AP Statistics Workbook | EBM · FTR · SE

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This workbook covers four critical AP Statistics topics. Each question is written in CollegeBoard free-response style. Type your answer, then tap Submit to see instant feedback and full explanations.

Questions

Minutes

Units

Unit 1 EBM — Error Bound / Margin of Error

The margin of error (EBM) quantifies the precision of a confidence interval estimate. It is the product of the critical value and the standard error of the statistic. A confidence interval takes the form: Point Estimate ± EBM.

★ Must Memorize

\( EBM = z^* \cdot \dfrac{\sigma}{\sqrt{n}} \) for means (σ known)
\( EBM = t^* \cdot \dfrac{s}{\sqrt{n}} \) for means (σ unknown, use \(t\))
\( EBM = z^* \cdot \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}} \) for proportions
Larger \(n\) → smaller EBM → more precise
Higher confidence level → larger \(z^*\) → larger EBM
To halve EBM, multiply \(n\) by 4

▶ Worked Example

A sample of 64 students has \(\bar{x} = 78\), \(s = 12\). Construct a 95% CI for the population mean and state the EBM.

EBM = \(t^* \cdot \frac{s}{\sqrt{n}} = 1.998 \cdot \frac{12}{\sqrt{64}} \approx 1.998 \cdot 1.5 \approx 3.0\). CI: (75.0, 81.0)

Unit 2 FTR — Fail to Reject H₀

When the p-value exceeds the significance level \(\alpha\), we fail to reject H₀. This does NOT mean H₀ is true; it means we lack sufficient statistical evidence to support H‡. The language of the conclusion must be in context.

★ Must Memorize

FTR → \(p\text{-value} > \alpha\)
Never say “accept H₀” — always “fail to reject H₀”
FTR does NOT prove H₀ true
Type II error: FTR when H‡ is actually true
Power = P(reject H₀ | H‡ true) = 1 − β
Increasing \(n\) reduces the chance of a Type II error

▶ Worked Example

A test yields \(p = 0.12\) with \(\alpha = 0.05\). State the conclusion in context (testing whether a new drug lowers blood pressure).

Since 0.12 > 0.05, we fail to reject H₀. At the 5% significance level, there is not sufficient evidence to conclude that the new drug lowers blood pressure.

Unit 3 SE — Standard Error

The standard error (SE) estimates the standard deviation of a sampling distribution. It measures how much a sample statistic typically varies from sample to sample.

★ Must Memorize

\(SE_{\bar{x}} = \dfrac{s}{\sqrt{n}}\) (single sample mean)
\(SE_{\hat{p}} = \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}\) (single proportion)
\(SE_{\bar{x}_1 - \bar{x}_2} = \sqrt{\dfrac{s_1^2}{n_1} + \dfrac{s_2^2}{n_2}}\) (two means)
\(SE_{\hat{p}_1-\hat{p}_2} = \sqrt{\hat{p}_c(1-\hat{p}_c)\left(\frac{1}{n_1}+\frac{1}{n_2}\right)}\) (pooled, for hypothesis test)
SE decreases as \(n\) increases (proportional to \(1/\sqrt{n}\))
SE is used in both CI construction and hypothesis tests

▶ Worked Example

A poll of \(n = 400\) finds \(\hat{p} = 0.60\). Calculate the standard error of \(\hat{p}\).

\(SE = \sqrt{\frac{0.60 \times 0.40}{400}} = \sqrt{0.0006} \approx 0.0245\)

Unit 4 S — Significance & P-values

The p-value is the probability of observing data as extreme as (or more extreme than) the sample data, assuming H₀ is true. Statistical significance is declared when \(p \leq \alpha\).

★ Must Memorize

P-value is NOT the probability that H₀ is true
Reject H₀ when \(p \leq \alpha\); FTR when \(p > \alpha\)
Common \(\alpha\) levels: 0.10, 0.05, 0.01
Statistical significance ≠ practical significance
Type I error (\(\alpha\)): reject H₀ when it is actually true
Type II error (\(\beta\)): FTR H₀ when H‡ is actually true
Decreasing \(\alpha\) reduces Type I error but increases Type II error

▶ Worked Example

A study gives \(z = 2.41\) (two-tailed). The p-value is 0.016. Interpret this at \(\alpha = 0.05\).

Since 0.016 < 0.05, we reject H₀. There is statistically significant evidence to support H‡ at the 5% significance level.

───── Practice Problems Below ─────

Final Results

0/20

EBM

0/5

FTR

0/5

Significance

0/5

Answer Key & Explanations