Measurement and Data>Simulating Random Samples to Understand Representativeness(TEKS.Math.8.11.C)
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Texas 8th Grade Math › Measurement and Data>Simulating Random Samples to Understand Representativeness(TEKS.Math.8.11.C)
A school has 800 students (200 in each grade). To estimate the fraction who prefer hot lunch, researchers randomly select 50 students from each grade (200 total) and repeat the simulation 20 times with different random seeds.
What should they expect from the 20 random samples?
All 20 estimates will be identical because random means no variability.
The samples will be biased toward cafeteria eaters even if selection is random.
Estimates will vary from sample to sample, but with 200 students they should cluster around the true population fraction.
Exactly one sample will match the true population fraction; the rest are invalid.
Explanation
In a random sample, every student has an equal chance to be selected. Repeated random samples will give slightly different results, but larger random samples (like 200) tend to be more stable and center around the true population value, allowing inference about the school.
A factory makes 5,000 items per day. To monitor defects, the quality team randomly selects 100 items each day for a week and records the defect rate.
If they instead selected only 20 random items per day, how would that affect representativeness?
Larger random samples (like 100) generally give more stable estimates of the defect rate than smaller random samples (like 20).
Smaller samples are more accurate because they are faster to collect.
If the sampling is random, sample size does not affect the reliability of the estimate.
Random samples guarantee that no defective items will be missed.
Explanation
Random sampling gives each item an equal chance to be chosen. While both 20 and 100 can be random, a larger random sample usually reduces variability in the estimate, making it more representative of the population and more reliable across repeated samples.
A city library has 12,000 members. To estimate what percent borrowed an e-book last month, staff take three different random samples of 150 members each and compute the percent for each sample.
Which statement best describes what repeated random sampling shows here?
If the three estimates differ, the sampling was not random.
The first sample is always the most accurate estimate of the population.
Each sample must match the population exactly if the sample is large.
The three estimates will differ slightly, but they should be centered around the true population percentage.
Explanation
Random sampling gives every member an equal chance to be chosen. Repeated random samples will vary from one another, but their estimates tend to center around the true population value, which is why random samples support inference.
A recreation center with 1,200 members wants the average weekly visits. Staff decide to survey every 10th person who walks in on Monday morning.
What does this sampling method suggest about randomness and representativeness?
It is random because choosing every 10th person makes selection unpredictable.
It may not be representative because not all members have an equal chance (e.g., people who do not come Monday mornings are excluded); a larger, truly random sample would be better.
It will perfectly match the population average because of the regular spacing.
Sample size does not matter if you always sample on the same day.
Explanation
Random sampling means every member has an equal chance of selection. Sampling only Monday morning visitors can be biased. Using a truly random method across the membership—and increasing the random sample size—improves representativeness and reliability.
A game company with 60,000 players wants to estimate the average satisfaction rating for a new update. They simulate 100 different random samples of 400 players and compute the sample average each time.
What can this simulation tell us about making inferences from samples?
If the 100 sample averages are identical, the population average must have changed.
You cannot learn anything about the population from samples because they are too small.
Random samples allow you to estimate the population average, and larger samples make the estimates less variable across repetitions.
Only a full census of all 60,000 players can estimate the population average.
Explanation
Random sampling gives each player an equal chance to be chosen, allowing inference about the population. Repeated random samples will vary, but their averages cluster near the true population value. Increasing the random sample size generally reduces variability and improves stability.