Confidence Intervals for the Mean of a Normal Population
Confidence intervals are a fundamental concept in statistical inference, providing a range of plausible values for a population parameter based on sample data. In the context of AHL 4.16, we focus specifically on confidence intervals for the mean of a normal population.
Basic Concept
A confidence interval for the population mean μ is an interval estimate that is likely to contain the true population mean with a certain level of confidence. It is typically expressed as:
$(\text{point estimate} - \text{margin of error}, \text{point estimate} + \text{margin of error})$
Where the point estimate is usually the sample mean $\bar{x}$, and the margin of error depends on the chosen confidence level, sample size, and the distribution used.
NoteThe confidence level, often denoted as (1-α), is typically expressed as a percentage (e.g., 95% or 99%) and represents the probability that the interval contains the true population mean.
Calculating Confidence Intervals
The formula for a confidence interval depends on whether the population standard deviation (σ) is known or unknown.
When σ is Known (Using Normal Distribution)
When the population standard deviation is known, we use the standard normal distribution (z-distribution). The confidence interval is calculated as:
$$\bar{x} \pm z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$$
Where:
- $\bar{x}$ is the sample mean
- $z_{\alpha/2}$ is the critical value from the standard normal distribution
- σ is the known population standard deviation
- n is the sample size
Suppose we have a sample mean of 75, a known population standard deviation of 10, a sample size of 36, and we want a 95% confidence interval.
The z-score for a 95% confidence level is 1.96.
CI = $75 \pm 1.96 \cdot \frac{10}{\sqrt{36}}$ = $75 \pm 3.27$ = (71.73, 78.27)
We can interpret this as: We are 95% confident that the true population mean falls between 71.73 and 78.27.
When σ is Unknown (Using t-Distribution)
When the population standard deviation is unknown, we use the t-distribution. The confidence interval is calculated as:
$$\bar{x} \pm t_{\alpha/2, n-1} \cdot \frac{s}{\sqrt{n}}$$
Where:
- $\bar{x}$ is the sample mean
- $t_{\alpha/2, n-1}$ is the critical value from the t-distribution with n-1 degrees of freedom
- s is the sample standard deviation
- n is the sample size
