AP Statistics Unit 7 Comprehensive Notes

Unit 7.1 – Introducing Statistics & Why Error Matters

• Central questions of statistical inference:
– How do we find numerical evidence for/against a claim?
– When is the evidence convincing as opposed to random error?
• Two competing explanations for any observed effect
– Chance variation created the difference (sampling/assignment variability)
– A genuine treatment/wording/condition effect exists
• Randomization & sampling are the tools that let us quantify how large chance-variation typically is, which is what p-values and confidence intervals measure.

Classroom Illustration – Bonus vs Rebate Experiment

• 47 college volunteers each received \$50.
– 25 were told the money was “bonus income.”
– 22 were told the money was a “tuition rebate.”
• After one week: mean amount spent
– Bonus group ≈ \$\bar x{BONUS} – Rebate group ≈ \$\bar x{REBATE}
• Observed statistic: $\bar x<em>{BONUS}-\bar x</em>{REBATE}=12.49$
• Two explanations

1. No wording effect; the \$12.49 arose from random assignment noise.
2. Bonus wording really encourages higher spending.
   • Random-assignment (permutation) simulation
   – Shuffle the 47 amounts into two groups of 25 and 22 many times.
   – Record each shuffled $\bar x<em>{BONUS}-\bar x</em>{REBATE}$.
   – Approximate $P(\bar x<em>{BONUS}-\bar x</em>{REBATE}\ge12.49)$ under H₀: no effect.
   • If that probability (p-value) is very small → rule out explanation 1 and accept explanation 2.

Key Takeaways for Evidence

• To claim an effect you must BOTH
– Show that the direction & size of the sample statistic are consistent with the claim.
– Show that such an extreme statistic is unlikely to appear by chance when H₀ is true.
• Eliminate random chance → remaining explanation is a real effect.

Unit 7.2 – Confidence Intervals for a Population Mean

Why not use z?

• Formula from proportions unit: $CI=\text{statistic}\pm z^{*}(\text{SD of statistic})$
• For means the ideal would be $\bar x\pm z^{*}\frac\sigma{\sqrt n}$ but (\sigma) is almost never known in practice.
• Substituting sample SD, $\bar x\pm z^{*}\frac s{\sqrt n}$ under-covers (captures the true mean < advertised level).
• Solution: replace the z-critical value with a t-critical value that is wider to compensate.

One-Sample t Interval

• Statistic = $\bar x$
• Standard error (SE): $SE<em>{\bar x}=\frac s{\sqrt n}$ • Critical value: $t^{*}$ from a t distribution with $df=n-1$ (found via technology or Table B). • Interval: $\bar x\pm t^{*}\,SE</em>{\bar x}$

Checking Conditions (1-sample mean)

1. Random sample or randomized experiment.
2. 10 % condition (if sampling without replacement): $n\le0.10N$.
3. Normality of sampling distribution
   – Large sample ((n>30)) → CLT
   – If (n<30), the sample graph must show no strong skew/outliers.

Example – Human Body Temperature simulation

• Even with population normality, replacing (\sigma) by s required t values to get correct 95 % capture rate.

Margin of Error & Critical Values

• General form: $\text{ME}=(\text{critical value})(\text{standard error})$
• Larger n → smaller SE → smaller ME.
• Lower confidence → smaller critical value → narrower CI.
• For means: $ME=t^{*}\dfrac s{\sqrt n}$

Interpreting Confidence Intervals & Levels (Unit 7.3)

Interpretation template

• Interval: “We are C % confident that the interval from ___ to ___ captures the [population parameter in context].”

Confidence level meaning

• “In repeated random sampling of size n, about C % of all constructed C % intervals will capture the true mean.”

Justifying a claim

• If the claimed value lies inside the CI → plausible (not enough evidence against).
• If claimed value lies outside CI → not plausible → evidence for alternative claim.

Example – Powdered Sugar Bags

• 95 % CI: (900.92 g, 912.68 g).
• Company claim 907 g lies inside → no evidence of under-filling.

Hypothesis Tests for a Population Mean (Units 7.4 & 7.5)

Hypotheses

• Null: $H<em>0:\ \mu=\mu</em>0$ (statement of no effect/equality)
• Alternative (research) can be
– \mu>\mu0 (right-tailed, one-sided) – $\mu<\mu</em>0$ (left-tailed)
– $\mu\ne\mu_0$ (two-sided)
• Direction must be chosen before data collection.

Test Statistic (1-sample)

$t=\frac{\bar x-\mu_0}{s/\sqrt n} \quad (df=n-1)$

P-value

• Probability, under H₀, of observing a statistic as extreme (or more) than the sample.
• Compute from t distribution with appropriate tails, matching Ha.

Decision rule

• If $p\le\alpha$, reject H₀ → evidence supports Ha.
• If p>\alpha, fail to reject H₀ → insufficient evidence.

Example – Vertical Jump (“Got Hops?”)

• (n=20,\ \bar x=15.8\,\text{in},\ s=2.33\,\text{in}). Test H₀: (\mu=15).
• $t=\dfrac{0.8}{2.33/\sqrt{20}}=1.53,$ df = 19.
• Two-sided p ≈ 0.142 > 0.05 → fail to reject. No convincing evidence that school mean differs from 15 in.

Matched-Pairs (Paired) Designs

• Same individuals receive both treatments or naturally paired (before/after, left/right, husband/wife).
• Work with differences $d<em>i$; treat them as a single sample. • CI or test uses one-sample t on $\bar d, s</em>d, n<em>d$. • Example – Bacon seasoning weight retention: H0:\ \mud=0,\ Ha:\ \mu_d>0.

Two-Sample Procedures (Units 7.6 & 7.7)

Conditions

1. Two independent random samples or randomized comparative experiment.
2. 10 % condition for each sample if sampling without replacement.
3. Normality for each group (large sample or graphical check).

Two-Sample t Interval (difference of means)

• Statistic: $\bar x<em>1-\bar x</em>2$ (order matters!).
• Standard error: $SE=\sqrt{\dfrac{s<em>1^2}{n</em>1}+\dfrac{s<em>2^2}{n</em>2}}$
• CI: $(\bar x<em>1-\bar x</em>2)\pm t^{*}\,SE$ where $t^{*}$ uses df found by technology (Welch-Satterthwaite).

Margin of Error formula

$ME=t^{*}\sqrt{\dfrac{s<em>1^2}{n</em>1}+\dfrac{s<em>2^2}{n</em>2}}$

Interpretation example – Argiope spiders

• 95 % CI for (female – male) length: (7.956 mm, 12.129 mm).
• Because 0 is not in the interval and entire interval positive → females are larger, on average.

Mis-ordering caveat

• Swapping subtraction order negates interval and flips interpretation.

Two-Sample t Test (difference of means)

Test Statistic

$t=\frac{(\bar x<em>1-\bar x</em>2)-0}{\sqrt{\dfrac{s<em>1^2}{n</em>1}+\dfrac{s<em>2^2}{n</em>2}}}$ with df from technology.

p-Value tails

• Ha: \mu1-\mu2>0 → $P(t\ge t<em>{obs})$ • Ha: $\mu</em>1-\mu<em>2<0$ → $P(t\le t</em>{obs})$
• Ha: $\mu<em>1\ne\mu</em>2$ → $2\times P(t\ge |t_{obs}|)$

Example – Contagious Yawning

• (\bar x{yawn}=3.64,\ s1=1.28, n1=14;\ \bar x{no}=1.46, s2=1.05, n2=13)
• $t=4.51,\ df\approx24.6$ → p < 0.001.
• Since p < 0.05, reject H₀ → significant evidence yawning is contagious.

Full write-up checklist (performing a significance test)

1. State parameters & both hypotheses (include inequality direction).
2. Plan/Identify method (2-sample t test) & check all conditions.
3. Do calculations: statistic, df, p-value (show or quote tech output).
4. Conclude: compare p to α, state decision, translate to context.

Complete Worked Example – Cholesterol Drug (2007B Q5)

• Group 1 (standard): (n1=50,\ \bar x1=10,\ s1=8) • Group 2 (new drug): (n2=50,\ \bar x2=18,\ s2=12)
• H₀: $\mu<em>{new}-\mu</em>{std}=0$; Ha: \mu{new}-\mu{std}>0
• $t=3.66, df\approx86$; p ≈ 0.0002.
• Because p < 0.05 → reject H₀. New drug more effective at lowering cholesterol.

Unit 7.10 – Selecting Inference Procedures

Recognising Paired vs Independent Data

• Paired: each observational unit contributes two measurements (before/after, left/right, matching). Data table shows rows with two numbers. Inference about single mean of differences (matched-pairs t).
• Independent samples: two separate groups, each unit supplies one measurement. Inference about difference of means (two-sample t).

General classification flow

1. Variable type
   – Categorical → proportion procedures ((z)).
   – Quantitative → mean procedures ((t)).
2. Number of groups
   – One → 1-sample CI/test.
   – Two independent → 2-sample CI/test.
   – Paired → matched-pairs (1-sample on differences).
3. Purpose
   – Estimate parameter → Confidence interval.
   – Decide about claim → Significance test.
4. Tail direction (when testing): left, right, or two-sided from the research question.

Sample “Name-that-procedure” answers

• TV vs reading minutes, same people → paired quantitative ⇒ matched-pairs t CI or test.
• Sierra vs Kirkwood lodging (Yes/No) → 2-prop z test.
• UCLA rural SAT only one sample quantitative → 1-sample t test.
• Factory music productivity (with/without) on each employee → paired t.
• Survival rate of new businesses → 1-prop z CI.
• Avg contacts per day (n=300) → 1-sample t CI.
• Chimps vs apes vocabulary (quantitative, independent) → 2-sample t.
• Instant coupons (categorical Bought/Not) independent groups → 2-prop z test.
• Home team win rate (categorical, one group) → 1-prop z test.
• Gene-modified plants insect counts (quantitative, ind.) → 2-sample t.

Formulas & Symbols Reference (all given on AP sheet)

• One-sample t CI: $\bar x\pm t^{*}\dfrac s{\sqrt n}$
• Two-sample t CI: $(\bar x<em>1-\bar x</em>2)\pm t^{*}\sqrt{\dfrac{s<em>1^2}{n</em>1}+\dfrac{s<em>2^2}{n</em>2}}$
• One-sample t statistic: $t=\dfrac{\bar x-\mu<em>0}{s/\sqrt n}$ • Two-sample t statistic: $t=\dfrac{(\bar x</em>1-\bar x<em>2)-0}{\sqrt{\dfrac{s</em>1^2}{n<em>1}+\dfrac{s</em>2^2}{n_2}}}$
• Margin of error general: $ME=(\text{critical value})(SE)$
• Relationship: quadrupling n halves ME because $ME\propto \dfrac1{\sqrt n}$.

Ethical & Practical Notes

• Always decide alternative hypothesis before data collection to avoid p-hacking.
• Clearly communicate assumptions; if conditions fail, results are unreliable.
• Random assignment allows causal inference; random sampling allows population generalization. Both allow t-procedures to be valid.
• Interpret results in context; avoid claiming certainty—always “convincing evidence” not proof.