Question

For her statistics project, Freda wants to find the proportion of high school students who believe learning math is
necessary, regardless of their future majors. She conducts a survery on her statistics classmates and finds that 82%
of them believe it is necessary to learn math. Is Freda's sample a random sample and a good representation of the
population in her study? If so, what can Freda conclude about the proportion of high school students who believe
learning math is necessary? If not, propose a random sample on which Freda should be conducting her survey.

Answer 1

No — Freda’s sample is not a random sample and is not a good representation of the population she wants to study.

Why not
- She surveyed her statistics classmates, a convenience sample. Those classmates are not a random draw from all high school students.
- Students taking a statistics class are likely more interested in or exposed to math and so are more likely to say math is necessary (selection bias).
- The sample is probably small and restricted to one class (limited coverage), so it can’t be used to make reliable inferences about the broader population of high school students.

Because the sample is biased, Freda cannot conclude that 82% of all high school students believe learning math is necessary.

What she should do instead (a practical random-sampling plan)
1. Define the population precisely: e.g., “all high school students in the school district (or state/country), regardless of future major.”
2. Choose a sampling frame: obtain lists of students from schools in the population (or lists of schools if using cluster sampling).
3. Select a random sample:
- Simple random sample: if a complete list of students is available, use a random-number generator to select students.
- Cluster sampling (practical): randomly select a number of schools, then randomly sample students within each selected school.
- Stratified sampling (improves precision): divide students by grade or region and take random samples within each stratum to ensure representation.
4. Decide sample size: to estimate a proportion with a 95% confidence level and ±5% margin of error, use n ≈ 385 (worst-case p = 0.5). If she expects p ≈ 0.82, n ≈ 227 suffices for ±5%. Inflate the sample to allow for nonresponse (e.g., if expected response rate is 80%, divide required n by 0.8).
5. Use neutral wording and ensure anonymity to reduce response bias (e.g., “Do you believe learning math in high school is necessary for your future? Yes / No”).

If Freda follows a proper random sampling procedure with an adequate sample size, then she can estimate the proportion and report a confidence interval (for example, with n ≈ 385 and a sample proportion p̂, a 95% CI is p̂ ± 1.96·sqrt(p̂(1−p̂)/n)).