5 Estimation and Confidence Intervals

Learn how confidence intervals estimate population quantities, how to choose and interpret interval methods, and how to use interval precision and assumptions in business decisions.

Estimates and interval width

A uses sample data to estimate an unknown population quantity, such as average order value or the proportion of customers who renew. A adds a range of plausible values to show the estimate's sampling uncertainty. Its general form is

\text{estimate} \;\pm\; (\text{critical value})(\text{standard error}).

The is the product of the critical value and . Larger samples usually reduce the , while higher confidence levels require larger critical values and therefore wider intervals.

Interpreting a

A 95% confidence procedure captures the true population parameter in about 95% of intervals produced over repeated samples, provided the procedure's assumptions hold. Once one interval has been calculated, the parameter is fixed: that interval either contains it or does not. Thus, saying that an interval is “95% confident” describes the method's long-run performance, not a 95% probability that this particular interval contains the parameter.

Intervals for a population mean

For a random, independent sample of size $n$ , with sample mean $\bar{x}$ and sample standard deviation $s$ , the usual two-sided for a population mean $\mu$ , when the population standard deviation is unknown, is

\bar{x} \pm t_{1-\alpha/2,\,n-1}\frac{s}{\sqrt{n}}.

Here, $1-\alpha$ is the , and the critical value comes from the $t$ -distribution with $n-1$ degrees of freedom. When the population standard deviation $\sigma$ is known, use $z_{1-\alpha/2}\sigma/\sqrt{n}$ instead. The is appropriate when the data are reasonably well behaved; especially with small samples, strong skew or outliers can undermine the method.

Example: mean order value

A retailer samples 36 orders and finds a mean order value of $52 with a sample standard deviation of $12. For a 95% interval, the critical $t$ -value with 35 degrees of freedom is about 2.03. The is $12/\sqrt{36}=2$ , so the interval is

52 \pm 2.03(2) = (47.94,\;56.06).

The retailer can report that the population mean order value is estimated to be between about $47.94 and $56.06, using a procedure with 95% long-run coverage.

Intervals for a population proportion

For a binary business outcome, such as whether a customer renews, let $x$ be the number of successes in a sample of size $n$ . The sample proportion is $\hat{p}=x/n$ . A common large-sample normal-approximation interval is

\hat{p} \pm z_{1-\alpha/2}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}.

This approximation requires enough successes and failures to be reasonable. It can perform poorly with small samples or proportions near zero or one; in those situations, a is often preferable.

Example: customer renewals

Of 200 sampled customers, 120 renewed, so $\hat{p}=0.60$ . The approximate 95% interval is

0.60 \pm 1.96\sqrt{\frac{0.60(0.40)}{200}} \approx (0.532,\;0.668).

The estimated renewal rate is 60%, with an approximate interval from 53.2% to 66.8%. Check the method's conditions before reporting this approximation.

Intervals for other quantities

The estimation logic also applies to other population quantities, but the appropriate and critical value depend on the quantity and study design.

Difference in means: To compare average sales per customer between two groups, estimate $\mu_1-\mu_2$ using the difference in sample means and its appropriate . For paired data, such as sales before and after a change for the same stores, analyze the within-store differences. A containing zero does not establish that the means are equivalent; it indicates that zero is among the values not ruled out by that interval.
Difference in proportions: To compare rates, such as returns in two shipping programs, use an interval for $p_1-p_2$ . Choose a method appropriate to the sample sizes and observed proportions; simple normal approximations may be inaccurate.
Regression coefficients: A for a slope estimates the change in the mean outcome associated with a one-unit increase in a predictor, under the fitted model. For simple linear regression, the usual interval is the estimated slope $b_1$ plus or minus a $t$ -critical value times its .
Variance or standard deviation: For normally distributed data, intervals for variance or standard deviation use the chi-square distribution rather than the mean's $t$ -interval.
A population total: If a simple random sample estimates a population mean and the population size $N$ is known, multiply both endpoints of the mean interval by $N$ to obtain an interval for the total, subject to the sampling design and any needed finite-population adjustment.

Using intervals in business decisions

A helps assess both the estimated effect and its precision. For example, a campaign's estimated increase in average revenue may be positive, but a wide interval could still include no increase or effects too small to justify the cost. Compare the interval with a , not just zero: an effect that is statistically plausible may still be too small to matter financially.

Intervals describe uncertainty from sampling under stated assumptions. They do not automatically correct for biased sampling, nonresponse, measurement errors, confounding, or a model that does not fit the data. A precise interval from unrepresentative data can still be misleading.

Putting interval estimates together

A combines a with a . Its describes long-run coverage, while its width reflects precision. Select a method suited to the parameter—such as a for a mean, an appropriate binomial interval for a proportion, or model-based intervals for comparisons and regression—and interpret the result in light of its sampling assumptions and business relevance.