How Confidence Level Changes a Proportion Interval

A proportion interval starts with a point estimate

Suppose a random sample contains $x$ observations with a characteristic out of $n$ total observations. The sample proportion

$$\hat p=\frac{x}{n}$$

is the point estimate of the unknown population proportion $p$. For the usual one-sample normal-approximation interval, the estimated standard error is

$$SE=\sqrt{\frac{\hat p(1-\hat p)}{n}}.$$

A confidence interval has the general form

$$\hat p\pm z^*SE,$$

where $z^*$ is the standard-normal critical value associated with the selected confidence level.

Higher confidence means a wider interval

With the sample fixed, $\hat p$, $n$, and the standard error do not change. Raising the confidence level increases $z^*$, which increases the margin of error and widens the interval.

For example, suppose 232 of 400 sampled students use a campus tutoring service. Then

$$\hat p=\frac{232}{400}=0.58, \qquad SE=\sqrt{\frac{(0.58)(0.42)}{400}}\approx0.02468.$$

The 90% interval is approximately

$$0.58\pm1.645(0.02468)=(0.539,0.621),$$

while the 95% interval is approximately

$$0.58\pm1.960(0.02468)=(0.532,0.628).$$

The 95% procedure pays for greater long-run capture by producing a wider range.

Interpret confidence as a property of the procedure

A practical interpretation is: “We are 95% confident that the population proportion lies between the two endpoints.” The repeated-sampling meaning is more precise: if the same sampling-and-interval procedure were repeated many times, about 95% of the resulting intervals would contain the fixed population proportion.

It is not correct to say that 95% of individual observations lie inside the interval. The interval estimates a population proportion, not the spread of individual data.

Check the conditions before generalizing

The normal approximation needs enough expected successes and failures for the sampling distribution of $\hat p$ to be approximately normal. The observations should also come from a random or otherwise defensibly representative process and be sufficiently independent.

When counts are small or sampling is biased, arithmetic alone does not rescue the inference. A different interval method or a better data-collection design may be needed.