Changing the Order Changes the Difference
In “Reading the Sign of the Interval for a Difference,” you used the signs of the endpoints to identify the direction of a plausible difference. Here the important extra step is to keep track of the order in that difference. The interval for population 1 minus population 2 and the interval for population 2 minus population 1 describe the same comparison from opposite directions.
Suppose \(\mu_A\) is the true mean for population A and \(\mu_B\) is the true mean for population B. The differences \(\mu_A-\mu_B\) and \(\mu_B-\mu_A\) have opposite signs because reversing subtraction changes the sign of the result. A positive value of \(\mu_A-\mu_B\), for example, is a negative value of \(\mu_B-\mu_A\). The comparison has not changed; only the order used to express it has changed.
The endpoints switch places as well as changing signs. If the original interval runs from \(L\) to \(U\), then \(-U\) is the lower endpoint of the reversed interval and \(-L\) is the upper endpoint. For instance, reversing \((2, 7)\) gives \((-7, -2)\), not \((-2, -7)\). The latter puts the endpoints in the wrong order.
Reversing the order does not change the confidence level or the units. It also does not change the interval’s width: \(U-L\) is the same as \((-L)-(-U)\). It changes the center’s sign and the direction described by the endpoints. You do not need to calculate a new critical value or standard error just to express the same interval in the opposite order.
Before interpreting either interval, name the first population and the second population in the order shown. Then read the result literally: “mean for the first population minus mean for the second population.” A consistent sentence keeps that order from the parameter through to the contextual interpretation.
A Reliable Way to Reverse and Interpret
Use this sequence when a question gives an interval in one order but asks about the opposite order. Keep the original interval visible until you have completed the reversal; that makes it easier to catch a sign or endpoint error.
Identify which population mean is subtracted from which. For example, \(\mu_A-\mu_B\) means the mean for A minus the mean for B.
For an original interval \((L,U)\), use \((-U,-L)\) for the reversed parameter. Negate both endpoints and put the smaller value first.
A positive value of the reversed difference means the second population’s mean exceeds the first population’s mean in the original order.
Include the confidence level, both populations in the requested order, the quantitative variable, the endpoints, and the units.
If the original interval contains zero, the reversed interval contains zero too. Reversal swaps which side of zero corresponds to which direction, but it cannot make zero disappear. As in “Reading the Sign of the Interval for a Difference,” an interval that includes zero does not prove the population means are equal.
Worked Examples
Worked Example: Reversing a Positive Difference in Battery Life
An invented comparison estimates mean operating time for two types of rechargeable batteries. Let population A be batteries of type L and population B be batteries of type M. Suppose a 90% confidence interval for \(\mu_A-\mu_B\) is \((1.6,\ 4.8)\) hours. A reader asks for the interval in the order \(\mu_B-\mu_A\).
Negate the original upper endpoint to get the new lower endpoint, then negate the original lower endpoint to get the new upper endpoint:
Both endpoints are negative. In the reversed order, the negative values mean type M’s population mean operating time is lower than type L’s. Equivalently, type M’s mean minus type L’s mean is plausibly between 1.6 and 4.8 hours below zero.
A complete interpretation in the requested order is: We are 90% confident that the true mean operating time for rechargeable batteries of type M minus the true mean operating time for rechargeable batteries of type L is between \(-4.8\) and \(-1.6\) hours.
The original interval said type L’s mean operating time exceeds type M’s by between 1.6 and 4.8 hours. The reversed interval says the same thing from type M’s perspective. The two sentences are consistent because the order and signs both change.
Worked Example: Reversing a Negative Difference in Water Use
An invented garden project compares mean daily water use for plants under two watering schedules. Let population 1 be plants on schedule Cedar and population 2 be plants on schedule Elm. A 95% confidence interval for \(\mu_1-\mu_2\) is \((-9.2,\ -2.7)\) liters per plant per day. Express it for \(\mu_2-\mu_1\).
Negate the upper endpoint, \(-2.7\), to get the lower endpoint, \(2.7\). Negate the lower endpoint, \(-9.2\), to get the upper endpoint, \(9.2\). Thus the reversed interval is:
In the original order, the interval was entirely negative, so the mean for Cedar was lower than the mean for Elm. In the reversed order, it is entirely positive: the interval says the mean for Elm exceeds the mean for Cedar.
A suitable interpretation is: We are 95% confident that the true mean daily water use per plant under schedule Elm minus the true mean daily water use per plant under schedule Cedar is between 2.7 and 9.2 liters.
Notice the units remain liters per plant per day. Reversing the groups does not change the variable or its units, and it does not change the confidence level. It changes which schedule’s mean is first in the subtraction.
Worked Example: Reversing an Interval That Includes Zero
An invented study compares the mean time to complete a digital puzzle using two interface designs. Let population A be users of design Fern and population B be users of design Moss. A 90% confidence interval for \(\mu_A-\mu_B\) is \((-1.3,\ 2.5)\) minutes. Find the corresponding interval for \(\mu_B-\mu_A\), then describe what it says.
Negating the upper endpoint gives the new lower endpoint, and negating the lower endpoint gives the new upper endpoint:
Zero remains strictly between the endpoints. In the reversed order, the interval includes plausible differences where Moss users’ mean completion time is lower than Fern users’, where the means are equal, and where Moss users’ mean completion time is higher.
A full interpretation is: We are 90% confident that the true mean puzzle-completion time for users of design Moss minus the true mean puzzle-completion time for users of design Fern is between \(-2.5\) and \(1.3\) minutes.
The interval does not establish which design has the lower population mean completion time. Reversing the order has not added information; it has simply expressed the same uncertainty in the opposite direction.
Worked Example: Matching the Wording to a Requested Order
An invented transportation comparison reports a 95% confidence interval of \((3,\ 11)\) minutes for mean commute time in neighborhood North minus mean commute time in neighborhood South. A question asks how much greater or less the mean is in South than in North.
The requested difference is South minus North, the reverse of the reported order. Reverse the endpoints and signs:
The reversed interval is entirely negative. It says the mean commute time in South is lower than the mean commute time in North. The size of the difference is between 3 and 11 minutes, but the signed interval in the requested order is \(-11\) to \(-3\) minutes.
A clear interpretation is: We are 95% confident that the true mean commute time for residents of South minus the true mean commute time for residents of North is between \(-11\) and \(-3\) minutes.
A common wording trap is to say that South’s mean is “between 3 and 11 minutes higher” than North’s. That contradicts the negative interval. South’s mean is plausibly 3 to 11 minutes lower; the signed difference South minus North is between \(-11\) and \(-3\) minutes.
Common Mistakes and AP Exam Tips
- Negating endpoints but leaving them in the old order: If the original interval is \((L,U)\), write the reversed interval as \((-U,-L)\). The first endpoint must be smaller than the second.
- Changing the words but not the signs: An interval for A minus B cannot be quoted unchanged as an interval for B minus A. Reverse both the parameter order and the endpoint signs.
- Changing only one endpoint: Reversal affects the entire difference, so negate both endpoints. Then check that the new interval has the same width as the original.
- Mixing the order within a sentence: Do not define the parameter as A minus B and then interpret it as B minus A. Name the first population, the second population, and the subtraction order consistently.
- Changing units or confidence level: Reversing the subtraction order changes neither. Keep the original variable’s units and the interval’s stated confidence level.
- Overstating an interval that contains zero: If zero remains between the endpoints after reversal, the interval still includes plausible differences in both directions. Do not claim that one population mean is definitely greater or that the means are equal.
For full credit, write the requested parameter first, transform the endpoints in the correct order, and interpret the result using the same population order. A useful check is to describe the original interval and the reversed interval in words: they should express the same comparison from opposite perspectives.
Check Your Understanding
For each question, pay attention to both the subtraction order and the endpoint order.
- A 90% interval for mean weekly screen time in group P minus group Q is \((0.8,\ 3.1)\) hours. What is the corresponding interval for group Q minus group P?
- A 95% interval for mean monthly rainfall in town Cedar minus town Birch is \((-14,\ -5)\) millimeters. In which town is the population mean rainfall higher according to the interval?
- A 90% interval for mean task time in format A minus format B is \((-2.4,\ 1.1)\) minutes. What is the interval for format B minus format A, and does it include zero?
- Why is \((-1.6,\ -4.2)\) not written in the correct endpoint order as a reversed interval?
- When an interval is reversed, which features stay the same: confidence level, units, width, and direction indicated by the signs?