Why wouldn't we always just use a 99% confidence interval?

To be more confident, you have to cast a wider net. A $99\\%$ confidence interval will be wider than a $95\\%$ confidence interval for the same data. If the interval is too wide (e.g., "I am $99\\%$ confident the average test score is between $10$ and $100$"), it becomes practically useless.

How does sample size change the interval?

A larger sample size gives you more information, which shrinks your margin of error. If you survey $1000$ people instead of $50$, your "net" can be much smaller and still maintain a $95\\%$ confidence level. Larger samples lead to narrower, more precise intervals.

Can I find the original estimate if I only have the interval bounds?

Yes! Because the estimate is exactly in the middle of the interval, you just average the upper and lower bounds. For an interval of $(10, 20)$, the estimate is $(10 + 20) / 2 = 15$.

How do you read a confidence interval?

When you read a confidence interval, you are looking at a range of plausible values for a hidden population number (like an average or a percentage) based on a sample. Instead of guessing one exact number, a confidence interval says, "Based on my data, I am highly confident the true number is somewhere between this lower limit and this upper limit."

Think of it like fishing. If you try to catch a fish with a single spear, you might miss. But if you throw a wide net, you are much more likely to capture the fish. The "spear" is a single point estimate, while the "net" is the confidence interval. Reading it correctly means understanding both the width of the net and how confident we are that the fish is actually inside it.

The two parts of an interval

A confidence interval is usually written in the format: Estimate $\\pm$ Margin of Error. The estimate (like a sample mean or proportion) is the dead center of your interval. The margin of error tells you how far you cast your net above and below that center.

For example, if a poll says $60\\%$ of students like pizza, with a margin of error of $5\\%$ , the interval is $55\\%$ to $65\\%$ . You read this as: "We are confident that the true percentage of all students who like pizza is between $55\\%$ and $65\\%$ ."

What the confidence level actually means

Most intervals come with a "confidence level," usually $95\\%$ . This does not mean the interval is $95\\%$ accurate. Instead, it describes the method used to make the interval.

If you were to take $100$ different samples from the same population and calculate a confidence interval for each one, about $95$ of those nets would successfully catch the true population parameter. When we say "we are $95\\%$ confident," we mean we used a net-throwing method that works $95\\%$ of the time.

Where students slip up

The most common mistake is saying, "There is a $95\\%$ probability that the true average is in this specific interval." In standard statistics, the true average is a fixed number—it is not moving around.

The interval you calculated is also fixed. The true number is either in your interval, or it isn't. The $95\\%$ refers to the reliability of your sampling process before you collected the data, not a probability about the specific numbers you ended up with. Always use the phrasing "We are $95\\%$ confident that the true parameter lies between..."

Worked through

A researcher samples $50$ high school students and calculates their average time spent on homework. The sample mean is $2.5$ hours. The $95\\%$ confidence interval is calculated to be $(2.1, 2.9)$ . How should you interpret this interval?

First, identify the parameter we are trying to estimate. Here, it is the true average time all high school students spend on homework.

Second, identify the bounds. The lower bound is $2.1$ hours and the upper bound is $2.9$ hours.

Finally, put it together using the standard statistical phrasing: "We are $95\\%$ confident that the true average time high school students spend on homework is between $2.1$ and $2.9$ hours."

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Where this comes from: OpenStax Introductory Statistics, Chapter 8: Confidence Intervals · Khan Academy, AP Statistics Unit 6: Inference for categorical data: Proportions