Lecture 7 – Inference for a Population Proportion (Complete Study Notes)

l    Lecture 1 – Sample Proportions

• Overarching course aim: introduce Statistics as the science of collecting, analysing & interpreting data.

• Last lecture recap: Central Limit Theorem (CLT), assumption checking, inference for a mean.

• This week’s roadmap

• Sample proportions

• Sampling distribution of counts & proportions

• Confidence intervals for a proportion

• Hypothesis tests for a proportion

Why move from means to proportions?

• Real-world variables are often binary (yes/no, success/failure) rather than quantitative & normal.

• CLT still provides an approximate normal distribution for the sample mean of Bernoulli trials when $n$ large.

• Statistical tools always rely on assumptions → must diagnose validity (e.g., normal quantile plot, sample size).

Motivating example – “Mike Promesses” poll

• Needs $p=0.5$ of votes to win.

• Poll: $n=200$ respondents, $\hat p_{\text{obs}}=0.54$.

• Desired outputs:

1. 95 % Confidence Interval (CI) for true population proportion $p$.

2. Hypothesis test $H<em>0:p\le 0.5$ vs Ha:p>0.5.

• R quick look: prop.test(x = 108, n = 200, p = 0.5, alternative="greater") → $\chi^2=1.125,\; p=0.1444$, CI $[0.479,1]$.

Binomial model (revision)

• Assumptions (7.1):

1. Fixed number of trials $n$.

2. Two outcomes (success/failure).

3. Constant success probability $p$.

4. Independent trials.

• PMF: $P(X=x)=\binom{n}{x}p^{x}(1-p)^{n-x},\;x=0,\dots,n.$

• Moments: $\mu<em>X=np,\;\sigma^2</em>X=np(1-p),\;\sigma_X=\sqrt{np(1-p)}.$

Recognising a binomial

• Whenever we “count how many ### in a random sample of fixed size”, the resulting RV is binomial.

• Examples: number of Australians favouring emission cuts, number of female students in SRS of size 4, etc.

Worked exercise – Australian Open Women final

• Assume $p=0.2$. Sample $n=4$ → $X\sim B(4,0.2)$.

• $P(X=0)=0.4096$ (verified by dbinom).

• $P(X=1)=0.4096$, full distribution table presented.

• Moments: $\mu<em>X=0.8,\;\sigma^2</em>X=0.64,\;\sigma_X=0.8$.

Definition 7.2 – Sample proportion

$\hat p=\dfrac{X}{n}\quad(\text{a statistic}).$

Lecture 2 – Sampling Distribution of Counts & Proportions

Learning outcomes

• [Ch7-L2-O1] Know the sampling distribution of $\hat p$.

• [Ch7-L2-O2] Understand normal approximation.

Exact distribution of $\hat p$

• If $X\sim B(n,p)$ then $\hat p=X/n$ takes values ${0/n,1/n,\dots,n/n}$ with same probabilities $P(\hat p=x/n)=P(X=x).$

• Moments:

$\mu<em>{\hat p}=p,$ $\sigma^2</em>{\hat p}=\dfrac{p(1-p)}{n},$ $\sigma<em>{\hat p}=\sqrt{\dfrac{p(1-p)}{n}}.$ • Standard error (unknown $p$): $SE</em>{\hat p}=\sqrt{\dfrac{\hat p(1-\hat p)}{n}}.$

Example – Lowy emission-reduction poll

• $n=1200,\; \hat p_{\text{obs}}=0.63.$

• Moments for $X$: $\mu=1200p,\; \sigma=\sqrt{1200p(1-p)}.$

• SE: $\text{se}=\sqrt{\dfrac{0.63(1-0.63)}{1200}}\approx0.01394.$

Normal approximation (Proposition 7.3)

• When $n$ large with $np\ge 10$ and $n(1-p)\ge10$, then

• $X\approx N\bigl(np,\,np(1-p)\bigr)$

• $\hat p\approx N\Bigl(p,\,\dfrac{p(1-p)}{n}\Bigr).$

• Continuity correction improves accuracy for discrete $X$:

P(X<k)=P\bigl(Y<k-0.5\bigr)\text{ for }Y\sim N(np,np(1-p)).

Example – 50 underground tanks

• Assume $p=0.25,\;n=50$.

• $\mu<em>X=12.5,\;\sigma</em>X=3.062.$

• P(X<10)=P\bigl(Z<\tfrac{10-12.5}{3.062}\bigr)\approx0.2072.

• Exact binomial: 0.1637; with continuity correction 0.164 – almost identical.

CLT connection

• Bernoulli variables $B<em>i\in{0,1}$ with $E(B</em>i)=p,Var(B_i)=p(1-p).$

• Sample mean $\bar B=\hat p$ → CLT ⇒ normal approximation.

Rule-of-thumb (Assumptions 7.2)

• Use normal approximation if both $np$ and $n(1-p)$ ≥ 10 (or with $\hat p_{\text{obs}}$ when $p$ unknown).

Data-science interlude – tidyverse

• Introduced pipe %>%, verbs: filter, mutate, arrange, select, relocate, summarise, group_by, join variants.

• Illustrated using palmerpenguins & tiny datasets (band_members, band_instruments).

Lecture 3 – Confidence Intervals for a Proportion

Learning outcomes

• [Ch7-L3-O1] Know CI formula (Wald).

• [Ch7-L3-O2] Check validity assumptions.

• [Ch7-L3-O3] Recognise common CI pattern.

Wald CI (Definition 7.4)

Given random SRS of size $n$ with $\hat p<em>{\text{obs}}$ and $C$ confidence level, $CI</em>C(p)=\Bigl[\hat p<em>{\text{obs}}\;\pm\;z^*\sqrt{\dfrac{\hat p</em>{\text{obs}}(1-\hat p_{\text{obs}})}{n}}\Bigr],$

where $z^<em>$ is the $C$ quantile of $N(0,1)$ (e.g., $z^</em>=1.96$ for 95 %).

Assumptions

1. Independent observations (SRS or sampling fraction ≤ $1/10$ of population).

2. Approximate normality: $n\hat p<em>{\text{obs}}\ge10,\;n(1-\hat p</em>{\text{obs}})\ge10.$

Political poll worked example

• $n=2951, \hat p=0.51$. 95 % CI: $[0.492,0.528].$

• Interpretation: “We are 95 % confident true Labor support lies between 49.2 % and 52.8 %.”

Additional examples

• Birth-in-spring (108/400) → check $np,n(1-p)$ before CI.

• Australian Open viewing (49/400) → CI \approx $[9\%,15.5\%].$

Pattern recognition

• All CIs so far: $\text{point estimate} \;\pm\; (\text{quantile})\times(\text{SE}).$

• Mean with known $\sigma$: $\bar X \pm z^<em>\dfrac{\sigma}{\sqrt n}$. • Mean with unknown $\sigma$: $\bar X \pm t^</em>\dfrac{S}{\sqrt n}$.

• Proportion: formula above (Wald).

Cautions

• Wald CI can be poor for small $n$; alternatives: Wilson, Jeffreys, Agresti-Coull (implemented in binomCI).

• CI margin covers random sampling error only – non-response & bias not captured.

• Avoid over-precision; round sensibly (e.g., 54 % \pm 7 %).

Lecture 4 – Hypothesis Tests for a Proportion

Learning outcomes

• [Ch7-L4-O1] Perform Z-test for proportion.

• [Ch7-L4-O2] Check assumptions.

• [Ch7-L4-O3] Link tests \leftrightarrow confidence intervals.

• [Ch7-L4-O4] Recognise Wald test statistic pattern.

Z-test statistic (Definition 7.5)

To test $H<em>0:p=p</em>0$ vs alternative $H<em>a$, use $Z=\dfrac{\hat p - p</em>0}{\sqrt{\dfrac{p<em>0(1-p</em>0)}{n}}}.$

If assumptions hold ⇒ $Z\sim N(0,1)$ approximately.

Why SE uses $p_0$ not $\hat p$

• Under $H<em>0$ we pretend $p=p</em>0$ (worst-case). Therefore variance uses $p_0$.

Assumptions

1. Independence (SRS & population ≥10×sample).

2. Approx normality with null value: $np<em>0\ge10,\;n(1-p</em>0)\ge10.$

Worked examples

1. Births in spring (108/400): - $p<em>0=0.25$, one-sided Ha:p>0.25.

   • $z_{\text{obs}}=0.923,\; p=0.178$ → fail to reject; no evidence.

2. Voting intention change 2013 vs 2016: - $n=2951, p_0=0.465, \hat p=0.51$, two-sided.

   • $z_{\text{obs}}=4.901, p\approx9.5\times10^{-7}$ → very strong evidence of change.

3. TV ratings (49/400) vs $p<em>0=0.0688$: - $z</em>{\text{obs}}=4.243, p=2.2\times10^{-5}$.

   • 95 % CI $[0.090,0.155]$ excludes 0.0688 ⇒ same conclusion – MATH1041 students differ.

Relationship test \leftrightarrow CI (Proposition 7.4)

• For two-sided tests at level $\alpha$, rejecting $H_0$ ⇔ null value outside $100(1-\alpha)\%$ CI.

• One-sided tests correspond to one-sided CIs.

Wald statistic pattern

$\text{Test statistic}=\dfrac{\text{estimator}-\text{hypothesised value}}{SE}.$

Applies to Z-tests (mean or proportion) & t-tests (when $\sigma$ unknown).

Continuity Correction Summary

• Used when approximating discrete binomial by continuous normal.

• Rule: replace integer cutoff $k$ with $k\pm0.5$ depending on inequality.

• Greatly improves tail probability accuracy (demonstrated with tank example).

Keywords & Take-away Concepts

• Proportion, sample proportion $\hat p$.

• Binomial distribution & parameters $(n,p)$.

• Sampling distribution & CLT.

• Normal approximation criteria $np, n(1-p)\ge10$.

• Standard error $SE_{\hat p}$.

• Wald confidence interval.

• Z-test for a single proportion.

• Continuity correction.

• Tidyverse data-wrangling verbs (filter, mutate, …).

Practical R Commands Cheat-Sheet

• Compute exact binomial prob.: dbinom(x,n,p); cumulative pbinom(k,n,p).

• Normal approx: pnorm(value, mean, sd) or qnorm(prob) for quantiles.

• prop.test(x,n,p0, alternative="two.sided", conf.level=0.95, correct=TRUE) – gives CI & test (with continua­­ty-corr $\chi^2$).

• binom.test(x,n,p=p0) – exact binomial test & exact CI.

• Standard CI by hand:

se <- sqrt(p_obs*(1-p_obs)/n)
  ci <- p_obs + c(-1,1)*z_star*se

Common Pitfalls & Best Practices

• Do NOT apply normal approximation when $np$ or $n(1-p)$ < 10. Use exact binomial methods instead.

• Interpret CIs in context; avoid overstating precision.

• Report assumptions explicitly (independence, sampling design, sample size adequacy).

• Remember CI ≠ probability interval for parameter; 95 % refers to long-run coverage.

• Wald CI unreliable for extreme $\hat p$ or small $n$ → prefer Wilson/Agresti-Coull.

Connections to Further Topics

• Two-sample comparisons of proportions will extend these ideas (not covered here).

• $\chi^2$ tests of independence generalise to multi-category tables.

• Generalised linear models (logistic regression) model proportions with covariates.

Self-Reflection Prompts (Exercise 7.2)

• Which part of the inference workflow (formulating $H<em>0$/$H</em>a$, checking assumptions, computing $SE$) still feels unclear?

• Identify a binary variable in your discipline and outline how you would design a study, compute $\hat p$, draw a CI, and test a claim.