📊Honors Statistics Unit 8 Review

A confidence interval gives you a range of plausible values for the true population mean, built from your sample data. The general formula for a confidence interval when you know (or can estimate) the population standard deviation is:

$\bar{x} \pm z^* \frac{\sigma}{\sqrt{n}}$

Here's what each piece means:

$\bar{x}$ : the sample mean (your best single estimate of the population mean)
$z^*$ : the critical value tied to your confidence level (1.96 for 95%, 2.58 for 99%)
$\sigma$ : the population standard deviation (if unknown, substitute the sample standard deviation $s$ )
$n$ : the sample size

The quantity $\frac{\sigma}{\sqrt{n}}$ is called the standard error. It measures how much the sample mean tends to vary from sample to sample. Larger samples produce smaller standard errors, which means more precise estimates.

Worked example: Suppose you sample 100 women and find a mean height of 64 inches with a standard deviation of 2.5 inches. To build a 95% confidence interval:

Calculate the standard error: $\frac{2.5}{\sqrt{100}} = \frac{2.5}{10} = 0.25$
Multiply by the critical value: $1.96 \times 0.25 = 0.49$
Build the interval: $64 \pm 0.49$

The 95% confidence interval is (63.51, 64.49) inches. That 0.49 inches is your margin of error.

Note: When $\sigma$ is truly unknown and you're using $s$ from your sample, you should technically use a $t^*$ critical value instead of $z^*$ . With $n = 100$ , the difference is very small, but for smaller samples it matters a lot.

Interpretation of confidence levels

Getting the interpretation right is one of the trickiest parts of this topic. A 95% confidence level does not mean there's a 95% probability that this particular interval contains the true mean. The true mean is a fixed number; it's either in the interval or it isn't.

The correct interpretation: if you repeated the sampling process many times and built a 95% confidence interval each time, about 95% of those intervals would capture the true population mean. It's a statement about the method's reliability, not about any single interval.

For the example above, you'd say: "We are 95% confident that the true mean height of women is between 63.51 and 64.49 inches." This means the procedure used to generate this interval captures the true mean 95% of the time in the long run.

Confidence intervals for population means, Estimating a Population Mean (3 of 3) | Concepts in Statistics

Factors affecting interval width

Three things control how wide or narrow your confidence interval turns out:

Confidence level: Higher confidence requires a wider net. A 99% interval is wider than a 95% interval because you need more room to be more sure.
Sample size ( $n$ ): Larger samples shrink the standard error (since $n$ is in the denominator under a square root), producing narrower intervals. Going from $n = 100$ to $n = 500$ cuts the standard error by more than half.
Population variability ( $\sigma$ ): If individual measurements are more spread out, the sample mean is less precise, and the interval widens. A population with $\sigma = 3$ inches produces a wider interval than one with $\sigma = 2$ inches.

To get a more precise estimate, you can increase your sample size or accept a lower confidence level. In practice, increasing sample size is usually the better choice because it improves precision without sacrificing confidence.

Inferences from confidence intervals

Confidence intervals let you do more than just estimate a single mean. You can use them to evaluate claims and compare groups:

If someone claims the average height of women is 65 inches, but your 95% interval is (63.51, 64.49), that claim falls outside the interval. This is evidence against the claim at the 5% significance level.
When comparing two populations, non-overlapping confidence intervals suggest a meaningful difference between the means. (Though overlapping intervals don't automatically mean there's no difference; a formal two-sample test is more reliable for that comparison.)

Connection to broader statistical inference

Confidence intervals are one of the two main tools of statistical inference, alongside hypothesis testing. Both rely on the Central Limit Theorem, which states that the sampling distribution of $\bar{x}$ becomes approximately normal as $n$ increases, regardless of the population's shape. This is what justifies using $z^*$ (or $t^*$ ) critical values in the formula.

Hypothesis testing and confidence intervals are closely linked: a 95% confidence interval contains exactly those values that a two-sided hypothesis test at $\alpha = 0.05$ would fail to reject. They're two ways of looking at the same underlying inference.

📊Honors Statistics Unit 8 Review