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
The t-distribution is used regardless of sample size when σ is unknown. This is a key point in AHL 4.16.
ExampleSuppose we have a sample mean of 65, a sample standard deviation of 8, a sample size of 25, and we want a 90% confidence interval.
The t-score for a 90% confidence level with 24 degrees of freedom is approximately 1.711.
CI = $65 \pm 1.711 \cdot \frac{8}{\sqrt{25}}$ = $65 \pm 2.74$ = (62.26, 67.74)
We can interpret this as: We are 90% confident that the true population mean falls between 62.26 and 67.74.
Interpreting Confidence Intervals
Understanding how to interpret confidence intervals is crucial. Here are some key points:
- Probability of Containing the True Mean: A 95% confidence interval, for instance, means that if we were to repeat the sampling process many times and calculate the confidence interval each time, about 95% of these intervals would contain the true population mean.
- Not About Individual Intervals: It's a common misconception that there's a 95% chance that a specific interval contains the true mean. Rather, it's about the long-run behavior of the method.
- Width of the Interval: The width of the confidence interval provides information about the precision of our estimate. Narrower intervals indicate more precise estimates.
- Factors Affecting Width: The width of the confidence interval is influenced by:
- Sample size (n): Larger samples lead to narrower intervals
- Confidence level: Higher confidence levels lead to wider intervals
- Population variability (σ or s): More variable populations lead to wider intervals
A common mistake is to interpret a 95% confidence interval as meaning there's a 95% chance the true population mean lies within that specific interval. This is incorrect. Once calculated, the interval either contains the true mean or it doesn't. The 95% refers to the reliability of the method, not the probability of a specific interval containing the mean.
Practical Applications
Confidence intervals have numerous real-world applications:
- Quality Control: In manufacturing, confidence intervals can be used to estimate the mean lifetime of products.
- Medical Research: Researchers might use confidence intervals to estimate the average effect of a new drug.
- Opinion Polls: Political analysts use confidence intervals to estimate the proportion of voters who support a candidate.
- Environmental Science: Scientists might use confidence intervals to estimate average pollution levels in a river.
A biologist is studying the length of a particular species of fish. From a sample of 50 fish, she calculates a 95% confidence interval for the mean length to be (22.3 cm, 24.7 cm).
Interpretation: We are 95% confident that the true mean length of this fish species in the population is between 22.3 cm and 24.7 cm. This means that if the biologist were to repeat this sampling process many times, about 95% of the calculated intervals would contain the true population mean length.
Choosing Between Normal and t-Distribution
The choice between using the normal distribution or t-distribution depends on whether the population standard deviation (σ) is known:
- Known σ: Use the normal distribution (z-distribution)
- Unknown σ: Use the t-distribution, regardless of sample size
In real-world scenarios, it's rare to know the population standard deviation. Therefore, you'll often find yourself using the t-distribution in practical applications.
Conclusion
Confidence intervals are a powerful tool in statistical inference, allowing us to estimate population parameters with a measure of reliability. By understanding how to calculate and interpret these intervals, students can apply this knowledge to a wide range of real-world problems, from scientific research to business decision-making. The ability to choose the appropriate distribution (normal or t) based on the available information is a crucial skill in this topic.
