Using Random Samples to Draw Inferences About a Population - Seventh Grade Mathematics

Statistical inferences project patterns discovered in representative random samples onto whole populations, using proportional scaling to estimate quantities and assess sample variability.

Making Population Inferences with Proportions

An inference is an educated, evidence-based conclusion drawn about a large population based on data collected from a representative random sample. When a sample is chosen randomly, the proportion of individuals in the sample exhibiting a specific characteristic closely mirrors the proportion of individuals in the entire population with that characteristic.

               THE STATISTICAL INFERENCE PROPORTION
       ( Sample Items with Trait ) / ( Total Sample Size )
                                =
     ( Estimated Population Trait ) / ( Total Population Size )

Cross-multiplication allows you to scale sample counts up to the population level:

Example: A random sample of 150 middle school students is surveyed.
45 students report that soccer is their favorite sport.
The entire school has an enrollment of 900 students.
Estimate how many students in the school favor soccer.

Setup Proportion:
  45 / 150 = x / 900
  150 * x = 45 * 900
  150 * x = 40,500
  x = 40,500 / 150 = 270 students

We infer that approximately two hundred seventy students in the school favor soccer.

Sample Variability: The Law of Fluctuating Samples

If three different researchers take three independent random samples of sixty students from the same school, will their samples produce identical counts? Almost certainly not. Sample results naturally fluctuate from sample to sample due to random chance. This phenomenon is called sample variability.

               SIMULATED SAMPLES (n = 60 from School of 1,200)
   Sample A: 18 prefer pizza  --> Estimated school total: (18/60) * 1200 = 360
   Sample B: 21 prefer pizza  --> Estimated school total: (21/60) * 1200 = 420
   Sample C: 19 prefer pizza  --> Estimated school total: (19/60) * 1200 = 380
   Mean Estimate across samples = (360 + 420 + 380) / 3 = 386.7 students

Although the three individual estimates vary between three hundred sixty and four hundred twenty, they cluster tightly around the true population proportion. Taking multiple random samples or combining samples produces a more reliable, stable estimate than relying on a single small sample.

Sample Size and Precision

The reliability of a statistical inference depends directly on the size of the random sample:

+-----------------------+-----------------------------+------------------------+
| Sample Size (n)       | Sample Variability          | Precision of Inference |
+-----------------------+-----------------------------+------------------------+
| Very Small (e.g., 10) | High variability (volatile) | Low precision          |
| Moderate (e.g., 100)  | Moderate variability        | Good precision         |
| Large (e.g., 1,000)   | Low variability (stable)    | High precision         |
+-----------------------+-----------------------------+------------------------+

As the sample size increases, the estimates drawn from different random samples become increasingly consistent, converging toward the true population parameter. However, a large biased sample is always inferior to a small random sample: increasing the size of a flawed sample merely magnifies the bias with greater confidence.

Exercises

Work through each problem carefully in your notebook before reading the provided solutions.

Exercise 1. A quality control inspector examines a random sample of two hundred smartwatches from a warehouse containing eight thousand smartwatches. The inspector finds that six watches have battery defects. Estimate the total number of defective smartwatches in the warehouse.

Exercise 2. A wildlife conservation biologist captures, tags, and releases one hundred fifty trout into a lake. Later, the biologist captures a random sample of one hundred twenty trout from the lake and discovers that eighteen of them are tagged. Use a proportion to estimate the total trout population in the lake.

Exercise 3. Two students conduct independent random surveys of fifty students each to estimate how many students in their seven-hundred-student school participate in music programs. Student A's sample finds sixteen musicians; Student B's sample finds twenty-two musicians. Calculate the population estimate from each survey, and find the mean of the two estimates.

Exercise 4. Explain why a random sample of one hundred registered voters is more reliable for predicting an election than an online voluntary survey of ten thousand website visitors.

Solutions

Solution to Exercise 1. Set up a proportion comparing the defective rate in the sample to the population: six defective watches out of two hundred sampled equals x defective watches out of eight thousand total. Cross-multiply: two hundred times x equals six times eight thousand. Six times eight thousand equals forty-eight thousand. Divide forty-eight thousand by two hundred: forty-eight thousand divided by two hundred equals two hundred forty. We estimate there are approximately two hundred forty defective smartwatches in the warehouse.

Solution to Exercise 2. This method is the capture-recapture technique. The proportion of tagged trout in the recapture sample equals the proportion of tagged trout in the entire lake: eighteen tagged recaptures out of one hundred twenty sampled equals one hundred fifty initially tagged trout out of total population N. Write the proportion: eighteen over one hundred twenty equals one hundred fifty over N. Cross-multiply: eighteen times N equals one hundred twenty times one hundred fifty. One hundred twenty times one hundred fifty equals eighteen thousand. Divide eighteen thousand by eighteen: eighteen thousand divided by eighteen equals one thousand. The estimated total trout population in the lake is one thousand trout.

Solution to Exercise 3. For Student A: sixteen out of fifty students equals x out of seven hundred. Solve for x: sixteen divided by fifty equals zero point thirty-two. Multiply zero point thirty-two by seven hundred: zero point thirty-two times seven hundred equals two hundred twenty-four students. For Student B: twenty-two out of fifty students equals y out of seven hundred. Solve for y: twenty-two divided by fifty equals zero point forty-four. Multiply zero point forty-four by seven hundred: zero point forty-four times seven hundred equals three hundred eight students. Find the mean of the two estimates: add two hundred twenty-four and three hundred eight to get five hundred thirty-two. Divide five hundred thirty-two by two: five hundred thirty-two divided by two equals two hundred sixty-six students.

Solution to Exercise 4. The validity of a statistical inference depends fundamentally on the representativeness of the sample, not merely its raw size. The random sample of one hundred voters gives every registered voter an equal probability of selection, ensuring that diverse perspectives and demographics are proportionally reflected without human selection bias. In contrast, the online survey of ten thousand visitors is a voluntary response convenience sample. It only captures visitors to that specific website who feel motivated to vote, completely excluding voters without internet access or visitors to opposing websites. A large sample cannot correct for systematic bias; therefore, the smaller random sample is far more reliable.