conclusion: Although Model B achieved a higher accuracy (90%) than Model A (88%) on this specific test set, the difference is not statistically significant at the 0.05 level. There isn't strong enough evidence to conclude that Model B would consistently outperform Model A on new, unseen data from the same distribution. The observed 2% difference could plausibly be due to the specific data points included in the test set (random chance). In practice, you might still choose Model B if it has other advantages (like being faster or simpler), but you wouldn't claim superior accuracy based solely on this result.

These examples illustrate how interpreting statistical outputs like point estimates, confidence intervals, and p-values allows us to draw more careful and informed conclusions from our data, which is essential when making decisions or evaluating models in machine learning and data analysis.