AP Statistics 5.1 Introducing Statistics: Why Is My Sample Not Like Yours? Study Notes
AP Statistics 5.1 Introducing Statistics: Why Is My Sample Not Like Yours? Study Notes- New syllabus
AP Statistics 5.1 Introducing Statistics: Why Is My Sample Not Like Yours? Study Notes -As per latest AP Statistics Syllabus.
LEARNING OBJECTIVE
- Given that variation may be random or not, conclusions are uncertain
Key Concepts:
- Introducing Statistics: Why Is My Sample Not Like Yours?
Introducing Statistics: Why Is My Sample Not Like Yours?
Introducing Statistics: Why Is My Sample Not Like Yours?
When we take random samples from the same population, we often notice that results are not identical. This variation is expected and is an important idea in statistics. Understanding this variation helps us ask the right questions and interpret data properly.
Key Concepts
- Sampling Variability: Different random samples from the same population will produce different statistics (like means or proportions).
- Expected Differences: Variation between samples is natural and does not mean that one sample is “wrong.”
- Investigating Variation: Observing how statistics vary helps us ask questions about how representative a sample is and whether differences are due to chance or some real effect.
- Key Question: Why do my sample results differ from yours, even if we sampled from the same population?
Identify Questions Suggested by Variation
- How much do sample statistics typically vary from sample to sample?
- Are the differences between two samples due to random chance or a meaningful difference in the population?
- How can we reduce variability between samples (e.g., increase sample size)?
- Does my sample give a reliable estimate of the population parameter?
Example
Two students each take a random sample of 50 students from the same school and ask whether they prefer online or in-person classes. One finds that 60% prefer online, while the other finds 52% prefer online.
Why are the results different, and what questions should we ask?
▶️ Answer / Explanation
Step 1: The results differ because of sampling variability — different random samples produce different statistics.
Step 2: We should ask:
- Are both samples large enough to reduce random error?
- Is the observed difference (60% vs 52%) just chance, or does it suggest a real difference in the population?
- Would taking more or larger samples give more consistent results?
Conclusion: The difference does not mean either sample is wrong; it reflects natural variation from random sampling.
Example
Two researchers each take a random sample of 30 apples from the same orchard to estimate the average weight. Researcher A finds a mean weight of 155 g, while Researcher B finds 162 g.
Why are their results different, and what questions should be asked?
▶️ Answer / Explanation
Step 1: The difference comes from sampling variability. Even when sampling from the same orchard, random samples will not give identical averages.
Step 2: Questions to ask:
- Is the difference (155 g vs 162 g) within the range we expect by chance?
- Would larger sample sizes give more consistent results?
- Could one sample have been biased (e.g., picked apples from a single tree)?
Conclusion: Different samples lead to slightly different estimates, but both provide information about the true average weight of apples in the orchard.
Example
A city council wants to estimate the proportion of residents who support building a new park. Two different survey teams each take a random sample of 100 residents. Team 1 finds 48% in favor, while Team 2 finds 55% in favor.
Why are the results different, and how should we interpret them?
▶️ Answer / Explanation
Step 1: The difference (48% vs 55%) reflects natural variation in random sampling.
Step 2: Questions to ask:
- Is the difference large enough to suggest real differences, or is it just due to chance?
- Would combining samples or increasing sample size give more stable results?
- Were both samples collected fairly, or was there a source of bias (e.g., only surveying in one neighborhood)?
Conclusion: The variation does not mean either survey is wrong — it shows why statisticians use repeated sampling or confidence intervals to capture the true population proportion.