7.3 Justifying a Claim About a Population Mean Based on a Confidence Interval

A confidence interval for a population mean gives you a range of plausible values for the true average. To justify a claim, you check whether the claimed value falls inside or outside that interval: if a value sits outside the interval, you have evidence against it, and if it sits inside, your data are consistent with it.

Why This Matters for the AP Statistics Exam

This topic shows up when a question hands you a confidence interval (or asks you to build one) and then asks whether the data support a specific claim about the population mean. You need to interpret the interval correctly in context and use it to back up or push back on a claim. The same reasoning applies to matched-pairs situations, where you treat the differences as a single sample of values. Free-response questions in this unit often ask you to interpret an interval and connect it to a claim, so being precise with your wording and your logic is important for clear exam work.

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Key Takeaways

• A confidence interval for a mean either captures the true population mean or it does not, since each interval depends on a random sample that changes from sample to sample.
• Interpret an interval as "We are C% confident the interval captures the true population mean of [context]," and include a reference to the sample and the population.
• If a claimed value falls outside the interval, you have evidence against that claim; if it falls inside, the data are consistent with it.
• Width of the interval shrinks as sample size grows; for a single mean the width is proportional to 1/√n.
• A higher confidence level makes the interval wider for the same sample.
• Matched pairs are handled as one sample of differences, then interpreted just like a single mean.

Justifying a Claim From an Interval

A claim about a population mean is a statement about the true average value of some quantity in a population. You usually only have a sample, so you build a confidence interval and use it to decide whether a claimed value is reasonable.

The logic is direct:

• If the claimed mean is inside your interval, the data are consistent with the claim. You do not have enough evidence to reject it.
• If the claimed mean is outside your interval, you have evidence that the claim may be wrong.

One thing to keep straight: a single interval either contains the true mean or it does not. The "C% confident" language describes the long-run method, not the probability for one specific interval.

Interpreting the Interval

A complete interpretation has two parts: the confidence level and the context, including a reference to the sample and the population.

Template:

"We are C% confident that the interval captures the true population mean of ___ (in context)."

For example, with a 96% confidence interval for the mean foot length of all footprints found in a cave, based on a random sample of footprints: "We are 96% confident that the mean foot length for all footprints found in the cave falls within the confidence interval" (based on 2000 FRQ 2).

How Sample Size and Confidence Level Affect Width

The margin of error has two moving parts, and your sample size affects both:

1. critical value (t*)
2. standard error

Critical Value

The critical value for a population mean is a t score, and t scores depend on degrees of freedom, which depend on sample size. As your sample size goes up, your degrees of freedom go up.

For example, with a sample size of 41, the degrees of freedom is 40. Using a t-table or your calculator's invT, the critical value is 2.021 for a 95% confidence interval.

If you increase the sample size to 51 (df = 50), the critical value becomes 2.009.

So as sample size increases, the critical value decreases.

Standard Error

The standard error is the sample standard deviation divided by the square root of the sample size: SE = s/√n.

Using the example above, if the standard deviation is 1.2, a sample size of 41 gives a standard error of about 0.1874.

Increasing the sample size to 51 changes the standard error to about 0.168.

So as sample size increases, the standard error decreases.

Putting It Together

Since both parts of the margin of error decrease with a larger sample, the margin of error shrinks overall. The interval gets narrower, honing in on the population mean you are estimating. For a single mean, the width is proportional to 1/√n.

Confidence level works in the opposite direction. For a given sample, raising the confidence level widens the interval, because you need a bigger critical value to be more confident.

Worked Example

Suppose a company lists 40 nuggets per bag, and you want to test that claim. You take a random sample of 30 bags and find an average of 41.4 nuggets with a standard deviation of 1.2.

Build a 95% one-sample t confidence interval:

point estimate ± (critical value)(standard error)

41.4 ± (2.05)(1.2/√30) = (40.951, 41.849)

The claimed value of 40 is not inside the interval. That gives you evidence against the listed value. In this case, every value in the interval is above 40, so the data suggest the bags actually contain more than 40 nuggets on average. You can interpret it as: "We are 95% confident the interval captures the true mean number of nuggets per bag for this brand."

Conditions Reminder

Before trusting an interval for a mean, check that:

• The data come from a random sample or randomized experiment, and when sampling without replacement, that n is at most 10% of the population.
• The sampling distribution of the sample mean is approximately normal. If the data are skewed, n should be greater than 30. If n is less than 30, the sample data should be free from strong skewness and outliers.

These conditions are the same ones you verify when constructing the interval in the first place.

How to Use This on the AP Statistics Exam

Free Response

• State the interval clearly, then connect it to the claim. Say directly whether the claimed value is inside or outside.
• Write a full interpretation: confidence level, "captures the true population mean," and context that references both the sample and the population.
• For matched pairs, define your order of subtraction and treat the differences as one sample before interpreting.

Problem Solving

• When a question asks how a wider or narrower interval would happen, point to sample size or confidence level. Bigger n narrows the interval; higher confidence widens it.
• Use the relationship "width proportional to 1/√n" when a question asks how much the interval would change if the sample size changes.

Common Trap

• Do not say there is a "95% probability the true mean is in this interval." The mean is fixed; the interval is what varies from sample to sample.
• "Cannot reject the claim" is not the same as "the claim is proven true." A value inside the interval is just consistent with the data.

Common Misconceptions

• "95% confident" means a 95% chance the true mean is in this one interval. A specific interval either contains the mean or it does not. The confidence level describes how often the method works over many samples.
• A value inside the interval proves the claim. Being inside only means the data are consistent with that value. You are not accepting it as true.
• Bigger samples always shift the interval toward the claim. A larger sample narrows the interval, but it does not push the center toward any particular value.
• A higher confidence level makes the estimate more accurate. Higher confidence widens the interval, which makes it less precise, not more.
• You can skip the context in your interpretation. A complete interpretation references the sample taken and the population it represents, not just numbers.

Related AP Statistics Guides

• Unit 7 Overview: Means
• 7.2 Constructing a Confidence Interval for a Population Mean
• 7.1 Introducing Statistics: Should I Worry About Error?
• 7.4 Setting Up a Test for a Population Mean
• 7.6 Confidence Intervals for the Difference of Two Means
• 7.5 Carrying Out a Test for a Population Mean