Question

A) 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 survey of her statistics classmates is not a random sample and is not a good representation of the population she wants to study.

Why not
- It’s a convenience sample (only her classmates).
- Classmates in a statistics class are likely not representative of all high school students (they may be more interested in math/statistics, come from similar grade levels or schools, etc.).
- Therefore the 82% only tells you about that class, not the broader population of high school students.

What she should do instead (random sampling options)
1. Simple random sample of students:
- Obtain a roster of high school students in the population of interest (e.g., a district or state).
- Assign each student a number and use a random-number generator to pick n students.
- Contact those students and administer the survey.

2. Stratified random sample (recommended if you want better precision across subgroups):
- Divide the population into meaningful strata (grade, school, region, urban/rural, gender, etc.).
- Take a random sample within each stratum proportional to its size (or use equal allocation if you want similar precision across strata).

3. Cluster sampling (practical if rosters are hard to get):
- Randomly select a number of schools (clusters), then randomly sample students within the selected schools (or survey all students in selected classrooms).

Sample size guidance
- For a 95% confidence interval with ±5% margin of error, n ≈ 385 (using conservative p = 0.5).
- If she wants to use p̂ = 0.82 as a planning value, n ≈ 385–400 still gives similar precision.

Example of what she could conclude if she used a proper random sample
- If Freda did a simple random sample of n = 400 students and observed p̂ = 0.82, a 95% confidence interval would be
0.82 ± 1.96·sqrt(0.82·0.18/400) ≈ 0.82 ± 0.038 → (0.782, 0.858).
- She could then say: “We are 95% confident that about 78.2% to 85.8% of high school students believe learning math is necessary.”

Practical notes
- Get school/district permission and parental consent as required.
- Minimize nonresponse bias: follow up with students who don’t respond and report the response rate.
- If resources are limited, cluster sampling by randomly selected schools/classrooms is an acceptable practical alternative.

In short: do not generalize the 82% from her statistics class to all high school students. Instead use a randomly selected sample (simple random, stratified, or cluster) of an appropriate size to make valid inferences about the target population.