Topic 7.7 Notes – Justifying a Claim About the Difference of Two Means Based on a Confidence Interval

What a Confidence Interval for μ₁ − μ₂ Represents

The parameter we care about is:

• μ₁ − μ₂ = the true difference between two population means
  (Always define the order. For example, mean exam score of online students − mean exam score of in-person students.)

The interval you computed earlier has the form:

$$(\bar{x}_1 - \bar{x}_2) \pm t^* \left(\text{SE of } \bar{x}_1 - \bar{x}_2\right)$$

That gives a range of plausible values for μ₁ − μ₂ based on your samples.

What the sign tells you:

• Entire interval above 0 → μ₁ > μ₂
• Entire interval below 0 → μ₁ < μ₂
• Interval contains 0 → a true difference of 0 is plausible

Remember, we use an interval because different random samples would give slightly different results. The interval captures that uncertainty.

How to Correctly Interpret the Interval

An AP-worthy interpretation has four pieces:

1. Confidence level
2. The parameter (μ₁ − μ₂)
3. The populations in context
4. A reference to the samples

Here’s what strong wording sounds like:

“Based on these samples, we are 95% confident that the true difference in the population mean number of hours of sleep for athletes and non-athletes (μ₁ − μ₂) is between 0.8 and 1.9 hours.”

That checks every box.

What “95% confident” actually means

It means:

• If we repeatedly took random samples of the same sizes
• And built intervals the same way
• About 95% of those intervals would contain the true value of μ₁ − μ₂

The idea is easier to see visually. In the graph below, each vertical line is a confidence interval from a different random sample. The horizontal line marks the true population value. Most intervals capture it. A few miss.

Repeated confidence intervals from 100 random samples

The parameter is fixed. The interval changes. That’s the part students mix up.

Avoid saying “There is a 95% probability the true difference is in this interval.” The probability statement applies to the method, not this specific interval.

Using the Interval to Justify a Claim

Most claims look like one of these:

• “There is no difference.” → μ₁ − μ₂ = 0
• “There is a difference.” → μ₁ − μ₂ ≠ 0
• “Group 1 has a higher mean.” → μ₁ − μ₂ > 0

Everything hinges on 0, because 0 means equal means.

If 0 is in the interval

You say:

“Because 0 is in the interval, we do not have convincing evidence that the population mean ___ differs from the population mean ___.”

That does not prove they’re equal. It means the data are consistent with equality.

If 0 is NOT in the interval

You say:

“Because 0 is not in the interval, we have convincing evidence that the population mean ___ is greater than (or less than) the population mean ___.”

Be consistent with your subtraction order. If you defined μ₁ − μ₂ as (North − South), keep that direction all the way through. Students lose points when they flip the comparison at the end.

Also, only claim causation if the data came from a randomized experiment. If it was observational, stick to “is associated with.”

How Sample Size Affects the Width

The width of the interval is:

$$2 \times \text{margin of error}$$

And the margin of error depends on:

• The standard error
• The critical value $t^*$

The standard error includes $n_1$ and $n_2$ in the denominator. When sample sizes increase:

• Standard error decreases
• Margin of error decreases
• Interval gets narrower

A narrower interval means a more precise estimate of μ₁ − μ₂.

If confidence level increases, $t^*$ increases, so the interval gets wider. More variability in the data also makes the interval wider.

On tests, they love asking what happens if both sample sizes double. The answer is always that the interval becomes narrower, assuming everything else stays the same.