07 Confidence Intervals

From point estimates to intervals

A confidence interval combines a sample-based estimate with information about how much that estimate may vary across random samples.

The basic structure

A Confidence interval has the form

The point estimate identifies the center of the interval, while the margin of error determines how far the endpoints extend from that center. For example, a sample mean estimates a population mean, but another random sample would usually produce a different sample mean. An interval communicates this uncertainty more fully than a single number.

A Point estimate is a single sample statistic used to estimate a population parameter:

• The sample mean $\bar{x}$ estimates the population mean $\mu$.

• The sample proportion $\hat{p}=x/n$ estimates the population proportion $p$.

• The sample standard deviation $s$ estimates the population standard deviation $\sigma$.

Takeaway: A point estimate gives the best single-number estimate from the sample; a confidence interval adds a range that reflects sampling uncertainty.

Confidence levels and critical values

The Confidence level describes the long-run performance of an interval-producing method. If the same population were sampled repeatedly and the same method were applied each time, approximately the chosen percentage of the resulting intervals would contain the true parameter.

For a confidence level of $1-\alpha$, $\alpha$ is the total probability placed in the two tails of the sampling distribution. Common standard-normal critical values are:

• $90\%$ confidence: $\alpha=0.10$ and $z^*=1.645$.

• $95\%$ confidence: $\alpha=0.05$ and $z^*=1.960$.

• $99\%$ confidence: $\alpha=0.01$ and $z^*=2.576$.

A $95\%$ confidence procedure does not mean that a particular completed interval has a $95\%$ probability of containing the fixed parameter. After the interval is calculated, it either contains the parameter or it does not. The $95\%$ describes the success rate of the method over repeated samples.

Holding sample size and variability constant, increasing the confidence level increases the critical value and makes the interval wider.

Takeaway: Higher confidence provides a more reliable long-run procedure but requires a wider interval.

How interval width is determined

The Margin of error is the distance between the point estimate and either endpoint of a symmetric interval. It is determined by the critical value and the standard error:

Thus, the interval can be written as

For many estimators, the standard error decreases in proportion to $1/\sqrt{n}$. This has an important planning consequence: reducing the margin of error by half generally requires about four times as many observations.

The margin of error tends to decrease when:

• the sample size increases;

• population variability decreases; or

• a lower confidence level is selected.

Do not confuse the margin of error with the full interval width. If the margin of error is $E$, the width of a symmetric interval is $2E$.

Takeaway: Sample size controls precision through a square-root relationship, while confidence level and variability also affect interval width.

Intervals for a population mean

For a population mean, the correct formula depends on whether the population standard deviation $\sigma$ is known.

Known population standard deviation

When $\sigma$ is known, use

The margin of error is

This method requires independent observations and an approximately normal sampling distribution for the mean. A normally distributed population makes the method exact; for large samples, the Central Limit Theorem often supports the normal approximation.

Unknown population standard deviation

When $\sigma$ is unknown, replace it with the sample standard deviation $s$ and use the t-distribution:

The degrees of freedom are

The heavier tails of the t-distribution account for the additional uncertainty caused by estimating $\sigma$. As $n$ increases, the t-distribution becomes increasingly similar to the standard normal distribution.

Worked example

Suppose $n=64$, $\bar{x}=72$, and $s=8$. For a $95\%$ interval with $df=63$, take $t^*\approx2.000$. Then

Therefore,

The appropriate interpretation is: using this sampling method, a $95\%$ confidence interval for the population mean is from $70$ to $74$ units.

Takeaway: Use $z^*$ when the population standard deviation is known; use $t^*$ with $n-1$ degrees of freedom when it is unknown.

Intervals for a population proportion

A population proportion $p$ is estimated by the sample proportion

where $x$ is the number of successes and $n$ is the sample size. For a sufficiently large sample, the introductory normal-approximation interval is

Its margin of error is

The approximation is most reliable when the expected numbers of successes and failures are both adequately large. For small samples or proportions near $0$ or $1$, a Wilson or exact binomial interval is generally preferable to the simple symmetric normal interval.

Worked example

In a random sample of $400$ voters, $240$ support a proposal. The sample proportion is

Using $z^*=1.96$ for a $95\%$ interval,

The interval is

Thus, the estimated population proportion is $60\%$, and the approximate $95\%$ interval extends from $55.2\%$ to $64.8\%$.

Takeaway: Check the success and failure counts before relying on the normal approximation for a proportion.

Planning sample size

Sample size must be planned before data collection by specifying the confidence level, the largest acceptable margin of error, and an estimate of population variability. Always round the result up, because rounding down could make the actual margin of error larger than requested.

Planning for a mean

When a planning value for the population standard deviation is available, use

For example, with $95\%$ confidence, $E=3$, and $\sigma=12$,

Rounding up gives $n=62$.

Planning for a proportion

With a prior planning estimate $p^*$, use

If no prior estimate is available, use $p^*=0.50$. This is conservative because $p(1-p)$ is largest at $p=0.50$, producing the largest required sample size.

For $95\%$ confidence and $E=0.03$ with no prior estimate,

Rounding up gives $n=1068$.

When sampling without replacement from a relatively small, known population, a finite population correction can reduce the required sample size:

Here, $n_0$ is the sample size calculated as if the population were very large, and $N$ is the population size.

Takeaway: Set the desired precision first, calculate the required size, and round up.

Assumptions and error checks

Before interpreting an interval, verify that the method matches the design and data.

Conditions to check

• Randomness: The sample should reasonably represent the target population.

• Independence: Observations should be independent, or the sampling design must account for dependence.

• Distributional conditions: The normal or t-based method should be appropriate for the sample size and population shape.

• Proportion counts: A normal approximation should not be used automatically when success or failure counts are very small.

• Target parameter: An interval for a mean estimates a population mean, not the range of individual observations.

Frequent interpretation and calculation errors

1. Treating a $95\%$ confidence level as a $95\%$ probability for one fixed interval.

2. Using $z^*$ when $\sigma$ is unknown and the sample is small, instead of using $t^*$.

3. Rounding a required sample size down.

4. Confusing the margin of error with the full interval width.

5. Assuming that a higher confidence level produces a narrower interval.

A useful final check is to identify the parameter, the point estimate, the critical value, the standard error, and the assumptions supporting the method. Together, these determine whether the interval is both numerically correct and appropriately interpreted.

Final takeaway: A confidence interval is meaningful only when its sampling method, formula, assumptions, and interpretation all agree.