8.6 Confidence Interval (Women's Heights)

Confidence Interval for Women's Heights

Confidence intervals for population means

A confidence interval gives you a range of plausible values for a population mean based on sample data. Instead of saying "the average height of women is 64 inches," you can say "we're 95% confident the true mean falls between 63.5 and 64.5 inches." That range accounts for the fact that your sample won't perfectly represent the whole population.

To build a confidence interval, you need four things: the sample mean ($\bar{x}$), the sample standard deviation ($s$), the sample size ($n$), and a confidence level (typically 90%, 95%, or 99%).

The formula is:

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

where $z^*$ is the critical value from the standard normal distribution that corresponds to your chosen confidence level. Common critical values: $z^* = 1.645$ for 90%, $z^* = 1.96$ for 95%, and $z^* = 2.576$ for 99%.

The sample mean ($\bar{x}$) serves as your point estimate for the population mean. The confidence interval builds outward from that point estimate by adding and subtracting the margin of error.

Factors affecting interval width

Two main factors control how wide or narrow your interval turns out:

Sample size ($n$):

• Larger samples produce narrower intervals because the standard error ($\frac{s}{\sqrt{n}}$) shrinks as $n$ grows. More data means less uncertainty.
• Smaller samples produce wider intervals. With less data, you're less sure where the true mean sits.

Confidence level:

• Higher confidence (like 99%) requires a larger critical value ($z^*$), which widens the interval. You're casting a wider net to be more sure you've captured the true mean.
• Lower confidence (like 90%) uses a smaller $z^*$, giving a narrower interval. You gain precision but accept a higher chance of missing the true mean.

There's a real tradeoff here: you can't get both a narrow interval and high confidence without increasing your sample size.

One more note: when sample sizes are small (generally $n < 30$) and you're using $s$ instead of a known population standard deviation, you should use the t-distribution instead of the normal distribution. The t-distribution uses degrees of freedom ($n - 1$) and produces slightly larger critical values, which makes the interval wider to account for extra uncertainty.

Margin of error in height estimates

The margin of error is the "±" part of your confidence interval. It represents the maximum expected difference between your sample estimate and the true population parameter.

$\text{Margin of Error} = z^* \frac{s}{\sqrt{n}}$

For example, if you report "64 inches ± 0.5 inches," the margin of error is 0.5 inches, and your confidence interval runs from 63.5 to 64.5 inches.

A smaller margin of error means a more precise estimate. You can shrink it in two ways:

• Increase the sample size ($n$). Since $n$ is under a square root in the denominator, you need to quadruple $n$ to cut the margin of error in half.
• Lower the confidence level. This reduces $z^*$, which shrinks the margin of error, but you're now less confident the interval captures the true mean.

When reporting results, always include the margin of error or the full interval so your reader understands how precise the estimate actually is.

Statistical inference and hypothesis testing

Confidence intervals are one of the two main tools of statistical inference, which is the process of drawing conclusions about a population from sample data.

The central limit theorem is what makes this all work: for sufficiently large samples, the distribution of sample means is approximately normal regardless of the population's shape. That's why you can use the normal (or t) distribution to build confidence intervals.

Hypothesis testing is the other major inference tool. While a confidence interval estimates a range for a parameter, a hypothesis test evaluates a specific claim about that parameter (for example, "is the mean height of women equal to 64 inches?"). The two methods are closely related: if a hypothesized value falls outside your 95% confidence interval, you'd reject that hypothesis at the 0.05 significance level.