Explain confidence interval with an example

Question

Explain confidence interval in simple terms and provide a concrete example.

Codemia · Accepted Answer

To explain the concept of a confidence interval (CI), we first need to establish the context and notation. Let's say TikTok wants to understand the average time users spend on the app daily. We will conduct a survey where we collect data from a random sample of users.

Let:  
- $ \mu $ = true average time spent on the app (unknown population mean)  
- $ ar{x} $ = sample mean of the time spent  
- $ n $ = sample size  
- $ s $ = sample standard deviation  
- $ t $ = critical value from the t-distribution for a given confidence level

The confidence interval can be formulated as:  
$$ CI = \bar{x} \pm t \cdot \frac{s}{\sqrt{n}} $$  
This interval provides a range of values within which we expect the true average time spent on the app, $ \mu $, to lie with a specified confidence level (e.g., 95%).

To compute the confidence interval, we will follow these steps:  
1. **Collect Data**: Gather a sample of user data on time spent using TikTok.  
2. **Calculate Sample Mean and Standard Deviation**: Compute $ \bar{x} $ and $ s $ from the sample.  
3. **Determine Sample Size**: Identify $ n $, the number of users surveyed.  
4. **Find Critical t-value**: For a 95% confidence level and degrees of freedom $ df = n - 1 $, look up the critical value from the t-distribution.  
5. **Compute the Confidence Interval**: Use the formula provided to calculate the interval.

Assume we surveyed 30 TikTok users, and found:  
- $ \bar{x} = 45 $ minutes (average time spent)  
- $ s = 10 $ minutes (standard deviation)  
- $ n = 30 $

For a 95% confidence level and $ df = 29 $, the critical t-value is approximately 2.045.  
Now we can calculate the confidence interval:  
$$ CI = 45 \pm 2.045 \cdot \frac{10}{\sqrt{30}} $$  
$$ = 45 \pm 2.045 \cdot 1.8257 $$  
$$ = 45 \pm 3.73 $$  
Thus, the confidence interval is approximately $$ (41.27, 48.73) $$. This means we are 95% confident that the true average time spent on TikTok by all users lies between 41.27 and 48.73 minutes.

In practical terms, this confidence interval means that TikTok can report with high confidence that the average time users spend on the platform is between 41.27 and 48.73 minutes. This information is crucial for marketing strategies, user engagement initiatives, and feature development, as it reflects user behavior and engagement levels. However, it’s important to remember that this is an estimate based on a sample, and not the exact value for the entire user base.

Understanding confidence intervals connects to several related concepts:  
1. **Statistical Significance vs Practical Significance**: Just because a confidence interval doesn’t include zero doesn’t mean the effect is practically significant.  
2. **Bayesian vs Frequentist Inference**: While we used frequentist methods for this CI, a Bayesian approach could provide a different perspective by incorporating prior beliefs.  
3. **Confounders**: Potential confounders, such as time of day or user demographics, could affect the average time spent. Addressing these in data collection or analysis is critical for accurate insights.  
4. **Regression Analysis**: Users may want to analyze how different factors (age, location, etc.) predict time spent, which ties into confidence intervals for regression coefficients.

Explain confidence interval with an example

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