Selecting an SRS With randInt on a Calculator

From Random Digits to Calculator Commands

In Selecting an SRS With a Random Number Table, we used fixed-size digit groups to select roster labels. A calculator can generate those labels directly as random integers. The key rule is the same: accept a valid label that has not been selected before, and skip a label that is already in the sample.

On a TI-84, the command randInt(lower, upper) generates a random integer from the lower bound through the upper bound, including both endpoints. Adding a third argument tells the calculator how many integers to generate: randInt(lower, upper, number of draws). The particular values produced can vary from one run to another.

For example, if a roster has 24 individuals labeled 1 through 24, the command \(randInt(1,24)\) generates one label in that range. The command \(randInt(1,24,6)\) generates a list of six labels in that range. A generated label may appear more than once. The calculator is making random draws that can include repeats; the sample we build by skipping repeats is without replacement.

Some calculators display labels without leading zeros. If a roster is labeled 01 through 24, the calculator may display 4 rather than 04. These represent the same individual. Keep the roster-label correspondence clear, and do not mistake a display format for a different label.

Set the Range and Process the Results

First, make sure the sampling frame is an appropriate list of the population you want to study, as discussed in Population, Sampling Frame, and Sample. Give every individual one unique label. If there are \(N\) individuals, labeling them 1 through \(N\) is a convenient choice.

Enter the smallest and largest labels as the lower and upper bounds. These bounds are inclusive: \(randInt(1,24)\) can return 1 or 24. If your roster is labeled 001 through 180, enter the numerical bounds 1 and 180. A command with the wrong upper bound could omit roster members or generate values that do not correspond to anyone.

You can generate values one at a time or in batches. With one-at-a-time generation, use \(randInt(1,N)\), check the result, and repeat the command as needed. With batch generation, use \(randInt(1,N,k)\) to request \(k\) values. In either case, preserve the order in which values are generated.

If your calculator generates values using the exact roster bounds, every output will be in range. Unlike a random number table, you should not expect out-of-range values from a correctly entered randInt command. A repeated value is still possible, though, so you must check for duplicates.

A Step-by-Step Calculator Routine

Worked Example: Selecting Five People from a Roster

Worked Example: Continue After a Batch Has Too Few Labels

Worked Example: Labels That Include Leading Zeros

Why the Final Sample Is Without Replacement

The raw calculator draws may repeat because each call can produce any label in the range. But when you accept only labels not already selected, each individual enters the final sample at most once. That is what “without replacement” means here: once an individual is selected, that person cannot be selected again to fill another place in the sample.

Skipping a repeated value does not mean replacing it with a label chosen by the sampler. You simply ignore the repeat and continue to the next random draw. Do not change a repeated 8 into a 9, for example, or restart because the output seems inconvenient. Either action could make some possible samples more likely than others.

This procedure also gives each possible group of the requested size an equal chance of selection, assuming the calculator’s random integer generator treats the roster labels fairly. One way to see why is to consider the order of the distinct labels: every possible ordered sequence of different labels is equally likely. Any particular group of \(n\) individuals can appear in \(n!\) different orders, so every group has the same number of ways to be selected.

This connects the calculator routine to the definition of an SRS in Simple Random Sample Defined. The important feature is not that one selected group looks balanced or representative. It is that the selection process gives every possible group of the chosen size an equal chance.

Common Mistakes and AP Exam Tips

• Counting repeated labels twice. A sample of six individuals must contain six different individuals. A repeated calculator output is skipped, and another value is generated if needed.
• Using the wrong bounds. Check that the lower and upper bounds match the smallest and largest roster labels. The endpoints are included.
• Generating only the requested number of values. A batch of \(n\) draws can contain repeats and may therefore produce fewer than \(n\) distinct labels. Generate additional values until the sample has \(n\) different individuals.
• Changing a repeat into a different label. Do not substitute a number of your choosing. Skip the repeat and let the next random draw determine the next candidate.
• Restarting after an unwanted result. Follow the output in order. Restarting because a label is surprising or inconvenient can change the selection chances.
• Confusing the raw draws with the final sample. The list from randInt may have duplicates. The final sample consists only of the requested number of distinct labels.
• Misreading leading zeros. A calculator display of 5 corresponds to roster label 005 if labels use three digits. Use the roster to identify the individual.

For a full-credit description, state how individuals are labeled, give the randInt bounds, and explain that values are processed in order. Say that repeats are skipped and more values are generated until the requested sample size of distinct individuals is reached. If asked why the result is an SRS, connect the chance-based selection process to the equal-chance definition of an SRS.