Justifying a Claim With a Confidence Interval for Two Means: Study Guide (Statistics 4.8)

What a two-sample interval can and cannot tell you

In Topic 4.7 you built intervals of the form (x̄₁ − x̄₂) ± t* √(s₁²/n₁ + s₂²/n₂). Now you need to say what such an interval means and use it to judge a claim.

Start from one fact. The true difference μ₁ − μ₂ is a fixed number. The interval is not fixed: it depends on which two samples you happened to get. A different pair of samples gives a different x̄₁ − x̄₂ and different standard deviations, so a different interval. Any one interval you calculate may or may not contain μ₁ − μ₂, and you cannot check, because you do not know μ₁ − μ₂.

That is why the word is confidence. Confidence describes the reliability of the method, not the success of one particular interval.

Interpreting the interval

An interpretation of a C% confidence interval for μ₁ − μ₂ has four parts:

1. the confidence level, C%;
2. the interval, from a to b, with units;
3. the difference in population means, with the order of subtraction;
4. the response variable and both populations (or treatments) in context.

A reliable template:

We are C% confident that the interval from a to b [units] captures the difference (group 1 minus group 2) in the mean [response] of all [population 1] and all [population 2].

For an experiment, replace “all [population]” with the treatments: “the true difference in the mean [response] for [subjects like these] given treatment 1 and given treatment 2”.

Avoid these:

• “…the difference in the sample means.” The sample difference x̄₁ − x̄₂ is the centre of the interval, so it is always inside. The interpretation is about population means.
• “C% of individuals differ by between a and b.” The interval is about means, not individual values.
• Leaving out the order of subtraction. An interval (−4, 9) means nothing until you know which group was subtracted from which.

Interpreting the confidence level

The confidence level answers: “How often does this method work?”

If we took many random samples of the same sizes from these two populations and built a C% interval from each pair, about C% of those intervals would capture the true difference μ₁ − μ₂.

So “95% confident” is a statement about the long-run success rate of the procedure, which gives you a reason to trust it. It is not the probability that this one interval is right. It also does not mean 95% of sample differences fall inside your interval.

A higher confidence level uses a larger t*, so the interval is wider. You buy more confidence with less precision (you met this for one mean in Topic 4.3).

Using an interval to justify a claim

The interval is a set of plausible values for μ₁ − μ₂. The most common claim is “the two population means are different”. Look at where 0 lies, because μ₁ − μ₂ = 0 means “no difference”.

Where the interval lies	What you can conclude
Entirely above 0 (both limits positive)	Convincing evidence that μ₁ > μ₂
Entirely below 0 (both limits negative)	Convincing evidence that μ₁ < μ₂
Contains 0	0 is plausible: not convincing evidence of a difference. This does not show the means are equal

Claims about a size. Some claims name a value other than 0, such as “on average, group 1 scores at least 10 points higher”. Translate the claim into a statement about μ₁ − μ₂ (here μ₁ − μ₂ ≥ 10), then compare it with the whole interval:

• every plausible value satisfies the claim → convincing evidence for it;
• the interval straddles the claimed boundary → not convincing evidence either way;
• no plausible value satisfies the claim → convincing evidence against it.

Order of subtraction. If you reverse the order, the interval for μ₂ − μ₁ is the interval for μ₁ − μ₂ with both signs changed and the limits swapped: (a, b) becomes (−b, −a). The conclusion does not change.

What kind of conclusion? (Reminder from Unit 1.) With independent random samples, you can generalise to the two populations sampled. With a randomized experiment, a difference shown by the interval can be attributed to the treatments (cause and effect), for subjects like those in the study.

Always finish with a sentence in context that refers to the interval.

Worked example 1: electricity use in two districts

Context. A fictional city council took independent random samples of households in two districts and recorded each household’s electricity use in one month, in kilowatt-hours (kWh). Each district has several thousand households, so the 10% condition holds, and both samples have at least 30 households.

District	n	x̄ (kWh)	s (kWh)
Ashgrove	45	412	88
Fenmoor	50	401	95

A 95% two-sample t-interval for μ_A − μ_F (Ashgrove minus Fenmoor) is (−26.3, 48.3) kWh. (Point estimate 11 kWh, SE 18.78 kWh, df 92.9, t* 1.986, margin of error 37.3 kWh.)

(a) Interpret the interval. We are 95% confident that the interval from −26.3 to 48.3 kWh captures the difference (Ashgrove minus Fenmoor) in the mean monthly electricity use of all households in Ashgrove and all households in Fenmoor.

(b) Interpret the confidence level. If the council took many pairs of random samples of 45 Ashgrove and 50 Fenmoor households and built a 95% interval from each pair, about 95% of the intervals would capture the true difference in mean monthly use.

(c) A councillor claims that Ashgrove households use more electricity, on average. The claim is μ_A − μ_F > 0. The interval contains 0, and also negative values. A difference of zero, or Fenmoor using more, is plausible. So the interval does not give convincing evidence that Ashgrove households use more electricity on average.

(d) A second councillor claims Fenmoor households use at least 50 kWh more per month, on average. Translate: μ_F − μ_A ≥ 50, which means μ_A − μ_F ≤ −50. Every value in the interval is greater than −50 (the lower limit is −26.3). So the interval gives convincing evidence against this claim.

Check. The interval for μ_F − μ_A would be (−48.3, 26.3) kWh. It still contains 0, and its upper limit 26.3 is below 50: same conclusions.

Worked example 2: an experiment on study schedules

Context. A fictional school asked 64 volunteer students to learn a topic for a quiz scored out of 50. The students were randomly assigned, 32 to a spaced schedule (short sessions over a week) and 32 to a block schedule (one long session).

Schedule	n	x̄ (points)	s (points)
Spaced	32	36.8	6.2
Block	32	31.5	7.0

The schedules were randomly assigned and both groups have 32 ≥ 30 students, so the conditions are met (the 10% condition is not needed in an experiment). A 99% two-sample t-interval for μ_S − μ_B is (0.91, 9.69) points. (Point estimate 5.3, SE 1.653, df 61.1, t* 2.659, margin of error 4.39.)

(a) Interpret the interval. We are 99% confident that the interval from 0.91 to 9.69 points captures the true difference (spaced minus block) in the mean quiz score for students like these who use the spaced schedule and who use the block schedule.

(b) Do the data give convincing evidence that the spaced schedule leads to a higher mean score? Yes. Both limits are positive, so every plausible value of μ_S − μ_B is greater than 0. Because the schedules were randomly assigned, we can conclude that the spaced schedule causes a higher mean score for students like these volunteers. The volunteers were not a random sample, so we should be careful about generalising to all students.

(c) A website claims spaced practice raises the mean score by more than 10 points. The whole interval lies below 10 (upper limit 9.69). So the interval gives convincing evidence against this claim, at the 99% level.

(d) Reverse the order. The 99% interval for μ_B − μ_S is (−9.69, −0.91) points. Both limits are negative: the block schedule has the lower mean. Same conclusion, different signs.

Check. At 95% the same data give (1.99, 8.61) points, and at 90% (2.54, 8.06). Lower confidence gives a narrower interval, and all three exclude 0.

Worked example 3: judging written interpretations

Question. Four students interpret the 95% interval (−26.3, 48.3) kWh from Worked example 1. Which statements are acceptable?

1. “We are 95% confident that the difference in the mean electricity use of the 45 and 50 households sampled is between −26.3 and 48.3 kWh.” Not acceptable. It describes the samples. The sample difference is exactly 11 kWh; there is nothing to estimate.
2. “There is a 95% chance that μ_A − μ_F is between −26.3 and 48.3 kWh.” Not acceptable. The 95% belongs to the method, not this interval.
3. “95% of Ashgrove households use between 26.3 kWh less and 48.3 kWh more than Fenmoor households.” Not acceptable. It talks about individual households, not means.
4. “We are 95% confident that the interval from −26.3 to 48.3 kWh captures the difference in mean monthly electricity use, Ashgrove minus Fenmoor, for all households in the two districts.” Acceptable. It has the level, the interval with units, the order, the response and both populations.

Common misconceptions

• “The interval contains 0, so the means are equal.” 0 is plausible, not proven. Other values in the interval are just as plausible.
• “Most of the interval is positive, so group 1 is higher.” If the interval contains 0, there is no convincing evidence of a difference, however much of it is positive.
• “A 95% confidence interval has a 95% probability of containing μ₁ − μ₂.” The method captures μ₁ − μ₂ about 95% of the time in the long run; one interval either does or does not.
• Interpreting the sample means. The interval estimates the difference in population means; x̄₁ − x̄₂ is always its centre.
• Forgetting the order of subtraction. A negative interval means the second group has the larger mean; you can only say which group if you stated the order.
• Claiming cause and effect from two random samples. Causal conclusions need random assignment of treatments.
• Ignoring the conditions. An interval built when the conditions fail (for example, small, strongly skewed samples) cannot support any claim, however tidy its limits look.

Where this leads

Next, Topic 4.9 sets up a significance test for μ₁ − μ₂. You will see that a two-sided test and a confidence interval usually tell a consistent story about whether 0 is plausible. Go to the next study guide when you are ready. First, try the practice questions, then use the revision notes and the checklist. To revise how the intervals are built, return to Topic 4.7.