Confidence intervals Knowledge-Based Study

Study notes

1. What is a Confidence Interval?

   In inferential statistics, we often want to estimate an unknown population parameter, such as the population mean ($\mu$) or population proportion ($p$). A single value calculated from a sample, such as the sample mean ($\bar{x}$), provides a point estimate. However, because different samples yield different values due to sampling variability, a point estimate alone does not convey uncertainty.

   A confidence interval (CI) provides an estimated range of plausible values for the population parameter, calculated at a specified confidence level (commonly 90%, 95%, or 99%).

   The Anatomy of a Confidence Interval

   Most confidence intervals share a common structure:

   Confidence Interval = Point Estimate $\pm$ Margin of Error

   The margin of error (MOE) quantifies the uncertainty and is calculated as:

   Margin of Error = Critical Value $\times$ Standard Error

   • Point Estimate: The sample statistic (e.g., $\bar{x}$ for a mean).
   • Critical Value: A factor determined by the sampling distribution and confidence level. For normally distributed data with known population standard deviation ($\sigma$), we use $z^*$ (e.g., $z^* = 1.96$ for a 95% confidence level). When $\sigma$ is unknown, the $t$-distribution critical value ($t^*$) is used with $n - 1$ degrees of freedom.
   • Standard Error (SE): The standard deviation of the sampling distribution, calculated as $\text{SE} = \frac{\sigma}{\sqrt{n}}$ (or $\frac{s}{\sqrt{n}}$ when estimating with sample standard deviation $s$).

   Constructing a 95% Confidence Interval for a Mean

   Suppose a researcher takes a simple random sample of $n = 64$ items from a large population. The sample mean is $\bar{x} = 50$, and the population standard deviation is known to be $\sigma = 16$.

   1. Identify the point estimate: $\bar{x} = 50$.
   2. Determine the critical value: For a 95% confidence level using the standard normal distribution, $z^* = 1.96$.
   3. Calculate the standard error:
      $\text{SE} = \frac{\sigma}{\sqrt{n}} = \frac{16}{\sqrt{64}} = \frac{16}{8} = 2$.
   4. Calculate the margin of error:
      $\text{MOE} = z^* \times \text{SE} = 1.96 \times 2 = 3.92$.
   5. Form the interval:
      $\bar{x} \pm \text{MOE} = 50 \pm 3.92$, which gives the interval $(46.08, 53.92)$.

   How to Interpret Confidence Intervals

   Correct interpretation is essential to statistical inference:

   • Correct: "We are 95% confident that the true population mean $\mu$ lies between 46.08 and 53.92." This means that if we were to take many independent random samples of size 64 and construct a 95% confidence interval from each sample, approximately 95% of those resulting intervals would contain the true population parameter.
   • Incorrect: "There is a 95% probability that the true population mean lies between 46.08 and 53.92." Once an interval is computed, the fixed population parameter $\mu$ is either inside it or not; the probability refers to the long-run performance of the estimation method, not the parameter itself.

   Factors Affecting Interval Width

   The width of a confidence interval depends on two key elements:

   • Confidence Level: Increasing the confidence level increases the critical value, producing a wider interval.
   • Sample Size ($n$): Increasing the sample size decreases the standard error, producing a narrower, more precise interval.