AP Statistics 3.1 Estimators- Exam Style Questions - MCQs - New Syllabus
▶️ Answer/Explanation
The 95% confidence interval for the 2022 proportion of single-person households is:
\((0.2813,\;0.3387)\)
The 2021 Census value was:
\(p=0.29\)
Notice that:
\(0.2813 < 0.29 < 0.3387\)
Since the 2021 value of 0.29 lies within the 95% confidence interval, it is a plausible value for the 2022 population proportion. Therefore, the data do not provide sufficient evidence that the proportion of single-person households changed between 2021 and 2022.
A confidence interval that contains the hypothesized value corresponds to failing to reject the null hypothesis in a two-sided significance test at the matching significance level.
Therefore, there is not sufficient evidence at the 0.05 significance level to conclude that a significant difference exists.
✅ Answer: (C)
▶️ Answer/Explanation
A good estimator should have two important properties:
• Unbiased — its sampling distribution should be centered at the true parameter value.
• Low variability — its sampling distribution should be as narrow as possible.
The true parameter value is:
\( \theta = 0 \)
Looking at the histograms:
• Estimator A is centered near 0 but has very large variability.
• Estimator C is not centered at 0 and has substantial variability.
• Estimator D has small variability but is centered around 2, showing bias.
• Estimator E has very small variability but is centered around \(-2\), also biased.
• Estimator B is centered near 0 and has the smallest variability among the unbiased estimators.
Therefore, Estimator B is the best estimator because it is approximately unbiased and has relatively low spread.
✅ Answer: (B)
▶️ Answer/Explanation
The original confidence interval is centered at the sample proportion:
\(\hat{p}=\frac{0.14+0.20}{2}=0.17\)
The corrected sample proportion is \(\hat{p}=0.27\). The standard error for a proportion is based on:
\(\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\)
Since \(0.27\) is closer to \(0.50\) than \(0.17\), the value of \(\hat{p}(1-\hat{p})\) is larger. A larger standard error produces a larger margin of error, so the revised confidence interval will be wider than the original interval.
✅ Answer: (D)