AP Statistics 8.1 Introducing Statistics: Are My Results Unexpected? Study Notes
AP Statistics 8.1 Introducing Statistics: Are My Results Unexpected? Study Notes- New syllabus
AP Statistics 8.1 Introducing Statistics: Are My Results Unexpected? 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: Are My Results Unexpected?
Introducing Statistics: Are My Results Unexpected?
Introducing Statistics: Are My Results Unexpected?
When analyzing categorical data, we often compare what we actually observe in a sample (observed counts) to what we would expect if a certain claim, theory, or model were true (expected counts). Variation between the observed and expected counts can help us decide whether results are consistent with chance variation or whether they suggest something more than random fluctuation.
Key Question:
Is the difference between the observed counts and the expected counts too large to be explained by chance alone?
Ideas suggested by variation:
- Are the differences between observed and expected counts due to random sampling variability?
- Do the results provide evidence that the actual distribution of the categorical variable is different from what was assumed?
- Is there an association or effect present that makes the observed data inconsistent with the expected model?
Important Note:
Variation between what we find (observed counts) and what we expect to find (expected counts) may be due to random chance, but if the variation is unusually large, this may suggest that the model or assumption used to generate the expected counts is not correct.
Example
A teacher expects that in her class of 30 students, 10 will prefer math, 10 will prefer science, and 10 will prefer history. After a survey, the observed counts are:
- Math: 8
- Science: 12
- History: 10
Does the variation between the observed and expected counts suggest anything unexpected?
▶️ Answer / Explanation
The expected counts were all 10. The observed counts show small differences (Math: -2, Science: +2, History: 0). These variations could plausibly be due to random chance. The differences are minor and do not necessarily indicate a pattern or problem with the initial expectation.
Conclusion: The variation between observed and expected counts may be random; nothing in the data strongly suggests an unexpected result.
Example
A survey expects that 25% of respondents prefer apples, 25% prefer bananas, 25% prefer oranges, and 25% prefer grapes. In a survey of 80 people, the observed counts are:
- Apples: 18
- Bananas: 22
- Oranges: 20
- Grapes: 20
Does the observed variation from the expected counts suggest anything unusual?
▶️ Answer / Explanation
The expected counts for each fruit are \(80 \times 0.25 = 20\). The observed counts differ by -2, +2, 0, 0. These small differences are likely due to random variation in the sample. There is no strong evidence from these numbers alone to suggest that the population preference differs from the expected 25% for each fruit.
Conclusion: The variation observed is minor and can reasonably occur by chance; the results are not unexpectedly different from the expectations.