Auditing a Reported T Interval

Why Audit a Reported Interval?

A confidence interval can look plausible and still be wrong. A critical value may not match the sample size or confidence level, a standard error may use the wrong denominator, or the displayed endpoints may not follow from the stated inputs. Even when the arithmetic is correct, a report can mislabel the confidence level or omit units that clarify what the interval estimates.

An audit checks whether the pieces of a reported interval agree with one another. It does not, by itself, establish that the sample represents a target population: the sampling design and data conditions must also be assessed. The aim is to trace the calculation from the stated model and assumptions through the final endpoints, and to describe any remaining limitations accurately.

Definition: For a sample of $n$ quantitative observations with sample mean $\bar{x}$ and sample standard deviation $s$, a two-sided one-sample t confidence interval for a population mean $\mu$ has the form $$ \bar{x}\pm t^*\frac{s}{\sqrt{n}}, $$ where $t^*$ is the positive critical value for the stated confidence level with $df=n-1$. The standard error is $SE=s/\sqrt{n}$, and the margin of error is $t^*SE$. The interval endpoints are $\bar{x}-t^*SE$ and $\bar{x}+t^*SE$.

State the Model Before Checking the Arithmetic

The one-sample t interval relies on data that can reasonably be treated as independent observations from the population of interest. Random sampling or an appropriate random assignment can support this condition, while sampling without replacement calls for checking that the sample is no more than 10% of the population. For a small sample, the population distribution should be approximately normal, with no strong skewness or outliers. With larger samples, the procedure is generally less sensitive to departures from normality, but severe outliers and dependence still deserve attention.

When the observations are independent and normally distributed, the standardized statistic based on $\bar{x}$, $s$, and $\mu$ follows a t distribution with $n-1$ degrees of freedom. The two-sided confidence level $C$ determines the tail area $\alpha=1-C$; $t^*$ is the value with cumulative probability $1-\alpha/2$. In an audit, therefore, the critical value must agree with both the degrees of freedom and the stated confidence level. It is not enough for it merely to look like a familiar number.

The interval concerns a population mean, so its units must be the units of the measured variable. The sample standard deviation and standard error have those units; the critical value has no units; and multiplying $t^*$ by $SE$ produces a margin of error in the original units. The endpoints must consequently use the same units as $\bar{x}$.

A Systematic Audit

Work through the report in a fixed order. This helps separate a wrong input from a multiplication or rounding error, and makes the source of a discrepancy easier to explain.

Check the conditions and identify the target.
Confirm that the observations and design support inference about the stated population mean, and identify the variable’s units.

Check degrees of freedom and critical value.
Calculate $df=n-1$, then verify that $t^*$ corresponds to that $df$ and the reported two-sided confidence level.

Recalculate the standard error.
Use $SE=s/\sqrt{n}$, not $s/n$, and retain enough digits for the later calculations.

Recalculate the margin and endpoints.
Find $t^*SE$, then subtract and add it to $\bar{x}$. Compare unrounded values first, allowing for reasonable final rounding.

Check the report’s wording.
Verify the units and confidence-level label, and distinguish computational consistency from whether the interval is justified by the design and data.

Use the Endpoints as Evidence

An interval’s endpoints provide a useful independent check on its reported center and width. For a symmetric one-sample t interval, the midpoint should equal $\bar{x}$, and the half-width should equal the margin of error. This follows directly from the endpoint formulas. If $L=\bar{x}-t^*SE$ and $U=\bar{x}+t^*SE$, then

\frac{L+U}{2} =\frac{(\bar{x}-t^*SE)+(\bar{x}+t^*SE)}{2} =\bar{x}, \qquad \frac{U-L}{2} =\frac{(\bar{x}+t^*SE)-(\bar{x}-t^*SE)}{2} =t^*SE.

Thus, if $n$, $s$, and $t^*$ are also reported, the interval’s half-width can be checked against $t^*s/\sqrt{n}$. Conversely, if the center and $SE$ are known, the endpoints imply a critical value:

t^*_{\text{implied}} =\frac{(U-L)/2}{SE}.

This reverse calculation is a diagnostic, not a substitute for checking the confidence level and degrees of freedom. A value implied by rounded endpoints may be slightly different from the exact critical value. Compare with reasonable rounding tolerance, and use unrounded calculations when available.

Worked Example: Checking the Critical Value and Endpoints

Suppose a hypothetical report describes a random sample of $n=16$ commute times, with $\bar{x}=12.40$ minutes and $s=3.20$ minutes. It claims to give a 95% t interval and lists $t^*=2.120$, $SE=0.800$ minutes, and endpoints 10.73 and 14.07 minutes. Assume the sample was independent, the population contained at least 160 commuters, and the small-sample distribution check found no strong skewness or outliers. The stated conditions are plausible for this calculation.

First, the degrees of freedom are $df=n-1=15$. The 95% two-sided critical value for $df=15$ is approximately 2.131, so the reported 2.120 does not match the stated sample size and confidence level. The standard error, however, is correct:

SE=\frac{s}{\sqrt{n}} =\frac{3.20}{\sqrt{16}} =\frac{3.20}{4} =0.800\text{ minutes}.

Even using the report’s stated critical value, its claimed endpoints do not follow from the stated inputs. The implied margin from those inputs is

2.120(0.800)=1.696\text{ minutes},

so those inputs produce $12.40\pm1.696$, or endpoints 10.704 and 14.096 minutes. To two decimal places these are 10.70 and 14.10 minutes, not 10.73 and 14.07. In fact, the reported endpoints have midpoint 12.40 and half-width 1.67, implying $t^*=1.67/0.800=2.0875$, not 2.120.

Finally, use the correct critical value for a 95% interval with 15 degrees of freedom:

ME=2.131(0.800)=1.7048\text{ minutes}, \qquad 12.40\pm1.7048=(10.6952,\ 14.1048)\text{ minutes}.

Rounded to two decimal places, the corrected interval is 10.70 to 14.10 minutes. The audit found two distinct issues: the reported critical value was not the correct one, and the reported endpoints did not even follow from the report’s own critical value and standard error.

Worked Audits of Common Errors

Worked Example: Detecting an Incorrect Standard Error

A hypothetical random sample of 25 independently selected packages has a mean fill weight of 68.4 grams and a sample standard deviation of 10.0 grams. The population is large enough to satisfy the 10% condition, and a plot shows no extreme outliers or severe skewness. A report gives a 90% interval of 67.716 to 69.084 grams, saying that $SE=0.400$ grams was used.

The conditions make the t interval reasonable. The degrees of freedom are $25-1=24$, and the 90% critical value is approximately 1.711. Recalculate the standard error:

SE=\frac{10.0}{\sqrt{25}}=\frac{10.0}{5}=2.00\text{ grams}.

The reported $SE=0.400$ is incorrect; it is what results from dividing $s$ by $n$, rather than by $\sqrt{n}$. The correct margin of error and interval are

ME=1.711(2.00)=3.422\text{ grams}, \qquad 68.4\pm3.422=(64.978,\ 71.822)\text{ grams}.

Thus the 90% t interval is approximately 64.98 to 71.82 grams. The claimed endpoints are centered at 68.400 grams and have a half-width of 0.684 grams, consistent with the erroneous calculation $1.711(0.400)=0.6844$, not with the correct standard error. This is a consequential error: using $s/n$ makes the interval much too narrow.

Worked Example: Finding a Confidence-Level Label Error

A hypothetical sample of $n=9$ independent measurements of a chemical concentration has $\bar{x}=5.60$ milligrams per liter and $s=1.20$ milligrams per liter. Assume the sampling method is appropriate, the population is large enough for the 10% condition, and the small-sample data show no strong skewness or outliers. A report labels the interval 4.856 to 6.344 milligrams per liter as a 95% interval.

The degrees of freedom are $9-1=8$, and $SE=1.20/\sqrt{9}=0.400$ milligrams per liter. The reported endpoints have midpoint 5.600 and half-width 0.744 milligrams per liter. Their implied critical value is

t^*_{\text{implied}}=\frac{0.744}{0.400}=1.860.

For $df=8$, a critical value of about 1.860 corresponds to a two-sided 90% interval, not a 95% interval. The displayed endpoints are therefore consistent with the wrong confidence-level label. For a 95% interval, the critical value is approximately 2.306, giving

ME=2.306(0.400)=0.9224\text{ milligrams per liter}, \qquad 5.60\pm0.9224=(4.6776,\ 6.5224)\text{ milligrams per liter}.

The corrected 95% interval is approximately 4.678 to 6.522 milligrams per liter. The issue was not the center or standard error; it was the mismatch between the stated confidence level and the width produced by the critical value.

What an Audit Can and Cannot Establish

A successful arithmetic audit establishes that the reported values are internally consistent with the stated one-sample t method, inputs, and confidence level. It does not prove that the interval is reliable for the population of interest. For example, a perfectly calculated interval based on a convenience sample may not represent that population, and dependence among observations can invalidate the standard error calculation. Report those limitations separately from calculation errors.

A confidence interval’s stated confidence level describes the long-run coverage of the method when its assumptions hold: across repeated sampling, the method would produce intervals containing the fixed population mean at the stated rate. It is not a probability assigned to the fixed mean being inside one already calculated interval. In an audit, check that the stated level matches the critical value; do not infer that the interval is valid merely because its endpoints are symmetric around the sample mean.

Precision also matters when comparing reported and recalculated values. A value shown to two decimal places may differ slightly from a calculation made with unrounded inputs. Recompute with the available precision, then round the final endpoints consistently. A small discrepancy explained by rounding is different from one caused by the wrong $df$, a wrong $SE$, or a mistaken confidence level.

Common Mistakes

Checking endpoints but not their inputs. A plausible-looking interval can use a critical value for the wrong degrees of freedom. Verify $df=n-1$ and the confidence level before accepting $t^*$.
Dividing by $n$ instead of $\sqrt{n}$. The standard error is $s/\sqrt{n}$. Dividing by $n$ understates uncertainty and usually yields an interval that is too narrow.
Calling every discrepancy an arithmetic error. First compare unrounded results and account for reported precision. Then identify whether any remaining mismatch comes from the inputs, formula, or label.
Forgetting units. The critical value is unitless, but $SE$, the margin of error, and the endpoints have the measured variable’s units.
Treating a correct calculation as proof of good inference. Arithmetic cannot repair a nonrepresentative sample, dependence, or serious violations of the conditions.

Key takeaway: A reported one-sample t interval should agree in every link of the calculation: assumptions and target, $df=n-1$, confidence-level critical value, $SE=s/\sqrt{n}$, margin $t^*SE$, symmetric endpoints around $\bar{x}$, and correct units. If the reported endpoints are available, their midpoint and half-width provide a direct cross-check.

Check Your Understanding

For each question, explain the check you would perform and distinguish a calculation problem from an assumption or labeling problem.

A report uses $n=12$, $s=6$ seconds, and $SE=0.50$ seconds. What standard error should it report, and what calculation may have produced 0.50?
An interval is reported as 20 to 28 kilograms, with $\bar{x}=24$ kilograms and a claimed margin of error of 3 kilograms. What can the midpoint and half-width tell you?
For a two-sided interval based on $n=20$, what degrees of freedom should be used to select the critical value?
A reported interval has the correct center and standard error, but the critical value corresponds to 90% while the report says 95%. What part of the report needs correction?
Why can a numerically correct interval still be unsuitable for inference about a population mean?

What You'll Learn