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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 survey 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.(4 points)
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No — Freda’s sample is not a random sample and is not a good representation of the population she wants (all high school students).
Why not:
- She surveyed only her statistics classmates (a convenience sample). That group is likely to be different from the general high‑school population (selection bias): students taking statistics are probably more favorable toward or more exposed to math.
- Because the sample is not randomly drawn from the population of all high‑school students, results cannot be generalized and the 82% may be biased.
What she can conclude (if the sample were random):
- If she had taken a proper random sample of high‑school students, she could report the sample proportion p̂ = 0.82 as an estimate of the population proportion and compute a confidence interval (e.g., a 95% CI = p̂ ± z*sqrt(p̂(1−p̂)/n)) to quantify sampling error.
Suggested random-sampling alternative (one practical option):
1. Define the population precisely (e.g., all students in grades 9–12 in the city/district).
2. Obtain a sampling frame (a roster of students by school).
3. Use either:
- Simple random sample: assign each student a number and use a random number generator to select n students; or
- Stratified random sample: divide students by school and/or grade (and/or intended major), then take random samples from each stratum proportionally — this ensures representation across subgroups.
- (If many schools) Cluster sample: randomly select a sample of schools, then randomly sample students within those schools.
4. Choose an appropriate sample size (for a ±5% margin of error at 95% confidence, n ≈ 384 is commonly used; required n depends on desired precision).
5. Administer the same question to the selected students and compute p̂ and a confidence interval.
Bottom line: Freda cannot generalize the 82% result to all high‑school students based on her classmates alone. She should use a random sampling method (simple or stratified) to produce an unbiased estimate.
Why not:
- She surveyed only her statistics classmates (a convenience sample). That group is likely to be different from the general high‑school population (selection bias): students taking statistics are probably more favorable toward or more exposed to math.
- Because the sample is not randomly drawn from the population of all high‑school students, results cannot be generalized and the 82% may be biased.
What she can conclude (if the sample were random):
- If she had taken a proper random sample of high‑school students, she could report the sample proportion p̂ = 0.82 as an estimate of the population proportion and compute a confidence interval (e.g., a 95% CI = p̂ ± z*sqrt(p̂(1−p̂)/n)) to quantify sampling error.
Suggested random-sampling alternative (one practical option):
1. Define the population precisely (e.g., all students in grades 9–12 in the city/district).
2. Obtain a sampling frame (a roster of students by school).
3. Use either:
- Simple random sample: assign each student a number and use a random number generator to select n students; or
- Stratified random sample: divide students by school and/or grade (and/or intended major), then take random samples from each stratum proportionally — this ensures representation across subgroups.
- (If many schools) Cluster sample: randomly select a sample of schools, then randomly sample students within those schools.
4. Choose an appropriate sample size (for a ±5% margin of error at 95% confidence, n ≈ 384 is commonly used; required n depends on desired precision).
5. Administer the same question to the selected students and compute p̂ and a confidence interval.
Bottom line: Freda cannot generalize the 82% result to all high‑school students based on her classmates alone. She should use a random sampling method (simple or stratified) to produce an unbiased estimate.
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