When the Population Count Is Missing
In Independence and the 10 Percent Condition, you learned to compare the sample size with the size of the population from which the sample was selected. But study descriptions do not always report that population size. If the source population is clearly much larger than the sample, you can still justify treating observations as approximately independent—provided you explain why the population must be more than ten times the sample size.
The key is not to guess an exact value for \(N\). Instead, identify the source population and use information in the description to support the comparison. A small random sample from a clearly broad population, such as a nationwide group of millions of users, may meet the condition even when its exact size is not reported. A vague phrase such as “a large group,” without further context, may not be enough.
For instance, if \(n=200\), the population must be at least \(2{,}000\) for the condition to hold. If the description says the sample came from a nationwide population of millions, that scope provides clear evidence that the source population exceeds the threshold. You do not need to know whether the population has exactly two million, three million, or another number.
A Practical Reasoning Sequence
When \(N\) is missing, work through the same basic logic as when it is provided, but support the population-size comparison with words rather than an invented calculation. First identify the population from which the sample was actually drawn. Then calculate the minimum population size that would meet the condition, \(10n\). Finally, ask whether the description gives convincing contextual evidence that the source population is at least that large.
Use the group eligible for selection, not a larger group the researcher hopes to describe.
Multiply the sample size by 10. This is the minimum \(N\) that meets the 10% condition.
Look for a clear scope or scale, such as a nationwide population described as numbering in the millions.
Explain that the source population is clearly more than ten times the sample, so the condition is met. Do not claim an exact \(N\) unless one is given.
This reasoning does not change the condition. It changes only how you provide evidence for it. When the exact population size is given, show the numerical comparison. When it is not, state the contextual evidence that makes the comparison convincing.
A useful written response connects all three pieces: the sample size, the actual source population, and why that population is clearly large enough. For example: “The sample of 250 was selected without replacement from users of a nationwide service with millions of users. That source population is clearly more than \(10(250)=2{,}500\), so the 10% condition is met.” This does not pretend to know the exact number of users.
Worked Examples
Worked Example: A Nationwide App Survey
A fictional app company selects a random sample of 300 active users without replacement from its nationwide user base. The description says the service has millions of active users but does not give an exact count. Can you justify the 10% condition?
Identify and find the threshold. The relevant population is the active users eligible to be selected, and \(n=300\). A population of at least \(10(300)=3{,}000\) users would meet the condition.
Use the context. The description says this is a nationwide service with millions of active users. A population numbering in the millions is clearly greater than the 3,000-user threshold. No exact value for \(N\) is needed to make that comparison.
Conclude. The sample of 300 is drawn without replacement from a nationwide user base described as numbering in the millions, which is clearly more than ten times the sample size. Therefore, the 10% condition is met, and the observations may be treated as approximately independent for an appropriate one-proportion inference procedure.
Worked Example: A Sample of Licensed Drivers
A fictional public agency randomly selects 180 licensed drivers without replacement from all licensed drivers in a large country. The study description gives no total number of licensed drivers. It does make clear that the source population includes drivers throughout the country, rather than drivers from one small town. Is the independence justification reasonable?
Find the threshold. For \(n=180\), the population must have at least \(10(180)=1{,}800\) licensed drivers to meet the 10% condition.
Evaluate the evidence. The exact count is absent, so it would be inappropriate to assign a specific value to \(N\). However, the sampling frame covers licensed drivers throughout a large country. That description supports the conclusion that the eligible population is clearly larger than 1,800.
Conclude. Given the stated nationwide scope, the source population is clearly more than ten times the sample of 180. The 10% condition is therefore reasonably justified, so the observations may be treated as approximately independent. If the description had only said “a large group of drivers,” without clarifying the scope of the sampling frame, that wording would provide weaker evidence.
Worked Example: A Broad Member Survey
A fictional organization randomly selects 90 current members without replacement for a survey. The description says its membership includes people from every region of the country and numbers in the hundreds of thousands, but it does not report the exact membership total. Does the condition hold?
Calculate the required population size. The sample size is \(n=90\), so the threshold is \(10(90)=900\) members. The source population must contain at least 900 members.
Check what the description establishes. The membership is described as numbering in the hundreds of thousands. Even without an exact total, that scale is clearly far above 900. The relevant group is current members, because they were eligible to be selected.
Conclude. The sample of 90 is a small fraction of a membership numbering in the hundreds of thousands, so the 10% condition is met. The missing exact total does not prevent the justification because the contextual scale clearly exceeds the required threshold.
Worked Example: When “Large” Is Not Enough
A fictional organization randomly selects 140 customers without replacement from a group described only as “a large customer list.” The exact number of eligible customers and the geographic or organizational scope of the list are not provided. Can you conclude that the 10% condition holds?
Find the threshold. For \(n=140\), the source population must contain at least \(10(140)=1{,}400\) customers.
Assess the description. The phrase “a large customer list” does not establish that at least 1,400 customers were eligible. It may sound substantial, but the description gives no count, lower bound, or scope that can be compared with the threshold.
Conclude. With the information provided, the 10% condition cannot be justified. This does not show that the condition fails; it means there is not enough evidence in the description to decide. A complete response would say the population size is not reported and the wording does not establish that it is at least 1,400, rather than assuming the condition holds.
What Counts as Convincing Context?
A qualitative justification should make the comparison understandable to someone who has not seen the population count. The context is strongest when it gives a clear scale or scope. Descriptions such as “millions of active users nationwide” or “hundreds of thousands of current members” establish a scale that can be compared with a modest threshold such as \(10n\). A specific lower bound also works: if a description says there are at least 8,000 eligible people and the sample is 500, then \(8{,}000 \geq 10(500)=5{,}000\).
By contrast, “a large group” or “a broad survey” may not be enough on its own. Those words do not tell you whether the population is at least ten times the sample. The question is not whether the source population sounds large in everyday language; it is whether the information given supports \(N \geq 10n\).
Keep the source population straight. If a sample was selected from subscribers to a service, use the service’s eligible subscribers as the population for the check. Do not rely on a much larger national population just because the company operates nationally. The larger group matters only if it is actually the group from which the sample was drawn.
Also keep this condition separate from other requirements. As discussed in Why Inference Procedures Need Conditions, the 10% condition does not establish that the sample was randomly selected, that the sample represents the intended target population, or that the Large Counts condition is met. Each condition answers a different question.
Common Mistakes and AP Exam Tip
- Making up an exact population size. If the description says “millions,” do not write that \(N\) equals a particular number of millions. Use the given scale to compare with \(10n\).
- Using the wrong population. Explain the size of the group from which the sample was selected, not a larger group mentioned elsewhere in the situation.
- Assuming that “large” proves the condition. Show the threshold \(10n\), then explain what specific description supports the claim that the source population exceeds it.
- Claiming the condition fails just because \(N\) is absent. Missing information does not prove the condition is false. It may be possible to justify it from context, or the description may simply be insufficient to decide.
- Claiming the entire inference is valid after checking independence. The 10% condition addresses approximate independence for sampling without replacement; it does not replace the Random condition or the Large Counts condition.
Key Takeaway
An unstated population count does not prevent you from checking independence if the study description clearly establishes that the source population is more than ten times the sample size. Calculate \(10n\), identify the group actually sampled, and explain how the stated scale supports the comparison. If the context is too vague, do not invent a count or claim the condition is met.
Check Your Understanding
For each situation, identify the sample-size threshold and decide whether the description supports the 10% condition.
- A random sample of 240 users is selected without replacement from a nationwide service described as having millions of active users. What minimum population size is required, and how can the condition be justified without an exact \(N\)?
- A random sample of 125 people is selected from a group described only as “a large organization.” What is the threshold, and is the description specific enough to verify the condition?
- A sample of 60 current members is selected from a membership described as numbering in the tens of thousands. Explain why the missing exact count does or does not prevent an independence justification.
- A random sample of 300 customers is selected from a list of active customers. The company serves a country of many millions of residents, but the size of its customer list is not reported. Which population size matters, and what information is still needed?
- Write an AP-style sentence justifying the 10% condition for a sample of 400 people if the source population is described as having at least 12,000 eligible people.