Data Science Confidence Intervals

Confidence intervals (CIs) are a key concept in data science, providing a range of values within which we expect a population parameter (like a mean or proportion) to lie, given a certain level of confidence. Here’s an overview

 

1. What Is a Confidence Interval?

  • Definition:
    A confidence interval gives an estimated range of values which is likely to include an unknown population parameter. The interval is computed from sample data and is associated with a confidence level (e.g., 95%).
  • Confidence Level:
    This is the probability (expressed as a percentage) that the confidence interval contains the true population parameter if you were to repeat the experiment many times. For example, a 95% confidence level means that if you repeated the study 100 times, about 95 of those confidence intervals would capture the true parameter.

2. Constructing a Confidence Interval

The basic formula for a confidence interval for a population mean is:

CI = \( \bar{x} \pm z^* \times \left( \frac{\sigma}{\sqrt{n}} \right) \)

  • \( \bar{x} \) is the sample mean.
  • \( z^* \) (or  \( t^* \) for small sample sizes) is the critical value from the standard normal (or t-) distribution corresponding to the desired confidence level.
  • \( \sigma \) is the population standard deviation (or sample standard deviation s when sigma is unknown).
  • \( n \) is the sample size.
  • The term \( \frac{\sigma}{\sqrt{n}} \) is known as the standard error.

For smaller sample sizes or when the population standard deviation is unknown, we use the t-distribution:

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

Where \( t^* \) is the critical value from the t-distribution with \( n \) – 1 degrees of freedom.

3. Steps to Compute a Confidence Interval

  1. Collect the Data:
    Gather your sample and compute the necessary statistics (mean, standard deviation, and sample size).
  2. Determine the Confidence Level:
    Choose your confidence level (commonly 90%, 95%, or 99%).
  3. Select the Appropriate Distribution:
    • Use the z-distribution if the population standard deviation is known or if \( n \) is large (usually \( n \gt 30 \) ).
    • Use the t-distribution if the population standard deviation is unknown and \( n \) is small.
  4. Calculate the Standard Error:
    Compute \( \frac{\sigma}{\sqrt{n}} \) or \( \frac{s}{\sqrt{n}} \).
  5. Find the Critical Value:
    • For a 95% confidence level, the z-critical value is approximately 1.96.
    • For the t-distribution, look up the value in a t-table based on your chosen confidence level and degrees of freedom (\( n – 1 \)).
  6. Compute the Margin of Error:
    Multiply the critical value by the standard error.
  7. Construct the Interval:
    Add and subtract the margin of error from the sample mean.

4. Example

Imagine you have a sample of 40 observations with:

  • Sample mean \( \bar{x} = 100 \)
  • Sample standard deviation \( {s} = 15 \)
  • Desired confidence level: 95%

Step-by-Step:

  1. Standard Error:  SE = \( \frac{s}{\sqrt{n}} = \frac{15}{\sqrt{40}} \approx \frac{15}{6.32} \approx 2.37 \)
  2. Critical Value:
    For a 95% confidence level with 39 degrees of freedom (using the t-distribution), assume  \( t^* \approx 2.02 \) (this value can vary slightly depending on the table).
  3. Margin of Error:   \( \text{Margin of Error} = 2.02 \times 2.37 \approx 4.79 \)
  4. Confidence Interval: 

So, you can be 95% confident that the true population mean lies between 95.21 and 104.79.

5. Why Confidence Intervals Matter in Data Science

  • Uncertainty Quantification:
    They provide a measure of uncertainty around estimates, giving more context than a simple point estimate.
  • Decision Making:
    By understanding the range within which a parameter lies, data scientists can make more informed decisions and recommendations.
  • Comparisons:
    Confidence intervals can be used to compare groups or treatments. If intervals do not overlap significantly, this may suggest a statistically significant difference.