AP Statistics 6.9 Justifying a Claim Based on a Confidence Interval for a Difference of Population Proportions Study Notes
AP Statistics 6.9 Justifying a Claim Based on a Confidence Interval for a Difference of Population Proportions Study Notes- New syllabus
AP Statistics 6.9 Justifying a Claim Based on a Confidence Interval for a Difference of Population Proportions Study Notes -As per latest AP Statistics Syllabus.
LEARNING OBJECTIVE
- An interval of values should be used to estimate parameters, in order to account for uncertainty.
Key Concepts:
- Interpret a Confidence Interval for a Difference of Proportions
- Justify a Claim Based on a Confidence Interval for a Difference of Proportions
Interpret a Confidence Interval for a Difference of Proportions
Interpret a Confidence Interval for a Difference of Proportions
A confidence interval for the difference in population proportions \( p_1 – p_2 \) provides a range of plausible values for the true difference between two populations. It reflects how confident we are that this interval captures the actual difference in proportions.
- The interval is based on the sample data (\( \hat{p}_1, \hat{p}_2 \)) and accounts for sampling variability.
- The form of the interval is:
\( (\hat{p}_1 – \hat{p}_2) \pm z^* \sqrt{\dfrac{\hat{p}_1 (1 – \hat{p}_1)}{n_1} + \dfrac{\hat{p}_2 (1 – \hat{p}_2)}{n_2}} \)
Interpretation:
- We are C% confident that the true difference in population proportions \( p_1 – p_2 \) lies within the calculated interval.
- In repeated random sampling with the same sample sizes, about C% of such intervals would capture the true difference.
- The interpretation must always refer to:
- the populations being compared,
- the parameters of interest (\( p_1 – p_2 \)), and
- the context of the situation.
Example:
A survey found that 60% of urban residents and 52% of rural residents support recycling incentives. A 95% confidence interval for \( p_1 – p_2 \) (urban − rural) is calculated as (0.01, 0.15).
Interpret the confidence interval.
▶️ Answer / Explanation
- We are 95% confident that the true difference in proportions of urban and rural residents who support recycling lies between 0.01 and 0.15.
- This means that the proportion of urban supporters is likely between 1% and 15% higher than that of rural supporters.
- In repeated random sampling, about 95% of such intervals would capture the true difference in support rates.
Justify a Claim Based on a Confidence Interval for a Difference of Proportions
Justify a Claim Based on a Confidence Interval for a Difference of Proportions
A confidence interval can be used to determine whether the data provide convincing evidence for a difference between two population proportions.
- If the confidence interval does not include 0:
- There is statistical evidence of a difference between the population proportions.
- The direction of the interval (positive or negative) indicates which proportion is greater.
- If the confidence interval includes 0:
- There is no significant difference between the population proportions.
- The observed difference could be due to random sampling variability.
Steps to Justify a Claim:
- State the claim being evaluated (e.g., “Group A has a higher proportion than Group B”).
- Examine whether 0 lies within the confidence interval for \( p_1 – p_2 \).
- Use the interval’s sign and range to determine whether the data support or fail to support the claim.
Example:
A researcher claims that a greater proportion of seniors than juniors prefer hybrid classes. A 95% confidence interval for \( p_1 – p_2 \) (seniors − juniors) is (0.03, 0.21).
▶️ Answer / Explanation
- The interval is entirely positive (0.03 to 0.21), meaning \( p_1 – p_2 > 0 \).
- This suggests that the proportion of seniors preferring hybrid classes is likely higher than that of juniors.
- Because 0 is not contained in the interval, the data support the researcher’s claim.
Conclusion: We are 95% confident that the proportion of seniors preferring hybrid learning is between 3% and 21% higher than that of juniors — justifying the claim.
Example:
A school claims that male and female students differ in their preference for online classes. A 95% confidence interval for \( p_1 – p_2 \) (male − female) is (−0.12, 0.08).
▶️ Answer / Explanation
- The interval includes 0, which means the true difference could be positive, negative, or zero.
- Therefore, there is insufficient evidence to conclude that the proportions differ significantly.
- The data do not support the school’s claim.
Conclusion: Because 0 lies within the interval, the difference in preference between males and females could be due to random variation.