9. Confidence Intervals

Core idea and interval structure

A confidence interval uses sample data to estimate an unknown population parameter while acknowledging sampling variability. Instead of giving only one number, it reports a range of plausible values.

The basic structure is

or equivalently, an interval from a lower bound to an upper bound. The point estimate is the sample statistic at the center of the interval, and the margin of error determines how far the endpoints extend from that estimate.

For a population proportion $p$, the point estimate is the sample proportion $\hat p$. For a population mean $\mu$, the point estimate is the sample mean $\bar{x}$.

The margin of error has the general form

This relationship explains the main factors affecting interval width:

• Greater variability increases the standard error and widens the interval.

• A smaller sample size generally increases the standard error and widens the interval.

• A higher confidence level requires a larger critical value and widens the interval.

• Increasing the sample size generally narrows the interval because standard errors decrease as $1/\sqrt n$.

Takeaway: A confidence interval combines an estimate with a quantified description of sampling uncertainty.

Confidence level and interpretation

The confidence level describes how well the interval-building method performs over many repeated random samples. A 95% confidence procedure produces intervals that contain the true parameter in about 95% of repeated samples of the same size and design.

A suitable interpretation of one computed interval is: We are 95% confident that the population parameter lies between the reported endpoints. The confidence level does not mean that there is a 95% probability that one already-computed interval contains the parameter. The parameter is treated as fixed, while the interval would change from sample to sample.

For two-sided intervals, commonly used critical values are approximately:

• 90% confidence: $z^*=1.645$

• 95% confidence: $z^*=1.960$

• 99% confidence: $z^*=2.576$

A 99% interval based on the same data is wider than a 95% interval because it uses a larger critical value. This is a trade-off: more confidence gives greater coverage in repeated sampling but less precision in each individual interval.

Takeaway: Always distinguish the long-run meaning of the confidence level from an incorrect probability claim about one fixed interval.

Intervals for a population proportion

For a categorical outcome, the sample proportion is

where $x$ is the number of observed successes and $n$ is the sample size. For a sufficiently large random sample, a one-proportion interval is

The estimated standard error is

and the margin of error is

Before using this normal-approximation formula, check:

1. Randomness: The data come from a random sample or randomized process.

2. Independence: Observations are independent. When sampling without replacement, a common guideline is that the sample is no more than 10% of the population.

3. Large counts: Both observed success and failure counts are at least 10:

   $$n\hat p\ge 10\quad\text{and}\quad n(1-\hat p)\ge 10.$$

For example, suppose $232$ of $400$ sampled adults report using a transportation app. Then

Using a 95% critical value of $1.96$, the estimated standard error is approximately $0.0247$, and the margin of error is approximately $0.048$. The interval is

In context, we are 95% confident that between 53.2% and 62.8% of adults in the target population use the transportation app. The interval can also be reported as 58.0% with a margin of error of about 4.8 percentage points.

Takeaway: A proportion interval requires an appropriate sampling process, independence, and enough observed successes and failures for the approximation to be reliable.

Intervals for a population mean

For a quantitative variable, the sample mean is

When the population standard deviation is unknown, the one-mean interval uses the t-distribution:

where $s$ is the sample standard deviation, $n$ is the sample size, and $t^*$ is based on $n-1$ degrees of freedom. The standard error is $s/\sqrt n$, and the margin of error is

If the population standard deviation $\sigma$ is known, a z-interval may instead be used:

For a one-mean t-interval, check randomness, independence, and the shape of the data. With a small sample, the quantitative variable should be approximately normal and free of extreme outliers or severe skewness. With a larger sample, the Central Limit Theorem generally makes the procedure more reliable, although extreme outliers can still distort the result.

For example, a random sample of $25$ packages has $\bar{x}=72.4$ hours and $s=8.5$ hours. The degrees of freedom are

Using $t^*\approx2.064$, the standard error is $8.5/\sqrt{25}=1.7$ hours and the margin of error is approximately $3.51$ hours. Thus,

We are 95% confident that the mean shipping time for all packages handled by the company is between 68.89 hours and 75.91 hours.

Takeaway: Use a t-interval when estimating a mean with an unknown population standard deviation, and verify that the data support the procedure.

Precision, decisions, and common errors

The width of an interval determines how precisely the parameter has been estimated. For a fixed confidence level, increasing the sample size decreases the margin of error. Because standard error contains $\sqrt n$ in the denominator, quadrupling the sample size approximately halves the margin of error.

For a mean, greater sample variability increases $s/\sqrt n$ and therefore increases the margin of error. For a proportion, the quantity $\hat p(1-\hat p)$ is largest when $\hat p$ is near $0.50$. When no prior estimate is available for planning a study, using $\hat p=0.50$ gives a conservative sample-size calculation.

An interval should be translated into the context of the question. A statement containing only numerical endpoints is incomplete. A useful conclusion identifies:

1. The population being studied.

2. The parameter being estimated.

3. The confidence level.

4. The interval in meaningful units or percentage terms.

Interval width also matters for decisions. If a company wants at least 60% customer approval and the interval is from 53.2% to 62.8%, the interval contains values both below and above 60%. The study therefore does not establish that approval exceeds 60%; more data or other evidence may be needed.

A confidence interval cannot repair a biased sample, nonresponse, dependence, poor measurement, or other flaws in study design. It addresses sampling variability, not every possible source of error. Avoid these common mistakes:

• Treating the confidence level as the probability for one fixed interval.

• Forgetting to check randomization, independence, sample size, or data shape.

• Using a z-interval for a mean when $\sigma$ is unknown and $s$ is used.

• Confusing the standard error with the margin of error.

• Reporting percentages without identifying whether they are decimals or percentages, or reporting means without units.

Takeaway: A statistically correct interval must also be communicated in context and judged for practical usefulness.