Random Sampling and Inference
A free Pre-Algebra lesson from the “Statistics and Probability” unit, with a worked example and practice problems including step-by-step solutions.
A random sample gives every member of the population an equal chance of being chosen — only random samples reliably represent the whole population. A biased sample over- or under-represents groups. To estimate a population total from a sample, set up a proportion: sample-count / sample-size = unknown / population. In Statistics and Probability, the goal is not just to get an answer but to recognize the structure of the problem, choose a reliable strategy, and explain why the result is reasonable. The practice set now includes targeted skill work, transfer questions, and mixed review so students build fluency and retention.
What you'll learn
- Recognize random vs biased samples
- Use a sample proportion to estimate a population total
- Understand that larger random samples are usually more reliable
Worked example
Problem. A random sample of 100 voters shows 60 support a measure. Estimate support among 1000 voters.
- Sample proportion: 60 / 100 = 0.6 (60%).
- Apply to the population: 0.6 * 1000 = 600.
- Connect the calculation back to Random Sampling and Inference so the method, not just the arithmetic, is clear.
- Check the result against the original question before writing the final answer.
Answer: 600
Practice problems
1. A random sample means every member of the population has:
Choices: Equal chance of being chosen · No chance · A higher chance if they speak up
Show solution
- Warm-up: First identify exactly what the question is asking: A random sample means every member of the population has:
- For data questions, identify what each statistic measures before calculating so the result matches the question.
- That is the definition of random sampling.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
- Write the final response in the form requested by the prompt.
Answer: Equal chance of being chosen
2. Surveying only students in the gym about favorite school activities is:
Choices: Random · Biased
Show solution
- Warm-up: First identify exactly what the question is asking: Surveying only students in the gym about favorite school activities is:
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Gym-goers may favor sports more than the general population.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
- Write the final response in the form requested by the prompt.
Answer: Biased
3. A sample of 100 randomly chosen voters predicts 60% support a measure. Estimate support in 1000 voters.
Show solution
- Warm-up: First identify exactly what the question is asking: A sample of 100 randomly chosen voters predicts 60% support a measure. Estimate support in 1000 voters.
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- 60% of 1000 = 600.
- Check the result by substituting or estimating: the response should match 600 and make sense in the original problem.
- Write the final response in the form requested by the prompt.
Answer: 600
4. Larger random samples generally give:
Choices: More accurate estimates · Less accurate estimates
Show solution
- Core Practice: First identify exactly what the question is asking: Larger random samples generally give:
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- More data narrows the margin of error.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
- Write the final response in the form requested by the prompt.
Answer: More accurate estimates
5. A capture-recapture sample of 50 fish has 5 tagged. The lake has 1000 fish total. Estimate the tagged population.
Show solution
- Core Practice: First identify exactly what the question is asking: A capture-recapture sample of 50 fish has 5 tagged. The lake has 1000 fish total. Estimate the tagged population.
- Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
- 5/50 = x/1000 means x = 100.
- Check the result by substituting or estimating: the response should match 100 and make sense in the original problem.
- Write the final response in the form requested by the prompt.
Answer: 100
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