Sampling

To decide what bias is most important, we focus on the way the sample is collected. A strong sample gives everyone in the population a fair chance to be included.

So we should examine the limits created by the specific place and time described in the situation.

The volunteer is at the park entrance at 10:00 AM on a Tuesday.

1. Place: Standing at the park entrance means only people who are already going into the park can be surveyed. That leaves out city residents who don’t go to that park at all.
2. Time: 10:00 AM on a Tuesday is during a typical work/school day. The survey is more likely to include people who are free then (for example, retirees or people not working at that time) and miss many residents who are at work or in class.

We can eliminate the other options by matching each one to what it actually means:

• Calculation error: This would be an arithmetic or processing mistake after collecting results. Here, the main problem happens before any calculations.
• Non-response bias: That’s when people who were selected don’t answer. In this situation, the larger issue is that many residents were never even reachable to be selected.
• Sampling variability: Random samples naturally differ from each other, but this problem isn’t mainly randomness—this sample isn’t random in the first place.

This is a clear example of Selection bias. The sampling method (specific location and weekday morning time) leaves out large groups of the city, so the survey results won’t reflect the full population.

To identify the bias, start by describing the sampling method. The researcher is essentially using convenience sampling—asking whoever is easiest to reach at a set place and time:

• Location: Park entrance
• Time: 10:00 AM on a Tuesday
• Sample rule: survey the first 50 people they see

This connects to Sampling Methods and Bivariate Data. Now we can eliminate choices that don’t fit.

A calculation error happens when someone computes statistics incorrectly (like adding wrong or calculating a wrong percentage). At this stage, the issue is with who gets asked, not with any math done afterward—so calculation error isn’t the main bias.

Sampling variability is about the normal random differences you’d see from one random sample to another. The question is asking for a systematic problem (bias). Since the method isn’t random and is restricted by time and place, variability isn’t the key issue here.

Non-response bias occurs when chosen participants don’t respond, and those non-responders differ from responders. But the scenario mainly shows a problem with who gets approached in the first place, not people refusing. So non-response bias is not the best answer.

Because surveying at 10:00 AM on a weekday misses many residents who are working or in school, the sample is systematically unrepresentative. That makes the biggest issue Selection bias.

To figure out the bias, first recognize the technique. The researcher decides to "stand at the park entrance at 10:00 AM on a Tuesday" and "survey the first 50 people they see."

That is convenience sampling. In Sampling Methods and Bivariate Data, convenience sampling is a non-probability method where participants are chosen because they are easy to access.

For valid conclusions, the sample should represent all city residents. But collecting responses at 10:00 AM on a Tuesday leaves out many groups, such as:

• People with standard daytime jobs
• Students who are in school
• Residents who don’t go near that particular park

Since the sample misses these groups, it is not representative and is strongly affected by selection bias.

Eliminate options that describe different issues:

• Sampling variability: expected random differences between random samples, not a built-in flaw in who gets chosen.
• Calculation error: mistakes in arithmetic/data processing, which hasn’t happened yet.
• Non-response bias: about selected people refusing; the bigger problem here is that many people weren’t likely to be selected at all.

Because convenience sampling here produces a group that doesn’t reflect the full range of city residents, the most significant bias is Selection bias.

With systematic sampling, you begin at a randomly chosen position in the sampling frame and then select every $k$-th item after that. The step size is found using:

$k=\frac{N}{n}$

where $N$ is the number of individuals in the population and $n$ is how many you want in the sample. This method is quick to run, but how well it represents the population depends on how the list is arranged.

The biggest weakness of systematic sampling is periodicity. This is when the population list has a hidden cycle that repeats, and the cycle happens to match the sampling interval $k$.

For instance, imagine taking every 5th entry from a timetable of cafeteria meals. If every 5th day is always “Pizza Day,” your sample will overinclude pizza and won’t reflect the full range of meals. An SRS does not follow a fixed spacing, so it is less likely to get trapped by that kind of repeating structure.

• Option A is incorrect because it uses absolute language (“definitely”). Systematic sampling can work well, but it is not guaranteed to beat an SRS, especially if the list order causes bias.
• Option D is incorrect because systematic sampling can be a probability method as long as the starting point is chosen randomly; the fixed interval does not make it non-probability by itself.

For Option C, an SRS normally needs a complete numbered list so every individual can be selected with equal chance. Systematic sampling is often used when working from a long physical flow or an ordered process (like choosing every 10th visitor), so saying SRS is preferred when a full list is missing mixes up the usual requirement.

Because a systematic sample can accidentally follow a repeating pattern in the list, it can be less representative than an SRS when the interval matches that pattern. That is exactly what Option B describes.

To decide which statement is true, compare Simple Random Sampling (SRS) with Systematic Sampling. These are standard methods in Sampling Methods and Bivariate Data.

Using the Process of Elimination, we can test each option against the definitions and known limitations of the methods.

In systematic sampling, you pick a random starting position and then select every $k^{th}$ element (with $k=\frac{\text{Population Size}}{\text{Sample Size}}$). Since the start is random and the rule is applied consistently, individuals have a calculable chance of selection.

So systematic sampling is a probability sampling method, not automatically a non-probability method.

To carry out an SRS, you typically must have a complete list of the population so you can label everyone and choose randomly.

Systematic sampling can sometimes be used even when you cannot easily list everyone in advance (for example, selecting every $10^{th}$ customer who enters a shop). So it is not correct to say SRS is preferred when a full list is unavailable.

In statistics, words like “always” or “definitely” are warning signs. A systematic sample may spread selections evenly, but it is not guaranteed to be more representative than an SRS. Everything depends on how the sampling frame is ordered.

Systematic sampling can be thrown off by periodicity: a repeating pattern in the list that matches the interval $k$.

For example, if a coach records fitness results by testing every $6^{th}$ practice session, and every 6th session is always a light recovery day, the sample will make the team look less fit than usual. In that specific situation, the systematic sample can be less representative than an SRS.

A simple random sample of size $n$ means you must choose $n$ separate individuals from the population. When an RNG is used, each generated number is meant to point to exactly one person in the sampling frame (the numbered list of everyone).

Option A refers to a "sampling interval $k$". That idea belongs to systematic sampling, where you pick every $k^{th}$ individual. An SRS made with an RNG does not rely on a fixed interval, so this option does not fit.

Option C talks about ensuring the "range of the population" is covered. Although random samples can end up representing the population, the reason we require distinct numbers is about getting $n$ different selections, not about forcing the data to spread across the range.

Option D claims you can avoid using a sampling frame. However, an RNG only works for sampling if you have a numbered sampling frame so you can match each output to a specific individual. Distinct numbers do not remove the need for that list.

If the RNG repeats a number (for example, 7 appears again), you have selected the same person twice, not two people. To truly obtain a sample of size $n$, you must end up with $n$ distinct individuals, so repeated numbers are ignored and new numbers are generated.

In a simple random sample (SRS), each member of the population has an equal chance of being selected. Using a Random Number Generator (RNG) usually means giving every person a unique number in a sampling frame (a complete list of the population). The aim is to choose $n$ different people for the study.

Option A uses the phrase "sampling interval $k$", which is a feature of systematic sampling (pick a start, then take every $k$-th member). Because the situation is simple random sampling, this reasoning is not relevant.

Option C suggests distinct numbers guarantee the range is covered. But distinctness only prevents duplicates. If the RNG repeats the same number, you have not expanded coverage of the range—you have just chosen the same individual again.

Option D says an RNG removes the need for a sampling frame. This is false: to interpret RNG outputs (like 5, 12, 88), you need a sampling frame that tells which person each number represents.

Because the sample size is $n$, you must end up with $n$ different individuals. If a number repeats, it points to the same person again. Therefore, the numbers must be distinct so each selected member is a unique individual from the sampling frame.

In statistics, a repeating feature in an ordered list is called periodicity. Here, the list has a clear cycle: every $8^{\text{th}}$ house is a corner house.

We are also told the selection interval is the same size: sampling interval k is also 8. So if the first chosen item is number $r$, the sample continues with $r+8$, $r+16$, and so on. In Sampling Methods and Bivariate Data, a key idea is that the way a list is organized can interact with a sampling method and create bias.

The Statistical Theory of Periodicity warns that systematic sampling can be biased when the interval $k$ matches (or lines up with) a repeating pattern in the population.

$\text{Sampling Interval}=\text{Pattern Period}=8$

Because these values are equal, the sample keeps landing on the same position within each block of 8. If the starting point happens to be a corner house, then every selected house will be a corner house. If the starting point is not a corner house, then none of the selected houses will be corner houses.

Option A claims the sample is perfectly representative. But in the full list, only $\frac{1}{8}=12.5\%$ of houses are corner houses. With this sampling setup, the sample would end up as either $100\%$ corner houses or $0\%$ corner houses, which clearly does not match the population.

Option B says the method becomes a Simple Random Sample (SRS). In an SRS, every possible sample of a given size can occur with equal probability. Here, the fixed step size plus the repeating pattern means many mixes (like getting a few corner houses but not all) cannot happen, so it is not an SRS.

Option D suggests systematic sampling removes bias. Actually, in this situation the systematic rule is what causes the problem, because the list has a cycle that matches the interval. Systematic sampling can work well when the list has no repeating structure, but it breaks down when the interval aligns with a pattern.

Because the sample will end up with either all corner houses or no corner houses, it will not reflect the true proportion in the population. So the most likely outcome is that the sample is biased because it will strongly over-count or under-count corner houses.

This situation includes periodicity: every $8^{\text{th}}$ house is a corner house, meaning houses 8, 16, 24, ... are corner houses.

A systematic sample with interval $k$ is also 8 means you choose a random start from 1 to 8, then keep selecting every 8th item after that.

Suppose the random starting house is House 1.

With $k=8$, the selected houses are: $\text{House 1, House 9, House 17, House 25, ...}$

None of these are corner houses (since corner houses are 8, 16, 24, ...). So the sample contains $0\%$ corner houses, far below the true $12.5\%$ rate.

Now suppose the random starting house is House 8.

With $k=8$, the selected houses are: $\text{House 8, House 16, House 24, House 32, ...}$

Every one of these is a corner house. So the sample contains $100\%$ corner houses, far above the true $12.5\%$ rate.

Comparing these outcomes:

• Start at 1: $0\%$ corner houses (too low).
• Start at 8: $100\%$ corner houses (too high).

Since the interval ($k=8$) matches the repeating pattern (every 8th), the sample depends heavily on the starting position and does not capture the real variety in the population.

From the scenarios, we can reject:

• Option A: Not correct. The sample can be extremely unbalanced, so it is not perfectly representative.
• Option B: Not correct. Systematic sampling only behaves like an SRS when there is no pattern linked to the ordering.
• Option D: Not correct. The systematic rule is what creates the bias here because it matches the periodic structure.

The tests show the sample will either completely miss corner houses or include only corner houses. That is a clear case of sampling bias caused by the sampling interval matching a periodic pattern in the list.

This is a systematic sampling question from Sampling Methods and Bivariate Data. The list has a repeating structure: every 8th house is a corner house, and the sampling step is the same: sampling interval k is also 8. So the ordering and the sampling plan line up exactly.

Consider Option B. A simple random sample (SRS) requires that every individual and every possible sample of size $n$ has an equal chance of being selected. Systematic sampling does not suddenly become an SRS, especially when a fixed interval interacts with a repeating pattern in the list.

Now look at Option A. Saying “perfectly representative” is already unlikely, and here it is clearly wrong. If the start is house 8, the sample is houses 8, 16, 24, ... which gives $100\%$ corner houses. If the start is house 1, the sample is houses 1, 9, 17, ... which gives $0\%$ corner houses. Neither matches the true $12.5\%$ corner-house rate.

Option D says bias disappears because the method is systematic. But when the list has periodicity and $k$ matches that period, systematic sampling can create periodicity bias rather than prevent bias.

After ruling out the other choices, Option C is left and it also matches the reasoning: because $k=8$ aligns exactly with the corner-house pattern, the sample will strongly over-represent corner houses or miss them entirely, so it is biased.