ENSH 304 Probability Notes

Chapter 3: Statistical Inference

Syllabus hours: 14 | Exam weight: 20 marks | Marks breakdown: estimation 4, confidence intervals 5, hypothesis tests 6, chi-square/ANOVA 5

Difficulty type: Highest priority | Version / Last Updated: 2026-04-18 | Not in syllabus: advanced Bayesian inference

Outcome: estimate parameters, build confidence intervals, run hypothesis tests, and interpret inference results in engineering settings.

1. Fundamental Concepts

A parameter describes the population; a statistic describes the sample.
An estimator is a rule; an estimate is the numerical value from that rule.
Confidence intervals give a plausible range for the parameter.
Hypothesis testing compares a null claim against a competing alternative.
Type I error rejects a true null; Type II error fails to reject a false null.
p-value measures how extreme the sample result is under the null hypothesis.

2. Core Methods and Formulas

When to use: use confidence intervals when you need a range estimate; use hypothesis tests when you need a decision on a claim; use z-tests when σ is known or n is large; use t-tests when σ is unknown and sample size is small.

When not to use: do not use a z-test with a tiny sample and unknown σ; do not use a confidence interval as a direct accept/reject rule unless the question explicitly asks for that interpretation.

\hat p=\frac{x}{n},\quad \hat q=1-\hat p

z=\frac{\bar x-\mu_0}{\sigma/\sqrt n},\quad t=\frac{\bar x-\mu_0}{s/\sqrt n}

t=\frac{\bar d}{s_d/\sqrt n}\quad \text{(paired t-test)}

z=\frac{\hat p-p_0}{\sqrt{p_0q_0/n}}

CI for mean: \bar x\pm z_{\alpha/2}\frac{\sigma}{\sqrt n}\quad or\quad \bar x\pm t_{\alpha/2,n-1}\frac{s}{\sqrt n}

CI for proportion: \hat p\pm z_{\alpha/2}\sqrt{\frac{\hat p(1-\hat p)}{n}}

\chi^2=\sum\frac{(O-E)^2}{E}

F=\frac{MS_{between}}{MS_{within}}

3. Standard Models / Topics

Topic 1: Point Estimation and Estimator Properties

Basic notes: estimation uses sample information to infer a population parameter. Good estimators are unbiased, consistent, efficient, and sufficient when possible.

Conditions / use: use point estimation when you need a single best value, not an interval.

Formula recap: $\hat{\mu}=\bar{x}$ , $\hat p=x/n$ , and for variance/standard deviation, the sample formulas depend on the data type and whether the question asks for a population or sample measure.

Seen-Before Check: “estimate the mean,” “estimate the proportion,” “properties of an estimator,” or “unbiased estimator” point to this topic.

PYQ pattern: identify estimator properties and compute a numerical point estimate from a sample.

[Core] Problem 1: Differentiate between parameter and statistic.

Answer: parameter belongs to the population; statistic belongs to the sample.

[Core] Problem 2: The mean of a sample of 50 observations is 23. Find the point estimate of the population mean.

Answer: point estimate of μ is 23.

[PYQ-Trap] Problem 3: Explain why sample mean is a better estimator than a single observation.

Answer: it uses more information, reduces random fluctuation, and is unbiased for the population mean under random sampling.

[Advanced] Problem 4: A sample of five values is 6, 8, 9, 7, 10. Estimate the population mean, sample variance, and standard error of the mean.

\bar x=(6+8+9+7+10)/5=8

s^2=\frac{(6-8)^2+(8-8)^2+(9-8)^2+(7-8)^2+(10-8)^2}{5-1}=\frac{10}{4}=2.5

s=\sqrt{2.5}\approx1.581,\quad SE(\bar x)=\frac{s}{\sqrt n}=\frac{1.581}{\sqrt 5}\approx0.707

Answer: point estimate of the mean is 8, sample variance is 2.5, and the standard error is approximately 0.707. This SE value will be used in confidence interval and hypothesis test calculations.

Interpretation checklist: state whether the estimate is for a population mean, proportion, or variance, and mention the sample size supporting it.

Topic 2: Confidence Intervals for Mean and Proportion

Basic notes: a confidence interval is a range of plausible values for the true parameter. A 95% interval does not mean the parameter has 95% probability of being inside one fixed interval; it means the method is correct 95% of the time in repeated sampling.

Conditions / use: use z-interval for known σ or large n; use t-interval for unknown σ and small n; use proportion intervals when the variable is binary.

Formula recap: $\bar x\pm z_{\alpha/2}\frac{\sigma}{\sqrt n}$ , $\bar x\pm t_{\alpha/2,n-1}\frac{s}{\sqrt n}$ , $\hat p\pm z_{\alpha/2}\sqrt{\frac{\hat p(1-\hat p)}{n}}$ .

Seen-Before Check: words like “construct 95% confidence interval,” “estimate the mean,” or “estimate the proportion” are direct triggers.

PYQ pattern: mean CI with σ known/unknown, and proportion CI from a sample count.

[Core] Problem 1: A sample of 35 meters has mean 240 and s=45. Find 90% and 99% confidence intervals for the mean.

CI_{90\%}=240\pm1.645\frac{45}{\sqrt{35}},\quad CI_{99\%}=240\pm2.576\frac{45}{\sqrt{35}}

Answer: compute the margins using the formulas and report both intervals separately.

[Advanced] Problem 2: A random sample of 500 pipes has 65 low-quality pipes. Construct 95% and 99% confidence intervals for the proportion of low-quality pipes.

\hat p=65/500=0.13,\quad CI=\hat p\pm z_{\alpha/2}\sqrt{\frac{\hat p(1-\hat p)}{n}}

Answer: plug in z-values 1.96 and 2.576 for 95% and 99% intervals.

Interpretation checklist: say what the interval says about the likely population mean/proportion and whether the interval is wide or narrow.

Topic 3: Hypothesis Testing Framework

Basic notes: hypothesis testing evaluates a null claim $H_0$ against an alternative $H_1$ using sample evidence. The decision is made either by the critical region method or by p-value.

Conditions / use: use this when the question asks to accept/reject a claim, compare a p-value, or test significance.

Formula recap: $z=\frac{\bar x-\mu_0}{\sigma/\sqrt n}$ , $t=\frac{\bar x-\mu_0}{s/\sqrt n}$ , $z=\frac{\hat p-p_0}{\sqrt{p_0q_0/n}}$ .

Seen-Before Check: “test the claim,” “significant difference,” “at 5% level,” or “reject the null hypothesis” are the classic triggers.

PYQ pattern: define hypotheses, compute test statistic, compare with critical value or p-value, and conclude in words.

[Core] Problem 1: State type I error and type II error.

Answer: Type I error rejects a true null hypothesis; Type II error fails to reject a false null hypothesis.

[Core] Problem 2: A manufacturer claims at least 95% of pumps meet specification. In a sample of 250 pumps, 228 meet specification. Test the claim at 1% level.

H_0:p=0.95,\quad H_1:p<0.95,\quad hat p=228/250=0.912

z=\frac{0.912-0.95}{\sqrt{0.95(0.05)/250}}

z\approx-2.76,\quad p\text{-value}\approx0.0029

Answer: compute z, compare with the 1% left-tail critical value, and the p-value is below 0.01, so reject H0.

[PYQ-Trap] Problem 3: A sample mean is larger than the claimed mean. Does that alone prove the claim is false?

Answer: no, because sampling variation and significance level must be considered through a formal test.

Interpretation checklist: always state the statistical decision and the real-world meaning separately.

Topic 4: Tests of Means

Basic notes: mean tests include single-sample tests, two-sample tests, paired t-tests, and one-way ANOVA when comparing several means. The choice depends on how many samples are involved and whether the data are paired.

Conditions / use: use single-sample tests for one mean, two-sample tests for independent groups, paired t-test for before/after or matched pairs, ANOVA for three or more means.

Formula recap: single-sample z/t, two-sample z/t, paired t on the differences $d_i$ , and $t=\frac{\bar d}{s_d/\sqrt n}$ for paired data; use $F=MS_{between}/MS_{within}$ for ANOVA.

Seen-Before Check: “before and after,” “two independent samples,” “which of three methods is better,” or “significant difference among means” are the clues.

PYQ pattern: choose the right mean test before doing any arithmetic.

[Core] Problem 1: A sample of 54 workers has mean weekly earnings 432.69, σ=33.90, and μ0=424.20. Test at 5% whether earnings have changed.

H_0:mu=424.20,\quad H_1:mu\neq424.20,\quad z=\frac{432.69-424.20}{33.90/\sqrt{54}}\approx1.84

Answer: compare z with ±1.96 and conclude the claim is not rejected at 5%.

[Advanced] Problem 2: Two drugs are tested on 7 patients each with observed weight-loss data. Test whether Drug A is more effective than Drug B.

Answer: use a two-sample t-test on independent samples if treated as separate groups.

[Advanced] Problem 3: For independent samples with n1 = n2 = 6, x̄1 = 15, x̄2 = 12.5, s1 = 2, and s2 = 2, test whether the means differ.

s_p=2,\quad t=\frac{15-12.5}{2\sqrt{1/6+1/6}}=2.17,\quad df=10

Answer: t is below the 5% two-tailed critical value of about 2.228, so the difference is not significant at 5%.

[Advanced] Problem 4: The average scores of three teaching methods are compared. What test should be used?

Answer: one-way ANOVA, because three means are being compared simultaneously.

[Advanced] Problem 5: For paired differences 2, 1, 3, 0, 4, test whether the average change is significant.

\bar d=2,\quad s_d=\sqrt{2.5}=1.581,\quad t=\frac{2}{1.581/\sqrt5}=2.83,\quad df=4

Answer: compare with t0.05,4; the paired result is significant.

[PYQ-Trap] Problem 6: For the three teaching methods, use data A = 5, 6, 7; B = 7, 8, 9; C = 9, 10, 11. Compute the ANOVA F-statistic.

\bar x_A=6,\quad \bar x_B=8,\quad \bar x_C=10,\quad \bar x=8

SSB=3[(6-8)^2+(8-8)^2+(10-8)^2]=24,\quad SSE=2+2+2=6

df_B=2,\quad df_W=6,\quad MSB=12,\quad MSW=1,\quad F=12

Answer: F = 12, which is greater than the 5% critical value for df (2,6), so reject the equal-means hypothesis.

Interpretation checklist: identify whether the test is about one mean, two means, paired differences, or multiple means, and then state what practical conclusion the test supports.

Topic 5: Chi-square Tests and Goodness of Fit / Independence

Basic notes: chi-square methods compare observed and expected frequencies. Goodness of fit checks whether a distribution matches data; independence checks whether two categorical variables are associated.

Conditions / use: use chi-square when data are counts in categories and expected counts are meaningful.

Formula recap: $\chi^2=\sum\frac{(O-E)^2}{E}$ and for a contingency table, expected frequency $E=\frac{(row\ total)(column\ total)}{grand\ total}$ .

Seen-Before Check: category tables, “independent or not,” “goodness of fit,” and “distribution of errors per page” are the clues.

PYQ pattern: chi-square of independence, goodness of fit, and sometimes ANOVA-style F comparison in one chapter set.

[Core] Problem 1: Define the chi-square distribution.

Answer: it is a distribution of the sum of squares of independent standard normal variables and is used in variance-based and categorical testing.

[Advanced] Problem 2: Test independence between smoking habit and hypertension using the given 2×3 table data.

Answer: compute expected counts, form $\chi^2$ , compare with the critical value, and conclude whether smoking and hypertension are independent.

[PYQ-Trap] Problem 3: A book page-error table is given. How do you test whether arrivals follow Poisson?

Answer: use a chi-square goodness-of-fit test with Poisson expected frequencies.

Interpretation checklist: state whether the variables are independent or whether the observed frequencies fit a proposed model.

4. Applied Problem Solving

[Core] Build a 95% confidence interval for a sample mean or proportion.
[Core] Run a z-test or t-test depending on the data and sample size.
[PYQ-Trap] Decide between paired t-test, two-sample t-test, ANOVA, and chi-square based on wording alone.

5. System-Level Understanding

Chapter 3 converts raw sample summaries into engineering decisions.
Chapter 2 supplies the sampling-distribution logic that makes CI and tests possible.
Inference language, especially significance and confidence, repeats across regression and quality control later.

6. Quick Reference

$z=\frac{\bar x-\mu_0}{\sigma/\sqrt n}$ when σ is known or n is large.

$t=\frac{\bar x-\mu_0}{s/\sqrt n}$ when σ is unknown and n is small.

$\chi^2=\sum\frac{(O-E)^2}{E}$ for categorical count tests.

Seen-Before Check: one mean, two means, paired data, many means, or categorical counts determines the test family immediately.

7. Exam Tips

Write the hypothesis pair first, then the test statistic, then the conclusion.
State whether the test is one-tailed or two-tailed before computing the critical value.
For intervals, say the interpretation in words after giving the numerical interval.
Seen-Before Check: if the question contains “construct,” “test,” or “independent,” classify the statistical tool before calculating.

8. Common Pitfalls

Using z when t should be used, or t when z should be used.
Confusing confidence level with significance level.
Forgetting to use sample-based expected frequencies in chi-square tables.
Calling a result “significant” without stating the practical meaning.

9. Tools and Guides

Calculator tips: keep z/t critical values and common χ² critical values available.
Cheat lines: inference = estimate, test, conclude; categorical = χ²; comparing three or more means = ANOVA.
Distribution selection rule: one mean → z/t; proportion → z; categories → χ²; multiple means → ANOVA.