Justifying a Claim Based on a Confidence Interval for a Mean: Study Guide (Statistics 4.3)

What a t-interval does and does not tell you

In Topic 4.2 you built intervals of the form x̄ ± t* × s/√n. This topic is about what such an interval means, and how to use it to make a decision.

The logic is the same as for proportions in Topic 3.4. The population mean μ is a fixed number. The interval changes from sample to sample, because x̄ changes. With means there is a second source of change: the sample standard deviation s also changes, so the width changes too. Any one interval you calculate may or may not contain μ, and you cannot tell which, because you do not know μ.

So for one calculated interval we talk about confidence, not probability. Confidence describes how reliable the method is.

Interpreting the interval

An interpretation of a C% interval has three parts:

1. the confidence level, C%;
2. the interval, from the lower limit a to the upper limit b, with units;
3. the parameter in context: a mean, the response variable and the population.

One-sample template:

We are C% confident that the interval from a to b [units] captures the true mean [variable] of all [population].

Matched-pairs template. For paired data the parameter is μd, the true mean of the differences. State the order of subtraction:

We are C% confident that the interval from a to b [units] captures the true mean difference in [variable] (first − second) for all [population or individuals like these].

The sample mean x̄ (or x̄d) sits at the centre of the interval, so the interval always contains it. An interpretation about x̄, or about “the 16 loaves we weighed”, is not an interpretation of the interval.

Interpreting the confidence level

If we took many random samples of the same size from this population and built a C% t-interval from each, about C% of those intervals would capture the true mean.

Figure 1 shows this. A computer took 20 random samples of size 10 from a normal population with μ = 50 and σ = 8, and built a 95% t-interval from each.

Three things to notice:

• 19 of the 20 intervals capture μ. Another run might give 18 or 20. The 95% is a long-run rate.
• Interval 18 looks ordinary. With only that sample, nothing would warn you that it missed.
• The intervals are not all the same width. A sample with a large s gives a wide interval; a sample with a small s gives a narrow one.

Using an interval to justify a claim

An interval is a set of plausible values for the parameter. Compare the claimed value with it.

Where the claimed value lies	What you can conclude
Outside the interval	Not plausible. Convincing evidence against the claim that the parameter equals it.
Inside the interval	Plausible. No convincing evidence against the claim. It is not proved: other values are plausible too.

For a directional claim (“more than 800 g”, “reduces the time”), check where the whole interval lies. If every plausible value satisfies the claim, the interval gives convincing evidence for it. If the interval straddles the boundary, it gives no convincing evidence either way.

The special role of 0 for matched pairs. A claim of “no difference on average” is the claim μd = 0.

• Interval entirely above 0: convincing evidence that the mean of (first − second) is positive.
• Interval entirely below 0: convincing evidence that it is negative.
• Interval contains 0: “no difference” is plausible, so there is no convincing evidence of a difference.

How the confidence level and the sample size change the interval

The margin of error is t* × s/√n.

Change (all else the same)	t*	Standard error s/√n	Margin of error and width
Increase the confidence level	increases	no change	increase
Decrease the confidence level	decreases	no change	decrease
Increase the sample size n	decreases slightly (more df)	decreases	decrease

The width is roughly proportional to 1/√n. Multiplying n by 4 roughly halves the width. With a t-interval it does a little better than half, because a larger sample also has more degrees of freedom and a smaller t*.

Worked example 1: judging a claim about a mean

Context. The fictional Hollins Bakery labels its large loaves “800 g”. A trading-standards officer weighs a random sample of 16 loaves from one week’s production of about 2,000. The weights show no strong skewness and no outliers. The sample mean is x̄ = 792.4 g and s = 11.2 g. Conditions: random sample; 16 ≤ 10% of 2,000 = 200; n < 30 but the sample data show no strong skew or outliers.

(a) 95% interval. df = 15, t* = 2.131. SE = 11.2/√16 = 2.8 g. MOE = 2.131 × 2.8 = 5.968 g. Interval: 792.4 ± 5.968 = (786.43, 798.37) g.

(b) Interpret it. We are 95% confident that the interval from 786.43 g to 798.37 g captures the true mean weight of all large loaves Hollins Bakery produced that week.

(c) Interpret the confidence level. If the officer took many random samples of 16 loaves from that week and built a 95% t-interval from each, about 95% of the intervals would capture the true mean weight of the loaves.

(d) Judge the label. 800 g is above the whole interval, so a mean of 800 g is not plausible. The interval gives convincing evidence that the true mean weight of that week’s large loaves is less than 800 g.

Check at other levels. At 90% the interval is (787.49, 797.31) g, which also excludes 800. At 99% (t* = 2.947, MOE = 8.251 g) it is (784.15, 800.65) g, which includes 800. At 99% confidence the label is still plausible. Always state the confidence level with the conclusion.

Worked example 2: a matched-pairs interval and a claim of no difference

Context. A fictional running club tested a new shoe. Each of 10 club members ran a 5 km time trial in their old shoes and in the new shoes, a week apart. A coin toss decided which shoes each runner wore first. Times are in seconds.

Runner	1	2	3	4	5	6	7	8	9	10
Old shoes	1512	1388	1645	1470	1590	1423	1555	1608	1497	1442
New shoes	1494	1363	1636	1439	1576	1401	1559	1581	1478	1430
d = old − new	18	25	9	31	14	22	−4	27	19	12

Step 1: one sample of differences. The data are paired (each runner gives two times), so work with d. x̄d = 173 ÷ 10 = 17.3 s and sd = 10.155 s. A positive d means the runner was faster in the new shoes.

Step 2: conditions. Random: the order of the shoes was randomly assigned. The 10% condition does not apply, because the runners were not sampled from a population. Sample data: there are only 10 differences, so check them. Ordered: −4, 9, 12, 14, 18, 19, 22, 25, 27, 31. Q1 = 12, Q3 = 25, IQR = 13, so the fences are −7.5 and 44.5. No difference is an outlier and there is no strong skew.

Step 3: 95% interval. df = 9, t* = 2.262. SE = 10.155/√10 = 3.211 s. MOE = 2.262 × 3.211 = 7.26 s. Interval: (10.04, 24.56) s.

Step 4: interpret. We are 95% confident that the interval from 10.04 to 24.56 seconds captures the true mean difference in 5 km time (old shoes − new shoes) for runners like these.

Step 5: judge three claims.

1. “The new shoes make no difference on average.” That is μd = 0. 0 is below the whole interval, so it is not plausible. There is convincing evidence that the mean time is lower in the new shoes. Because the order was randomly assigned, this supports a cause-and-effect conclusion, but only for runners like these volunteers, not for all runners.
2. “The new shoes save about 20 seconds.” 20 is inside the interval, so it is plausible. There is no convincing evidence against it, but it is not proved.
3. “The new shoes save at least 30 seconds on average.” Every plausible value is below 30, so the interval gives convincing evidence against this claim.

Check. The mean of the old times is 1513.0 s and the mean of the new times is 1495.7 s. Their difference, 17.3 s, equals x̄d, as it must. But you cannot get sd from the two separate standard deviations; you need the differences themselves.

Worked example 3: predicting the effect of changes

Use the Hollins Bakery sample (x̄ = 792.4 g, s = 11.2 g, n = 16). The 95% interval had MOE 5.968 g and width 11.94 g.

(a) 99% instead of 95%, same data. t* rises from 2.131 to 2.947; SE stays 2.8 g. MOE = 8.251 g, which is 1.382 times as large. Wider interval.

(b) 90% instead of 95%. t* = 1.753, MOE = 4.909 g. Narrower interval.

(c) n = 64 instead of 16, same x̄ and s, 95%. The sample is 4 times larger, so SE = 11.2/√64 = 1.4 g, half of 2.8 g. Also df = 63, so t* falls to 1.998. MOE = 1.998 × 1.4 = 2.798 g, giving (789.60, 795.20) g. The new width, 5.60 g, is 0.469 of the old width: a little less than half.

Lesson. More confidence costs width. Only a larger sample gives more confidence and a narrow interval together.

Conditions still matter

The long-run capture rate only holds if the Topic 4.2 conditions are met. Without random sampling or random assignment, the confidence level has no meaning. With a small sample from a strongly skewed population, the true capture rate can be well below C%. The margin of error covers only random variation; it cannot fix a biased sample or a poor measuring method.

Common misconceptions

• “There is a 95% probability that μ is between a and b.” The calculated interval either contains μ or not. Say “95% confident” and explain the 95% as the long-run capture rate.
• “95% of the loaves weigh between 786.43 and 798.37 g.” The interval estimates the mean weight. Individual loaves vary far more than this.
• “The interval captures x̄.” It always does; x̄ is its centre. The interpretation is about μ.
• “800 g is inside the 99% interval, so the loaves really average 800 g.” Inside means plausible, not proved.
• Forgetting the order of subtraction in a matched-pairs interpretation. “(10.04, 24.56)” means nothing until you say it is old − new.
• “0 is in the interval, so there is no difference.” It means there is no convincing evidence of a difference.
• “Doubling n halves the width.” You need about 4 times the sample size.
• “A higher confidence level changes the standard error.” The confidence level changes only t*.

Where this leads

Topic 4.4 asks the same kind of question with a significance test instead of an interval: Setting Up a Test for a Population Mean or Population Mean Difference. Try the practice questions now, then use the revision notes and the checklist to consolidate. To review how the interval is built, return to Topic 4.2.