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 start by defining it within the framework of statistical inference. A confidence interval provides a range of values, derived from a dataset, that is likely to contain the true population parameter (e.g., the mean) with a specified level of confidence. Mathematically, if we denote the sample mean as $ \bar{x} $, the sample size as $ n $, the standard deviation as $ s $, and the critical value from the z-distribution (for large samples) or t-distribution (for smaller samples) as $ t^* $, the confidence interval can be expressed as:

$$ CI = \bar{x} \pm t^* \left( \frac{s}{\sqrt{n}} \right) $$

This notation indicates that the CI is centered around the sample mean and extends $ t^* \left( \frac{s}{\sqrt{n}} \right) $ units in both directions.

To compute a confidence interval, we follow these steps:
1. **Collect Data**: Gather a random sample from the population.
2. **Calculate Sample Mean $ \bar{x} $**: Sum the values of the sample and divide by the number of observations.
3. **Calculate Standard Deviation $ s $**: Use the formula $ s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} $ where $ x_i $ are the sample values.
4. **Determine the Sample Size $ n $**: Count the number of observations in the sample.
5. **Select the Confidence Level**: Common levels are 90%, 95%, and 99% which correspond to certain critical values from the t or z distributions.
6. **Calculate the Margin of Error**: Compute $ ME = t^* \left( \frac{s}{\sqrt{n}} \right) $ using the appropriate critical value.
7. **Construct the Confidence Interval**: Finally, form the interval $ \bar{x} \pm ME $.

Let's say we conducted a survey on Oracle employees to find out their average work hours per week. We randomly sampled 30 employees and found:
- Sample Mean ($ \bar{x} $): 40 hours
- Sample Standard Deviation ($ s $): 5 hours
- For a 95% confidence level, the critical value ($ t^* $) for 29 degrees of freedom is approximately 2.045.

Now, we calculate:
1. **Margin of Error**:
   $$ ME = 2.045 \left( \frac{5}{\sqrt{30}} \right) \approx 2.045 \times 0.9129 \approx 1.87 $$

2. **Confidence Interval**:
   $$ CI = 40 \pm 1.87 \rightarrow (38.13, 41.87) $$

Thus, we can say with 95% confidence that the average work hours for all Oracle employees lie between approximately 38.13 and 41.87 hours per week.

The result of the confidence interval indicates that we are 95% confident that the true average work hours of all Oracle employees falls within the range of 38.13 to 41.87 hours per week. This means that if we were to take many samples and construct confidence intervals in the same way, about 95% of those intervals would contain the true population mean. However, it is crucial to note that this does not guarantee that the true mean is in this specific interval; it reflects the uncertainty inherent in sampling.

Understanding confidence intervals leads to several important discussions in statistics. For instance, one could explore the difference between Bayesian and frequentist approaches to inference, where Bayesian methods might provide a credible interval instead of a confidence interval. Additionally, one could discuss potential confounding variables that might affect the average work hours, such as remote work policies or departmental differences, and how to account for those when interpreting the results. Finally, the distinction between practical significance (the real-world relevance of the findings) and statistical significance (the likelihood that the findings are not due to chance) is another critical area for further exploration.

Explain confidence interval with an example

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