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One-sample t confidence intervals · Tutorial 636 of 1000

Recovering x-bar and Margin of Error From an Interval

Use a confidence interval’s endpoints to find its center, \(\bar{x}\), and its margin of error, then apply those values to follow-up questions.

Intermediate 9 min read

What You'll Learn

  • Find the midpoint of an interval to recover the sample mean.
  • Find the half-width to recover the margin of error.
  • Reconstruct an interval from its center and margin of error.
  • Use recovered values to answer contextual follow-up questions.
  • Recognize how rounded endpoints affect the precision of your answers.

Recovering the Center and Margin of Error

A confidence interval for a population mean is often written using its lower and upper endpoints. But those endpoints also contain two useful pieces of information: the sample mean \(\bar{x}\), which is the interval’s center, and the margin of error, which is the distance from that center to either endpoint. Recovering them is a matter of finding the interval’s midpoint and half-width.

This builds on “The Form of a One-Sample t Interval” and “Margin of Error for a Mean,” where a one-sample t interval was written as \(\bar{x}\pm ME\). Here, we work in the reverse direction: start with the endpoints and recover \(\bar{x}\) and \(ME\). These calculations do not require knowing the sample size or confidence level.

Formula: If a confidence interval has lower endpoint \(L\) and upper endpoint \(U\), then its midpoint is the sample mean and its half-width is the margin of error:
$$ \bar{x}=\frac{L+U}{2} \qquad\text{and}\qquad ME=\frac{U-L}{2} $$

The midpoint averages the two endpoints. The half-width subtracts the lower endpoint from the upper endpoint and divides by two. Both \(\bar{x}\) and \(ME\) have the same units as the quantity being measured. The margin of error is a distance, so it is nonnegative.

A quick check is to add and subtract the recovered margin of error from the recovered midpoint. You should get back the original endpoints:

$$ L=\bar{x}-ME \qquad\text{and}\qquad U=\bar{x}+ME $$

For instance, if the endpoints are 12 and 20, the midpoint is 16 and the half-width is 4. Checking gives \(16-4=12\) and \(16+4=20\). This also helps catch a common mix-up: the total interval width is \(U-L\), while the margin of error is only half of that width.

A Reliable Calculation Routine

First label the lower and upper endpoints, keeping their units. Then calculate the midpoint and half-width separately. Do not try to infer the sample mean from whichever endpoint looks more convenient, and do not report the full width as the margin of error.

1
Identify the endpoints.
Write \(L\) for the lower endpoint and \(U\) for the upper endpoint, in that order.
2
Find the midpoint.
Add the endpoints and divide by two. This recovers \(\bar{x}\), the sample mean that centers the interval.
3
Find the half-width.
Subtract the lower endpoint from the upper endpoint and divide by two. This recovers \(ME\).
4
Check and interpret.
Verify that \(\bar{x}-ME=L\) and \(\bar{x}+ME=U\), then answer the question in context and include units.

If a question asks for the interval’s total width as well as its margin of error, report both distinctly. The total width is \(U-L\), or equivalently \(2ME\). A wider interval has a larger margin of error; its midpoint may be unchanged.

An interval displayed to a limited number of decimal places may have rounded endpoints. In that case, the midpoint and margin of error calculated from those displayed values are approximate. Keep enough digits during the calculation, and round the final answer to a precision supported by the information given. If an exact comparison depends on a value very close to a rounded endpoint, use unrounded endpoints if they are available.

Recovering \(\bar{x}\) and \(ME\) does not reveal every detail of how the interval was made. For example, the endpoints alone do not tell you the sample size, sample standard deviation, degrees of freedom, or confidence level. Nor can the arithmetic confirm that the sampling and shape conditions for a one-sample t interval were met; those depend on how the data were collected and on the sample.

Worked Examples

Worked Example: Estimating Mean Filtered-Water Output

A fictional testing team reports a confidence interval from 12.6 to 16.2 liters for the population mean amount of filtered water produced in a specified period. Recover the sample mean and margin of error. Then explain what the center and margin of error represent.

Let \(L=12.6\) liters and \(U=16.2\) liters. The midpoint is:

$$ \bar{x}=\frac{L+U}{2} =\frac{12.6+16.2}{2} =\frac{28.8}{2} =14.4\text{ liters} $$

The margin of error is the half-width:

$$ ME=\frac{U-L}{2} =\frac{16.2-12.6}{2} =\frac{3.6}{2} =1.8\text{ liters} $$

Check the recovery: \(14.4-1.8=12.6\) liters and \(14.4+1.8=16.2\) liters. The interval’s total width is \(16.2-12.6=3.6\) liters, twice the margin of error.

The sample mean amount of filtered water produced was 14.4 liters. The margin of error is 1.8 liters, the distance from that sample mean to either endpoint. It is not the total length of the interval.

Worked Example: Recovering Values and Comparing a Claimed Mean

A fictional recreation center reports a confidence interval of 84.7 to 91.3 minutes for the population mean time visitors spend in its climbing area. Find \(\bar{x}\), \(ME\), and the total width. Then determine whether a claimed population mean of 85 minutes lies inside the interval.

The endpoints are \(L=84.7\) minutes and \(U=91.3\) minutes. First, recover the sample mean:

$$ \bar{x}=\frac{84.7+91.3}{2} =\frac{176.0}{2} =88.0\text{ minutes} $$

Next, calculate the margin of error and total width:

$$ ME=\frac{91.3-84.7}{2} =\frac{6.6}{2} =3.3\text{ minutes} $$
$$ \text{Total width}=91.3-84.7=6.6\text{ minutes} $$

The claimed value of 85 minutes is inside the interval because \(84.7<85<91.3\). In “Using a Confidence Interval to Test a Claimed Mean,” we learned how to use a matching interval for a two-sided test. If this is the appropriate interval for that test, 85 minutes being inside it means we fail to reject the claim at the matching significance level. It does not prove that the true mean is 85 minutes.

The recovered sample mean is 88.0 minutes, and the margin of error is 3.3 minutes. Notice that the claim is not being compared with the midpoint alone: it is compared with both interval endpoints.

Worked Example: Using the Midpoint to Recover a Missing Endpoint

A fictional garden supply cooperative reports that an interval for the population mean mass of a certain type of seed packet has midpoint 4.02 grams and margin of error 0.74 gram. Find the interval endpoints. Then compare a proposed value of 4.50 grams with the interval.

The lower endpoint is the midpoint minus the margin of error:

$$ L=\bar{x}-ME =4.02-0.74 =3.28\text{ grams} $$

The upper endpoint is the midpoint plus the margin of error:

$$ U=\bar{x}+ME =4.02+0.74 =4.76\text{ grams} $$

The reconstructed interval is 3.28 to 4.76 grams. A quick check confirms that its midpoint is \((3.28+4.76)/2=8.04/2=4.02\) grams and its half-width is \((4.76-3.28)/2=1.48/2=0.74\) gram.

The proposed value, 4.50 grams, lies inside the interval because \(3.28<4.50<4.76\). If the interval is being used for the matching two-sided test described in the earlier tutorial, this value is not excluded. The interval arithmetic alone does not establish that 4.50 grams is the true population mean.

Common Mistakes and AP Exam Tips

  • Reporting the full width as the margin of error. The full width is \(U-L\). Divide it by two to get \(ME\). A full-credit answer identifies the half-width as the margin of error.
  • Using subtraction to find the midpoint. The midpoint comes from adding the endpoints and dividing by two. Subtraction is used for the width, not the center.
  • Subtracting in the wrong order. Use upper endpoint minus lower endpoint for a positive width: \(U-L\). Then divide by two.
  • Leaving off units or context. If an interval is for minutes, both the sample mean and margin of error are in minutes. State what the mean estimates and describe the margin as a distance.
  • Claiming the endpoints reveal the confidence level or sample size. They do not. More information about the procedure or sample is needed to identify those details.
  • Treating rounded endpoints as exact. When only rounded endpoints are shown, the recovered values are based on those displayed numbers and may be approximate. Avoid suggesting more precision than the endpoints support.
  • Calling the margin of error a likely error in the sample mean. It is the distance from the center of the interval to an endpoint. It does not mean the sample mean is known to be wrong by exactly that amount.

For a full-credit response, show the midpoint calculation and the half-width calculation, label each result, and include the units. If the question asks for a contextual explanation, identify \(\bar{x}\) as the sample mean and \(ME\) as the interval’s distance from its center to either endpoint. When making a follow-up decision about a claimed mean, use the endpoints and the matching two-sided interval rule from “Using a Confidence Interval to Test a Claimed Mean.”

Key takeaway: From endpoints \(L\) and \(U\), recover the sample mean with \((L+U)/2\) and the margin of error with \((U-L)/2\). Check by subtracting and adding the margin of error to the midpoint.

Check Your Understanding

Show your calculations and include units where the question provides them.

  1. A confidence interval for a population mean is 28 to 40 seconds. Find its midpoint and margin of error.
  2. An interval has endpoints 63.5 and 70.1 kilograms. Find its total width and its margin of error.
  3. A confidence interval has midpoint 215 milliliters and margin of error 12 milliliters. Find both endpoints.
  4. A claimed mean is 52, and a reported interval runs from 49 to 56. Is the claim inside the interval? What additional information would be needed to connect this comparison to a two-sided test decision?
  5. Why might the midpoint and margin of error calculated from displayed endpoints be approximate rather than exact?

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