Why Check the 10% Condition?
In “Checking the Random Condition,” we looked at how the data were collected and what the random process supports. For a mean inference based on a sample drawn without replacement, there is another question: is the sample small compared with the population? This check helps justify treating the observations as approximately independent.
When sampling without replacement, each selected person or item is no longer available for the next draw. So, strictly speaking, the draws are dependent. If the sample is only a small part of the population, however, removing one observation has little effect on the options for later draws. The 10% condition is a practical rule for deciding when that dependence is small enough to use the usual mean-inference methods.
As covered in “The 10% Condition for Sample Means,” the condition applies to a simple random sample from a finite population. Here we focus on how to check it from the information in a study description and how to explain the result. Passing this check does not establish the random condition or the Normal/Large Sample condition; those require their own checks.
How to Check It
Start by identifying the population the sample was drawn from, then find its size \(N\). Compare the sample size \(n\) with that population size. You can calculate the sample’s percentage of the population, or calculate the minimum population size needed to meet the condition. Both approaches give the same answer.
If the sample percentage is 10% or less, the condition is met. If it is greater than 10%, the condition is not met. When using the second approach, compare the actual population size with \(10n\): \(N\geq 10n\) meets the condition, while \(N<10n\) does not.
Use the population from which units were actually sampled—not simply the largest population mentioned in the question. For example, if a sample was selected from a list of registered members, the relevant \(N\) is the number of members on that list. Also be precise about the target: a list might not include everyone in a broader population. That coverage issue is separate from the arithmetic of the 10% condition.
The 10% check is for sampling without replacement from a finite population. If the description does not say how sampling occurred, do not assume a method.
Find the sample size and the size of the population from which the sample was selected. Use counts that refer to the same sampling setting.
Check whether \(n/N\leq 0.10\), or equivalently whether \(N\geq 10n\).
Say whether the condition is met and connect that decision to treating observations as approximately independent for the mean procedure.
Worked Examples
Worked Example: A Sample Exactly at the 10% Limit
A clinic has 1,250 adult members on its current patient list. A researcher selects a simple random sample of 125 members without replacement and records each person’s wait time at the clinic. The researcher plans to use a one-sample t procedure to estimate the population mean wait time. Check the 10% condition.
State. The parameter of interest is the mean wait time, \(\mu\), for adult members on the clinic’s patient list. We need to check whether the sample observations can be treated as approximately independent for the mean procedure.
Plan. The sample was selected without replacement from a finite population. For the 10% condition, compare \(n\) with \(0.10N\), or check whether \(N\geq 10n\). The sample size is \(n=125\), and the population size is \(N=1{,}250\).
Do. Using the percentage method:
The equivalent check gives \(10n=10(125)=1{,}250\), and \(N=1{,}250\). Thus \(N\geq 10n\) is true, with equality.
Conclude. The sample is exactly 10% of the population, so it meets the “no more than 10%” condition. For this mean procedure, the dependence from sampling without replacement is small enough to treat the observations as approximately independent. This conclusion checks only the 10% condition; the random and Normal/Large Sample conditions still need separate attention.
Worked Example: A Sample That Is Too Large a Fraction
A recreation department has 600 people registered for a weekend program. It selects 75 people without replacement to ask about their weekly exercise time. The department wants to use a mean-inference procedure for the registered participants. Does the sample meet the 10% condition?
Identify the counts. Here, \(n=75\) and \(N=600\). The relevant population is the 600 registered participants from whose list the sample was selected.
Check the fraction. The sample percentage is
Since \(12.5\%>10\%\), the condition is not met. Check the same result another way: \(10n=10(75)=750\), but the population has only \(N=600\) people, so \(N<10n\).
Conclusion. This sample is more than 10% of the registered participants, so the 10% condition is not satisfied. The usual mean-inference calculation relying on approximate independence is not justified by this condition check. To meet the rule with a sample of 75, the population would need to contain at least 750 people.
Worked Example: The Population Size Is Not Given
A parks team reports that it selected 42 neighborhood residents without replacement and recorded their daily walking time. The description does not state how many residents are in the population from which the sample was selected. Can you verify the 10% condition?
Identify what is known. The sample size is \(n=42\), and the sample was drawn without replacement. The population size \(N\) is not provided, so the sample percentage cannot yet be calculated.
Find the threshold. For a sample of 42, the population must contain at least ten times that many residents:
If \(N=500\), for example, the sample percentage would be
and the condition would be met. But if \(N=400\), the percentage would be
and the condition would not be met. These comparisons show why the missing population size matters: the condition’s result depends on \(N\).
Conclusion. From the description alone, the 10% condition cannot be verified. The population size must be at least 420 for a sample of 42 to meet it. A complete response should not guess the number of neighborhood residents or claim that the condition holds without that information.
Worked Example: Sampling With Replacement
A quality-control technician repeatedly selects one item from a large shipment, records its weight, and returns it to the shipment before making the next selection. The technician plans to take 100 measurements. Is the 10% condition the relevant check for whether the selections are independent?
The items are returned before the next selection, so the sampling is with replacement. The 10% condition is specifically for sampling without replacement from a finite population; it is not the check needed here. Returning each item means that the selection process does not remove it from the options for later draws. If the draws are made independently as described, the 10% condition does not need to be applied, even if 100 is more than 10% of the shipment.
This does not automatically establish every condition for a mean procedure. The technician would still need to consider how the items were selected and the shape of the relevant distribution. The example illustrates why identifying “with” or “without” replacement comes before doing the 10% arithmetic.
Common Mistakes and AP Exam Tips
- Using the wrong population size. Use the population from which the sample was actually drawn. If the sample came from a list of 600 registered participants, do not substitute a larger number of people who live in the same area but were not on that sampling list.
- Reversing the ratio. The sample fraction is \(n/N\), not \(N/n\). Divide the sample size by the population size, then multiply by 100% if you want a percentage.
- Checking only whether \(n\) is “small.” State the comparison. For example: “\(75/600=12.5\%\), which exceeds 10%, so the condition is not met.” That makes the decision verifiable.
- Treating 10% as a minimum. The sample must be no more than 10% of the population. A smaller percentage meets the condition; a percentage greater than 10% does not.
- Using the rule for sampling with replacement. First check whether sampled units are returned before another draw. The 10% condition concerns sampling without replacement.
- Claiming that passing the condition proves all observations are independent. With sampling without replacement, the observations are not literally independent. The condition supports treating them as approximately independent for the usual inference procedure.
- Assuming a missing population size is large enough. If \(N\) is not given, you can state the minimum required size, \(10n\), but you cannot decide whether the condition holds without knowing or establishing the relevant \(N\).
- Letting this check stand in for every condition. The 10% condition addresses approximate independence. As “Why Conditions Matter in Mean Inference” explains, it does not replace checking randomness or the Normal/Large Sample condition.
A concise AP-style explanation names the sampling method and population, shows the comparison, and says what follows: “Because the sample of 75 was selected without replacement from 1,000 eligible members, \(75/1{,}000=7.5\%\), which is no more than 10%. The 10% condition is met, so the observations can be treated as approximately independent for the mean procedure.” If the population size is unknown, say that the condition cannot be verified and identify what information is needed.
Check Your Understanding
For each situation, decide whether the 10% condition is met, or state what information is missing. Explain your comparison.
- A community center has 900 registered members and selects 81 members without replacement for a survey. What percentage of the population is sampled?
- A school has 1,400 students and selects a simple random sample of 140 without replacement. Does the condition hold at the boundary?
- A group selects 55 people without replacement from a population of 500. What is the sample percentage, and does it meet the condition?
- A sample of 36 residents is selected without replacement, but the population size is not reported. What is the minimum population size needed to meet the condition, and can you decide from the information given?
- A technician selects an item, records its measurement, returns it, and then selects again. Should the 10% condition be checked for this sampling method? Explain.