Lecture 8 Notes – Statistical Inference for Two Parameters

Lecture 1: Two-Sample t-Test

• Scope & Goals

• Compare the means $\mu<em>1,\,\mu</em>2$ of two independent populations.

• By the end you should be able to

– derive / recall the test statistic for a two-sample t-test;

– calculate $100(1-\alpha)\%$ CIs for $\mu<em>1-\mu</em>2$.

• Position in the “road-map” of inference

┌── One variable ──┐ ┌──────────── Two variables ───────────┐

│categorical→$CI\ (p)$ │ │both categorical→$\chi^2$│

│quantitative→$CI\ (\mu)$│ │one cat.+one quant.→ two-sample or paired t │

│ │ │both quantitative→simple regression $t$ │

• Prototype examples

– “Pregnancy & Smoking” (nicotine vs control; guinea-pig maze errors).

– “Do ravens fly to gunshots?” (later turns out to require paired test).

• Terminology & Hypotheses

Population 1: RV $X<em>1\sim(\mu</em>1,\sigma<em>1)$; Population 2: $X</em>2\sim(\mu<em>2,\sigma</em>2)$.

Null $H<em>0: \mu</em>1=\mu<em>2\;(!\Leftrightarrow!\mu</em>1-\mu_2=0)$. One- or two-sided alternatives chosen a priori:

Ha:\ \mu1<\mu2,\;\mu1>\mu2,\;\text{or}\;\mu1\neq\mu_2.

• Assumptions ("pooled" version)

1. Two simple random samples; observations inside and across samples independent.

2. Equal population SDs: $\sigma<em>1=\sigma</em>2=\sigma$ (checked via similarity of $s<em>1,s</em>2$).

3. Either both samples are large (CLT) or both parent distributions \( \approx \) Normal.

• Pooled variance estimator & standard error

$S<em>p^2=\frac{(n</em>1-1)S<em>1^2+(n</em>2-1)S<em>2^2}{n</em>1+n<em>2-2}$ $SE(\bar X</em>1-\bar X<em>2)=S</em>p\sqrt{\frac1{n<em>1}+\frac1{n</em>2}}$

• Test statistic

$T=\frac{\bar X<em>1-\bar X</em>2}{S<em>p\sqrt{\dfrac1{n</em>1}+\dfrac1{n<em>2}}}\;\stackrel{H</em>0}{\sim}t<em>{\,(n</em>1+n_2-2)}$

• P-value rules

• Right-tail (Ha: \mu1>\mu2): $P(T\ge t</em>{obs})$

• Left-tail (Ha: \mu1<\mu2): $P(T\le t</em>{obs})$

• Two-tail ($H<em>a: \mu</em>1\neq\mu<em>2$): $2P(T\le -|t</em>{obs}|).$

Robust to mild non-normality as long as $\bar X<em>1-\bar X</em>2$ is \($\approx\) Normal.

• Confidence interval (level C)

CIC(\mu1-\mu2)=(\bar X1-\bar X2)\;\pm\;t^{\,df}\,Sp\sqrt{\dfrac1{n1}+\dfrac1{n2}}$with$df=n1+n2-2$and$t^*$the two-sided *t*-quantile covering area C.</p><p>• <strong>Worked example – Guinea-pig nicotine study</strong> <br></p><p>Data summary: <br></p><p>│Group│$n$│$\bar x$│$s$│ → Control: 10, 23.4, 12.3; Treatment: 10, 44.3, 21.5. <br></p><p>– Pooled$Sp=17.515$(R). –$t{obs}=\dfrac{23.4-44.3}{17.515\sqrt{0.1+0.1}}=-2.668

– One-sided P-value \( \approx \) 0.0078 \( \Rightarrow \) strong evidence nicotine increases errors.

– 95 % CI: (-20.4,-4.4) \($ \to \) errors higher by 4–37 on average.

• Unequal variances?

– If \sigma1\neq\sigma2$but$n1=n2$, pooled test is still fine. <br></p><p>– Large imbalance in both$n$and$\sigma \($ \to \) transform data or use Welch (not in MATH1041 curriculum).

Lecture 2: Paired t-Test

• Motivation – samples are dependent (matching, before/after, twins…). Independence assumption of two-sample test fails.

• Examples

– Ravens before/after gunshot (12 pairs).

– Speed-camera counts before vs after installation (4 pairs).

• Strategy – reduce to a single sample of differences Di=X{1i}-X{2i}$. Population parameter of interest:$\muD=E(D)$.</p><p>• <strong>Assumptions</strong></p><ol><li><p>$n$paired differences are independent.</p></li><li><p>Distribution of$D \($ \approx \) Normal (or n large for CLT).

• Test statistic & CI

T{paired}=\frac{\bar D}{SD/\sqrt n}\;\stackrel{H0}{\sim}t{\,n-1}$with$H0:\muD=0$. <br></p><p>$CIC(\muD)=\bar D\;\pm\;t^{*\,\,n-1}\,\dfrac{S_D}{\sqrt n}.

• R implementation

t.test(x_after, x_before, paired=TRUE, alternative="greater")

• Raven example recap

Differences: mean \bar D=1.083$, SD$SD=1.884$,$n=12$.$t{obs}=1.995$,$p=0.036 \($ \to \) moderate evidence more ravens post-shot.

One-sided 95 % CI: (0.11,\infty)$($\ge$0.11 raven increase).</p><p>• <strong>Speed-camera example</strong> <br></p><p>Differences huge (all negative),$t_{obs}=4.95$,$p=0.008$, reduction$\ge$2222 cars/month (one-sided 95$\alpha$, Type II error$\beta$, Power$1-\beta$. <br></p><p>– Blood-pressure example:$\sigma=12$,$n=100/100 \($ \Rightarrow \) SE=1.7$. <br></p><p>– Rejection region for two-sided$\alpha=0.05: \(|\text{diff}|>3.33\).

– If true diff = -3 \( \to \) power \( \approx \) 42 % .

– Formula manipulation shows n\approx251$per group needed for 80$n$.</p><p>– <em>Conditional</em> distribution: distribution of one variable at a fixed level of the other, e.g.$P(\text{North}|\text{Moderately})=28/84=0.33$.</p><p>• <strong>Graphical displays</strong> <br></p><p>– Side-by-side bar charts (split by levels). <br></p><p>– Clustered bar charts (grouped). <br></p><p>– Bar chart of conditional proportions (stacked with 100$ \times \) Stress Level (4 levels).

• Interpretation – Compare conditional distributions; if patterns differ, variables likely associated (quantified formally in Lecture 4).

Lecture 4: Chi-Square Test of Independence

• Purpose – Test H_0$of <em>no association</em> (independence) between two categorical variables in an$r\times c$table.</p><p>• <strong>Assumptions</strong></p><ol><li><p>Observations independent (usually SRS or randomised design).</p></li><li><p>Expected count in every cell$\ge10$(rule of thumb). Combine levels if violated.</p></li></ol><p>• <strong>Expected counts formula</strong> <br></p><p>$E_{ij}=\frac{(\text{row }i\text{ total})(\text{column }j\text{ total})}{n}$</p><p>• <strong>Test statistic</strong> <br></p><p>$X^2=\sum{i=1}^r\sum{j=1}^c\frac{(O{ij}-E{ij})^2}{E{ij}}$Under$H0: X^2\sim\chi^2_{df},\;df=(r-1)(c-1).$</p><p>• <strong>P-value</strong> – Right-tail probability$P(\chi^2{df}\ge X^2{obs})$.</p><p>• <strong>Chi-square distribution facts</strong> <br></p><p>– Domain$(0,\infty)$, right-skew. <br></p><p>– Mean$=df$, variance$=2df$.</p><p>– As$df\uparrow the curve becomes more symmetric.

• Worked example – Hemisphere \( \times \) Stress

– 4 \( \times \) 2 table, df=3$. <br></p><p>–$X^2_{obs}=6.59$,$p=0.086 \($ \Rightarrow \) weak evidence of association.

– All expected counts \ge10 \( \to \) assumptions OK.

• Cancer Type \( \times \) Site example

– 3\( \times \)3 table, one expected count <10 \\($\to \\) caution noted. <br></p><p>–$X^2_{obs}=6.56$,$df=4$,$p=0.161 \( \to \) no evidence of association.

• Airline Lateness example

– Original 3\( \times \)2 table had small counts; merged to 2\($ \times \)2.

– X^2_{obs}=13.17$,$df=1$,$p<0.001 \($ \to \) very strong association.

• Software summary (R)

– Build table: tab<-table(var1,var2) or as.table(cbind(...)).

– chisq.test(tab) returns statistic, df, p-value, expected counts, residuals.

– addmargins(tab) appends row/column totals.

• Quick reference: R functions vs data type

│One categorical│prop.test() │

│One quantitative│t.test() (or z.test if \sigma known) │

│Cat + Quant│t.test(independent or paired) │

│Cat + Cat│chisq.test() │

Bonus: Non-Parametric Sign Test

• When? – Very small paired sample, distribution of differences skewed or with outliers \($ \to \) Normal assumption dubious.

• Procedure

– Define X =$number of positive differences among$n$pairs. <br></p><p>– Under “no effect” median 0,$X\sim Bin(n,0.5)$.</p><p>– Compute$P-value from tail(s) of Binomial.

• Speed-camera revisited (n = 4)

– All four sites show reduction \($ \to \) x_{obs}=4$.</p><p>–$P=0.0625 (best possible with n=4).

– Conclusion: only weak evidence; paired t test was more powerful.

• Trade-off – Non-parametric tests safer but usually less powerful and target medians, not means.

Chapter Wrap-Up & Keywords

• You can now:

– Perform and interpret two-sample & paired t tests.

– Build and read two-way tables, compute marginal / conditional distributions.

– Conduct \chi^2 tests for independence and check their assumptions.

– Understand basics of statistical power & sample-size planning.

– Recognise situations needing non-parametric alternatives.

• Core formulas:

– Two-sample T statistic & CI (pooled).

– Paired T statistic & CI.

– Expected counts E_{ij}$.</p><p>–$\chi^2$statistic and degrees-of-freedom.</p><p>• Keywords: two-sample test, pooled variance, paired, two-way table, joint / marginal / conditional distribution, observed vs expected counts,$\chi^2$$ distribution, sign test, statistical power.