How unit 3 is examined
This unit covers the logic of a test (hypotheses, critical region, errors, power, p-value), then the standard tests for mean, proportion, variance, paired data and one-way ANOVA; no past questions are recorded, so learn the formulas.
Null and Alternative Hypothesis
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Definition. <mark>The null hypothesis $H_0$ is the statement of no change or no difference that is tested, and the alternative $H_1$ is what we accept if the sample rejects $H_0$.</mark>
Key points.
- $H_0$ always carries the equality, for example $H_0: \mu=\mu_0$.
- $H_1$ is two-tailed ($\mu\ne\mu_0$) or one-tailed ($\mu>\mu_0$ or $\mu<\mu_0$), and this fixes the critical region.
- A simple hypothesis fixes the parameter completely; a composite one does not.
- We only reject $H_0$ or fail to reject it; we never prove it.
Testing Procedure (Critical region)
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Definition. <mark>The critical region is the set of values of the test statistic for which $H_0$ is rejected.</mark>
Steps.
Step 1: State H0 and H1.
Step 2: Fix the level of significance alpha.
Step 3: Choose the test statistic and find the critical value.
Step 4: Compute the statistic from the sample.
Step 5: Reject H0 if it falls in the critical region, else accept.
Key points.
- A two-tailed test puts $\alpha/2$ in each tail; a one-tailed test puts $\alpha$ in one tail.
- For $\alpha=0.05$ the normal critical value is $|z|=1.96$ (two-tailed) and $1.645$ (one-tailed).
Type I and Type II errors
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Definition. <mark>A Type I error is rejecting a true $H_0$, and a Type II error is accepting a false $H_0$.</mark>
Key points.
- $P(\text{Type I})=\alpha$ and $P(\text{Type II})=\beta$.
- Reducing $\alpha$ for a fixed sample size increases $\beta$, so both cannot be cut together.
- Only a larger sample reduces both.
| Decision | $H_0$ true | $H_0$ false |
|---|---|---|
| Reject $H_0$ | Type I error | Correct |
| Accept $H_0$ | Correct | Type II error |
Level of significance and Power of a test, p-value
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Definition. <mark>The level of significance $\alpha$ is the maximum probability of a Type I error we tolerate, and the power is $1-\beta$, the probability of rejecting a false $H_0$.</mark>
Key points.
- The usual levels are 5% and 1%.
- The p-value is the probability, under $H_0$, of a statistic at least as extreme as the one observed; for a symmetric null distribution a two-tailed p-value is $2P(Z>|z|)$.
- Reject $H_0$ if p-value $<\alpha$.
Tests for mean and proportion
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Definition. <mark>These tests compare a sample mean or proportion with a claimed value, or two samples with each other.</mark>
Formula. Single mean, $\sigma$ known or $n>30$: $z=\dfrac{\bar x-\mu_0}{\sigma/\sqrt n}$. Small $n$, $\sigma$ unknown: $t=\dfrac{\bar x-\mu_0}{s/\sqrt n}$ with $n-1$ d.f. Two means: $z=\dfrac{\bar x_1-\bar x_2}{\sqrt{\sigma_1^2/n_1+\sigma_2^2/n_2}}$. Proportion: $z=\dfrac{p-P_0}{\sqrt{P_0Q_0/n}}$.
Key points.
- Two-sample small-sample $t$ uses pooled $s_p^2=\dfrac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}$ with $n_1+n_2-2$ d.f.
- Two proportions: $z=\dfrac{p_1-p_2}{\sqrt{\hat P\hat Q(1/n_1+1/n_2)}}$, where $\hat P$ is the pooled proportion.
Tests for variance
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Definition. <mark>A single variance is tested with the chi-square statistic, and two variances are compared with the F statistic.</mark>
Formula. $\chi^2=\dfrac{(n-1)s^2}{\sigma_0^2}$ with $n-1$ d.f.; $F=\dfrac{s_1^2}{s_2^2}$ with $(n_1-1,\,n_2-1)$ d.f., the larger variance placed in the numerator.
Key points.
- Both tests assume a normal population.
- $\chi^2$ is not symmetric, so the two tails use different critical values.
Tests for mean and correlation coefficient for paired sample
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Definition. <mark>A paired test works on the differences $d_i$ of matched observations, such as before and after readings on the same units.</mark>
Formula. $t=\dfrac{\bar d}{s_d/\sqrt n}$ with $n-1$ d.f., where $s_d^2=\dfrac{\sum (d_i-\bar d)^2}{n-1}$. For $H_0:\rho=0$: $t=\dfrac{r\sqrt{n-2}}{\sqrt{1-r^2}}$ with $n-2$ d.f.
Key points.
- For large $n$ the paired statistic is treated as $z$.
- For $H_0:\rho=\rho_0$ use Fisher's $z=\tfrac12\ln\dfrac{1+r}{1-r}$, approximately normal with variance $1/(n-3)$.
Analysis of Variance (one way)
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Definition. ==One-way ANOVA tests $H_0:\mu_1=\dots=\mu_k$ by comparing the variation between groups with the variation within groups using an F test.==
Formula. $F=\dfrac{MSB}{MSW}=\dfrac{SSB/(k-1)}{SSW/(N-k)}$, where $SST=SSB+SSW$.
Key points.
- The assumptions are normal populations, equal variances and independent samples.
- Reject $H_0$ if $F$ exceeds $F_\alpha(k-1,\,N-k)$.
Last-minute revision
- $H_0$ holds the equality; $H_1$ decides one or two tails.
- Type I error is rejecting a true $H_0$, with probability $\alpha$.
- Type II error is accepting a false $H_0$, with probability $\beta$; power is $1-\beta$.
- Critical $z$ at 5%: 1.96 two-tailed, 1.645 one-tailed.
- Reject $H_0$ if p-value $<\alpha$.
- Single mean: $z=(\bar x-\mu_0)/(\sigma/\sqrt n)$; use $t$ with $n-1$ d.f. for small $n$.
- Pooled variance $s_p^2$ has $n_1+n_2-2$ d.f.
- Variance: $\chi^2=(n-1)s^2/\sigma_0^2$; two variances $F=s_1^2/s_2^2$.
- Paired $t=\bar d/(s_d/\sqrt n)$ with $n-1$ d.f.
- Correlation: $t=r\sqrt{n-2}/\sqrt{1-r^2}$ with $n-2$ d.f.
- ANOVA: $F=MSB/MSW$ with $(k-1,\,N-k)$ d.f.
Memory hooks
- "Null is Nothing new": $H_0$ always says no difference.
- Type I is the False alarm (alpha); Type II is the Miss (beta).
- Power equals one minus beta.
- Paired means one sample of differences, so $n-1$ d.f.
- ANOVA is Between over Within.
Coverage checklist
- Null and Alternative Hypothesis: no past questions.
- Testing Procedure (Critical region): no past questions.
- Type I and Type II errors: no past questions.
- Level of significance & Power of a test, p-value for symmetric null distribution: no past questions.
- Tests for mean and proportion (single sample, two sample; exact & large sample): no past questions.
- Tests for variance (single sample and two samples): no past questions.
- Tests for mean and correlation coefficient for paired sample (Exact & Large sample): no past questions.
- Analysis of Variance (one way): no past questions.