Topic 4.3 Notes – Justifying a Claim Based on a Confidence Interval for a Population Mean or Population Mean Difference

What a Confidence Interval for a Mean Says

Here, the parameter is either the population mean $\mu$ or, for matched pairs, the population mean of differences $\mu_d$.

For one sample, the interval has form

$$\bar{x}\pm t^*\frac{s}{\sqrt{n}}$$

For matched pairs, you first compute a difference for each pair in a stated order, then use

$$\bar{x}_d\pm t^*\frac{s_d}{\sqrt{n}}$$

In that paired formula, $n$ is the number of pairs, not the total number of observations.

A confidence interval gives a range of plausible values for the population parameter.

• The midpoint is the point estimate, so it equals $\bar{x}$ or $\bar{x}_d$.
• The half-width is the margin of error.
• The units of the interval and parameter match the response variable. If the variable is minutes, the interval is in minutes.

For paired data, the subtraction order matters a lot. If you define difference as before minus after, then a negative interval means the after values tend to be larger.

The picture below connects this idea to repeated sampling. Each horizontal interval comes from a different sample, and most of them capture the true mean $\mu$.

Repeated confidence intervals for a population mean

This topic assumes the interval was built correctly and conditions were already checked.

Interpreting the Interval and the Confidence Level

If a $C\%$ confidence interval is $(a,b)$, the right interpretation is:

• “We are $C\%$ confident that the interval from $a$ to $b$ contains the true population mean $\mu$ of …”
• For matched pairs, say $\mu_d$ and include the subtraction order.

Example in context:

• “We are 95% confident that the interval from 4.1 to 6.3 minutes contains the true mean bus delay for all weekday trips on this route.”

For paired data:

• “We are 95% confident that the interval from $-18$ to $-5$ seconds contains the true mean difference in completion time, defined as condition A minus condition B, for adults like those in the study.”

The confidence level is about long-run success of the method.

• In repeated random sampling, about $C\%$ of intervals from this method would capture the true parameter.
• The parameter stays fixed.
• The interval changes from sample to sample.
• Your one computed interval either contains the true value or it doesn’t.

What you should not say:

• “There is a $C\%$ probability the true mean is in this interval.”
• “$C\%$ of individuals are between the endpoints.”
• “This is an interval for the sample mean.”

For matched pairs, $\mu_d=0$ means no population mean difference.

Using the Interval to Judge a Claim

This is one of the most tested skills in this topic.

1. Identify the claim’s value, usually $k$ or $0$.
2. Compare that value to the whole interval.
3. Write a conclusion in context using evidence language.

Here’s the decision chart:

• If $k$ is outside the interval, there is evidence against $\mu=k$.
• If the entire interval is above $k$, there is evidence that $\mu>k$.
• If the entire interval is below $k$, there is evidence that $\mu<k$.
• If $k$ is inside the interval, the data are consistent with $\mu=k$. There is not convincing evidence of a difference.

For paired differences:

• If $0$ is outside the interval, there is evidence of a nonzero mean difference.
• If the whole interval is negative or positive, that gives direction.
• Then translate the sign using your subtraction order.

Example: if $d=\text{A}-\text{B}$ and the interval for $\mu_d$ is $(-18,-5)$, then $0$ is outside the interval and the whole interval is negative. That supports a mean difference, with A lower than B.

Use wording like:

• “The interval provides convincing evidence that…”
• “The interval does not provide convincing evidence that…”

A value equal to an endpoint counts as in the interval.

How Confidence Level and Sample Size Change the Interval

The margin of error is

$$t^*\frac{s}{\sqrt{n}}$$

The full width is

$$2t^*\frac{s}{\sqrt{n}}$$

If you raise the confidence level for the same sample:

• $t^*$ gets bigger
• margin of error gets bigger
• interval gets wider
• midpoint stays the same

If you increase sample size:

• $\frac{s}{\sqrt{n}}$ gets smaller
• the interval tends to get narrower

Width is approximately proportional to $\frac{1}{\sqrt{n}}$.

• Double $n$ $\rightarrow$ width becomes about $\frac{1}{\sqrt{2}}$ times as large
• Quadruple $n$ $\rightarrow$ width is about half as large

A larger sample improves precision. It does not guarantee the interval center moves closer to a claimed value.

Scope and Common Mistakes

Your conclusion is only valid if the $t$-interval procedure was appropriate.

• For matched pairs, conditions apply to the differences, not the two raw lists separately.
• Random sampling supports generalizing to the population.
• Random assignment supports cause-and-effect in experiments.

Common mistakes:

• Mixing up $\mu_d$ with $\mu_1-\mu_2$ from two independent samples
• Forgetting to define subtraction order
• Talking about individual values instead of the population mean
• Judging a claim from the sample mean alone instead of the entire interval