FAD1015: Comprehensive Rapid-Fire Drill — Full Syllabus

Objective: Identify every weak spot across the entire FAD1015 syllabus.
Target: 2–3 min per problem. If you stall >4 minutes, skip and mark it.
Total problems: 64
Estimated time: 2–3 hours

Cheat Sheet (Memorize First)

Probability & Counting

Concept Formula Notes
Permutation (no rep) $\displaystyle P(n,r)=\frac{n!}{(n-r)!}$ Order matters
Permutation (rep) $n^r$ Reuse allowed
Identical objects $\displaystyle\frac{n!}{n_1!,n_2!,\cdots,n_k!}$ $n_i$ = count of each identical type
Circular $(n-1)!$ Rotations same
Circular (flip same) $\frac{(n-1)!}{2}$ Necklace/ring
Combination $\displaystyle C(n,r)=\binom{n}{r}=\frac{n!}{r!(n-r)!}$ Order doesn't matter
Addition rule $P(A\cup B)=P(A)+P(B)-P(A\cap B)$ Subtract intersection
Conditional $\displaystyle P(B\mid A)=\frac{P(A\cap B)}{P(A)}$
Independence $P(A\cap B)=P(A)P(B)$ Also $P(A\mid B)=P(A)$
Total probability $P(E)=\sum P(E\mid A_i)P(A_i)$ Partition of sample space
Bayes' theorem $\displaystyle P(A_i\mid E)=\frac{P(E\mid A_i)P(A_i)}{\sum P(E\mid A_j)P(A_j)}$ Reverse conditional

Distributions

Distribution PMF/PDF Mean Variance Notes
Binomial $\binom{n}{x}p^x q^{n-x}$ $np$ $npq$ $n$ fixed, $p$ constant, independent
Poisson $\frac{\lambda^x e^{-\lambda}}{x!}$ $\lambda$ $\lambda$ Rare events in interval
Normal $\frac{1}{\sigma\sqrt{2\pi}}e^{-(x-\mu)^2/2\sigma^2}$ $\mu$ $\sigma^2$ $Z=\frac{X-\mu}{\sigma}$
Uniform $U(a,b)$ $\frac{1}{b-a}$ $\frac{a+b}{2}$ $\frac{(b-a)^2}{12}$ Constant PDF
Exponential $\lambda e^{-\lambda x}$ $1/\lambda$ $1/\lambda^2$ Memoryless

Binomial ↔ Poisson ↔ Normal

  • Poisson approx to binomial: $n>20$, $np<5$ (or $nq<5$), $\lambda=np$
  • Normal approx to binomial: $np>5$ and $nq>5$, $\mu=np$, $\sigma=\sqrt{npq}$
  • Continuity correction: $P(X\le k)$ → $P(Y < k+0.5)$; $P(X\ge k)$ → $P(Y > k-0.5)$
  • Binomial table: gives $P(X\ge r)$ (upper tail). To get $P(X\le r)=1-P(X\ge r+1)$

Random Variables

Property Discrete Continuous
PDF conditions $f(x)\in[0,1]$, $\sum f(x)=1$ $f(x)\ge0$, $\int f(x),dx=1$
CDF $F(t)=\sum_{x\le t} f(x)$ $F(t)=\int_{-\infty}^t f(x),dx$
$P(a<X\le b)$ $F(b)-F(a)$ $F(b)-F(a)$ (inequality doesn't matter)
Mean $\mu=\sum xf(x)$ $\mu=\int xf(x),dx$
Variance $\sigma^2=E(X^2)-\mu^2$, $E(X^2)=\sum x^2f(x)$ $\sigma^2=E(X^2)-\mu^2$, $E(X^2)=\int x^2f(x),dx$
Linear transform $E(aX+b)=aE(X)+b$ Same
Var linear $\text{Var}(aX+b)=a^2\text{Var}(X)$ Same

Sampling & Estimation

  • Sampling dist of mean: $\bar{X}\sim N(\mu,\sigma/\sqrt{n})$ if pop normal or $n\ge30$ (CLT)
  • Standard error: $\sigma_{\bar{X}}=\sigma/\sqrt{n}$
  • CI ($\sigma$ known): $\bar{x}\pm z_{\alpha/2}\cdot\sigma/\sqrt{n}$
  • CI ($\sigma$ unknown): $\bar{x}\pm t_{\alpha/2,n-1}\cdot s/\sqrt{n}$
  • Sample size for mean: $n=(z_{\alpha/2}\sigma/E)^2$
  • Critical z: 90% → 1.645, 95% → 1.96, 99% → 2.576

Hypothesis Testing (4-Step)

Step Action
1 State $H_0$, $H_1$ ($=/\neq$, $\le/>$, $\ge/\lt$)
2 Compute $z=\frac{\bar{x}-\mu_0}{\sigma/\sqrt{n}}$ or $t=\frac{\bar{x}-\mu_0}{s/\sqrt{n}}$
3 Compare: $p\le\alpha$? Statistic in rejection region?
4 Conclusion in context
  • $\sigma$ known → z-test. $\sigma$ unknown, $n\ge30$ → z-test with $s$. $\sigma$ unknown, $n<30$ → t-test, $df=n-1$.
  • CI method: if $\mu_0$ is inside CI → fail to reject $H_0$

Matrices

Operation Formula R Code
Determinant (2×2) $\vert A\vert = ad-bc$ det(A)
Inverse (2×2) $A^{-1}=\frac{1}{\vert A\vert}\begin{pmatrix}d&-b\-c&a\end{pmatrix}$ solve(A)
Matrix mult $C_{ij}=\sum_k A_{ik}B_{kj}$ A %*% B
Transpose $(A^T){ij}=a{ji}$ t(A)
Create matrix — matrix(data, nrow, ncol, byrow=TRUE)
Bind rows — rbind(r1, r2)
Bind columns — cbind(c1, c2)
Diagonal — diag(A)
Solve system $X=A^{-1}B$ solve(A, b)

Part A: Probability & Counting

Target: 2 min per problem.

Set A1 — Formula Identification (4 problems)

Identify which formula/concept applies. Do NOT compute.

  1. A bag has 5 red and 3 blue marbles. You draw two without replacement. The probability the second is blue given the first was red. Which concept is this?
  2. Events A and B have $P(A\cap B)=P(A)P(B)$. What property do A and B satisfy?
  3. In a factory, 40% of items come from Machine X (3% defective) and 60% from Machine Y (2% defective). An item is defective. Which theorem do you use to find the probability it came from Machine X?
  4. $P(A\cup B)=0.8$, $P(A)=0.5$, $P(B)=0.4$. What formula do you use to find $P(A\cap B)$?

Score: ___/4

Set A2 — Calculations (4 problems)

  1. A coin is tossed 3 times. What is the probability of getting at least one head?
  2. $P(A)=0.3$, $P(B)=0.6$, $P(A\cap B)=0.2$. Find $P(A\cup B)$.
  3. In a class of 30, 18 take Chemistry, 15 take Physics, and 10 take both. What is the probability a randomly chosen student takes neither?
  4. Events A and B are independent with $P(A)=0.4$ and $P(B)=0.5$. Find $P(A\cap B)$.

Score: ___/4

Part B: Binomial Distribution — LEAK Q1

Target: 2–3 min per problem.

Set B1 — Characteristics & Identification (2 problems)

  1. Which of the following is NOT a characteristic of a binomial distribution?

    • (i) Fixed number of trials
    • (ii) More than two possible outcomes per trial
    • (iii) Independent trials
    • (iv) Constant probability of success

  2. A fair die is rolled 10 times and we count the number of times a 6 appears. Identify: (i) $n$, (ii) $p$, (iii) the distribution of $X$.

Score: ___/2

Set B2 — Direct Calculations (2 problems)

  1. $X\sim B(8,0.3)$. Find $P(X=2)$ using the formula.
  2. $X\sim B(5,0.2)$. Find $P(X\ge1)$.

Score: ___/2

Set B3 — Table Reading & p>0.5 Flip (2 problems)

  1. $X\sim B(10,0.8)$. Given a binomial table for $p=0.2$ only, explain how you would find $P(X\ge7)$.
  2. $X\sim B(12,0.65)$. Find the mean and standard deviation.

Score: ___/2

Part C: Poisson & Uniform Distributions — LEAK Q2

Target: 2–3 min per problem.

Set C1 — Poisson Calculations (3 problems)

  1. Calls arrive at a rate of $\lambda=4$ per hour. Find $P(X=2)$ in one hour.
  2. Accidents occur at $\lambda=3$ per week. Find $P(X\le1)$ in one week.
  3. For a Poisson distribution with $\lambda=5$, what are the mean and variance?

Score: ___/3

Set C2 — Binomial → Poisson Conversion (2 problems)

  1. $X\sim B(100,0.03)$. Can we use Poisson approximation? If so, state $\lambda$.
  2. $X\sim B(60,0.95)$. Convert so Poisson approximation applies. State the new $p$ and $\lambda$.

Score: ___/2

Set C3 — Uniform Distribution (5 problems)

  1. $X\sim U(10,30)$. Find $P(X<15)$.
  2. $X\sim U(0,50)$. Find the mean and standard deviation.
  3. $X\sim U(2,12)$. Find $P(4<X<8)$.
  4. $X\sim U(a,b)$ has mean $25$ and standard deviation $\approx 14.43$. Find $a$ and $b$.
  5. $X\sim U(a,b)$ has mean $10$ and range length $b-a=12$. Find $a$, $b$, and $P(X>14)$.

Score: ___/5

Part D: Normal Distribution — LEAK Q4

Target: 2–3 min per problem.

Set D1 — Standard Normal (2 problems)

  1. $Z\sim N(0,1)$. Find $P(Z>1.96)$.
  2. $Z\sim N(0,1)$. Find $P(-1.5<Z<0.5)$.

Score: ___/2

Set D2 — Normal Probabilities (2 problems)

  1. $X\sim N(100,15^2)$. Find $P(X>130)$.
  2. $X\sim N(50,10^2)$. Find the 90th percentile.

Score: ___/2

Set D3 — Normal Approximation to Binomial (2 problems)

  1. $X\sim B(200,0.4)$. Can we use normal approximation? If so, state $\mu$ and $\sigma$.
  2. $X\sim B(150,0.3)$. Using normal approximation, write the continuity-corrected expression for $P(X\le45)$.

Score: ___/2

Part E: Discrete & Continuous Random Variables — LEAK Q3, Q4

Target: 2–3 min per problem.

Set E1 — Discrete PDF/CDF (2 problems)

  1. Given the discrete PDF: $P(X=0)=0.1$, $P(X=1)=0.3$, $P(X=2)=0.4$, $P(X=3)=0.2$. Find $P(X\ge2)$.
  2. For the distribution above, find $F(2)$, the CDF at $x=2$.

Score: ___/2

Set E2 — Continuous PDF/CDF (2 problems)

  1. $f(x)=\frac{3}{8}(1-x^2)$ on $[-1,1]$. Verify this is a valid PDF (check the two conditions).
  2. For a continuous random variable with $f(x)=kx$ on $[0,2]$, find $k$.

Score: ___/2

Set E3 — Mean & Variance (2 problems)

  1. Discrete: $P(X=1)=0.2$, $P(X=2)=0.5$, $P(X=3)=0.3$. Find $E(X)$ and $\text{Var}(X)$.
  2. Given $E(X)=5$ and $\text{Var}(X)=4$, find $E(3X+2)$ and $\text{Var}(3X+2)$.

Score: ___/2

Set E4 — Find Unknown Constants (2 problems)

  1. Continuous: $f(x)=cx^2$ on $[0,3]$. Find $c$ so this is a valid PDF.
  2. Discrete: $P(X=x)=kx$ for $x=1,2,3,4$. Find $k$ and then $P(X<3)$.

Score: ___/2

Part F: Exponential Distribution

Target: 2 min per problem.

Set F1 — Exponential Calculations (2 problems)

  1. Waiting time follows $Exp(0.5)$. Find $P(X>3)$.
  2. Service time has mean 10 minutes. Find $\lambda$ and $P(X<5)$.

Score: ___/2

Set F2 — Memoryless Property & Rate Adjustment (2 problems)

  1. Battery life is exponential with mean 100 hours. The battery has already lasted 80 hours. What is the probability it lasts at least another 50 hours?
  2. A machine has $\lambda=2$ failures per week. What is the probability of no failures in two weeks?

Score: ___/2

Reader's Note — Part G: This section tests your ability to move from population parameters to sample statistics. The CLT is the engine behind everything here — understand that $\bar{X}$ is itself a random variable with its own distribution. G2 checks whether you can distinguish $\sigma$ known (z) from $\sigma$ unknown (t) for CIs. G3 is the classic exam trap: always round sample size up. The formulas look similar across sets; the key differentiator is whether $\sigma$ is given or you have $s$.

Part G: Sampling & Estimation

Target: 2–3 min per problem.

Set G1 — Sampling Distribution / CLT (2 problems)

  1. A population has $\mu=50$, $\sigma=12$. A sample of $n=36$ is taken. What is the distribution of $\bar{X}$? State its mean and standard error.
  2. For the above, find $P(\bar{X}>53)$.

Score: ___/2

Set G2 — Confidence Intervals (2 problems)

  1. $n=25$, $\bar{x}=80$, $s=10$. Find the 95% CI for $\mu$. ($t_{0.025,24}=2.064$)
  2. $n=100$, $\bar{x}=500$, $\sigma=40$. Find the 99% CI for $\mu$.

Score: ___/2

Set G3 — Sample Size Determination (2 problems)

  1. How large a sample is needed for a 95% CI with margin of error $E=3$ if $\sigma=12$?
  2. If we want 99% confidence instead of 95% (same $E$ and $\sigma$), will the sample size increase or decrease?

Score: ___/2

Reader's Note — Part H: This is a leaked topic — expect exam questions in this exact style. H1 tests whether you can correctly select z vs t and write $H_0/H_1$ in symbols (a common place to lose marks). H2 is the full 4-step workflow; the most common error is computing the test statistic correctly but misinterpreting the comparison. H3 is about reading R output — know that $df = n-1$, and that CI exclusion of $\mu_0$ is equivalent to rejecting $H_0$. Pay attention to tail direction when interpreting $p$-values.

Part H: Hypothesis Testing — LEAK Q5

Target: 3–4 min per problem.

Set H1 — Test Selection & Hypotheses (3 problems)

For each: (i) z-test or t-test? (ii) State $H_0$ and $H_1$ in symbols.

  1. $n=40$, $\bar{x}=102$, $s=15$, $\sigma$ unknown. Test if mean differs from 100 at $\alpha=0.05$.
  2. $n=16$, $\bar{x}=7.2$, $s=2.1$, $\sigma$ unknown. Test if mean is less than 8 at $\alpha=0.05$.
  3. $n=50$, $\bar{x}=98$, $\sigma=10$ (known). Test if mean is greater than 95 at $\alpha=0.01$.

Score: ___/3

Set H2 — Full Hypothesis Tests (3 problems)

  1. $n=36$, $\bar{x}=28$, $\sigma=9$. Test $H_0:\mu=25$ vs $H_1:\mu>25$ at $\alpha=0.05$. Compute $z$, compare to $z_{0.05}=1.645$, and conclude.
  2. $n=20$, $\bar{x}=48$, $s=5$, $\sigma$ unknown. Test $H_0:\mu=50$ vs $H_1:\mu\neq50$ at $\alpha=0.05$. $t_{0.025,19}=2.093$. Compute $t$ and conclude.
  3. From a test, $p$-value $=0.023$ and $\alpha=0.05$. Do you reject $H_0$? What if $\alpha=0.01$?

Score: ___/3

Set H3 — R Output Interpretation (2 problems)

  1. R output:

            One Sample t-test
data:  sample
t = 2.456, df = 24, p-value = 0.0215
alternative hypothesis: true mean is not equal to 50
95 percent confidence interval:
  50.23 54.77
sample estimates:
mean of x
  52.5

    (a) What is $n$? (b) At $\alpha=0.05$, reject $H_0$? (c) Does the CI contain $\mu_0=50$?

  2. R output:

            One-sample z-Test
data:  data
z = -1.85, p-value = 0.0643
alternative hypothesis: true mean is less than 100

    (a) Left-tailed, right-tailed, or two-tailed? (b) At $\alpha=0.05$, reject $H_0$? (c) At $\alpha=0.10$, reject $H_0$?

Score: ___/2

Part I: Matrices & R Programming — LEAK Q6 HIGHEST PRIORITY

Target: 2–3 min per problem.

Set I1 — Matrix Operations by Hand (2 problems)

Given $A=\begin{pmatrix}1&3\2&4\end{pmatrix}$, $B=\begin{pmatrix}2&0\1&5\end{pmatrix}$:

  1. Compute $\det(A)$ and $\det(B)$.
  2. Compute $A^{-1}$ (2×2 inverse formula).

Score: ___/2

Set I2 — R Code (3 problems)

Write the R code to accomplish each task.

  1. Create a $3\times3$ matrix M from numbers 1 to 9 filled column-wise.
  2. Using rbind(), create a matrix S with rows (10,20,30) and (40,50,60). Then assign row names "R1","R2" and column names "A","B","C".
  3. Given matrix A and matrix B (same dimensions), write R code to compute: (i) matrix multiplication, (ii) determinant of A, (iii) inverse of A, (iv) transpose of B.

Score: ___/3

Set I3 — Predict R Output (3 problems)

  1. X <- matrix(1:4, nrow = 2)
Y <- matrix(c(2,0,0,2), nrow = 2)
X %*% Y

    What is the output?

  2. M <- rbind(c(1,2,3), c(4,5,6))
t(M)

    What is the output?

  3. A <- matrix(c(4,3,2,1), nrow = 2)
det(A)
solve(A)

    What are $\det(A)$ and $A^{-1}$?

Score: ___/3

Final Scorecard

Part Topic Problems Raw Score
A Probability & Counting 8 ___/8
B Binomial Distribution 6 ___/6
C Poisson & Uniform 10 ___/10
D Normal Distribution 6 ___/6
E Random Variables (PDF, CDF, Mean, Var) 8 ___/8
F Exponential Distribution 4 ___/4
G Sampling & Estimation 6 ___/6
H Hypothesis Testing 8 ___/8
I Matrices & R 8 ___/8
TOTAL Full Syllabus 64 ___/64
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Proficiency Benchmarks

  • 45/64 (70%) — Proficient on most topics. Review the parts you scored lowest on.
  • 54/64 (85%) — Solid across the board. Your weak spots are small.
  • 60/64 (93%) — Exam-ready. Any mistake is a careless slip.

Speed Benchmarks

  • <90 min: Excellent mechanical fluency.
  • 90–150 min: Good pace. Review patterns that slowed you down.
  • >150 min: You're stalling. Drill the specific parts you scored lowest on again.

Part-by-Part Diagnosis

If you scored low in... Your weak spot is... Suggested action
Part A Probability fundamentals Review Week 2–3 lectures, Tutorial 3–4
Part B Binomial distribution Review L13, Tutorial 13, Q1 leak style
Part C Poisson/Uniform Review L14, L17, Poisson approx + rate adjustment
Part D Normal distribution Review L15–16, z-table practice, continuity correction
Part E Random Variables Review Week 4–6, Tutorial 5–6, PDF/CDF mechanics
Part F Exponential distribution Review L17, memoryless property
Part G Sampling/Estimation Review L20–22, Tutorial 9–10
Part H Hypothesis Testing Review L23–24, Tutorial 11, the 4-step framework
Part I Matrices/R Review L27–30, Tutorial 12/14, highest priority per leak

Error Log Template

After grading, list every wrong problem number with a one-word reason:

Problem Reason
e.g. 28 forgot cont. corr.

Re-solve all wrong problems immediately with notes, then again in 24 hours without notes.

Answer Key

Set A1

  1. Conditional probability $P(\text{blue}|\text{red})$
  2. Independence
  3. Bayes' theorem
  4. Addition rule: $P(A\cap B)=P(A)+P(B)-P(A\cup B)$

Set A2

  1. $1-P(\text{no heads})=1-(1/2)^3=1-1/8=7/8$
  2. $P(A\cup B)=0.3+0.6-0.2=0.7$
  3. Neither: $30-(18+15-10)=7$. $P=7/30$
  4. $P(A\cap B)=0.4\times0.5=0.2$

Set B1

  1. (ii) — must have exactly two outcomes per trial
  2. (i) $n=10$, (ii) $p=1/6$, (iii) $X\sim B(10,1/6)$

Set B2

  1. $P(X=2)=\binom{8}{2}(0.3)^2(0.7)^6=28\times0.09\times0.1176=0.296$
  2. $P(X\ge1)=1-P(X=0)=1-(0.8)^5=1-0.3277=0.6723$

Set B3

  1. Flip: let $Y=10-X\sim B(10,0.2)$. Then $P(X\ge7)=P(Y\le3)$ from table
  2. $\mu=np=12\times0.65=7.8$, $\sigma=\sqrt{12\times0.65\times0.35}=\sqrt{2.73}=1.652$

Set C1

  1. $P(X=2)=\frac{4^2 e^{-4}}{2!}=\frac{16e^{-4}}{2}=8e^{-4}=0.1465$
  2. $P(X\le1)=P(X=0)+P(X=1)=e^{-3}+3e^{-3}=4e^{-3}=0.199$
  3. Mean $=5$, Variance $=5$

Set C2

  1. Yes. $n=100>20$, $np=3<5$. $\lambda=np=3$
  2. Flip failures: $p_{\text{fail}}=0.05$, $\lambda=60\times0.05=3$

Set C3

  1. $P(X<15)=\frac{15-10}{30-10}=5/20=0.25$
  2. $\mu=\frac{0+50}{2}=25$, $\sigma=\frac{50-0}{\sqrt{12}}=50/3.464=14.43$
  3. $P(4<X<8)=\frac{8-4}{12-2}=4/10=0.4$
  4. $a+b=2(25)=50$, $b-a=14.43\sqrt{12}\approx50$. Solve: $a=(50-50)/2=0$, $b=(
  5. 

, 

50+50)/2=50$. So $U(0,50)$. 25. $a+b=2(10)=20$, $b-a=12$. Solve: $a=(20-12)/2=4$, $b=(20+12)/2=16$. $P(X>14)=\frac{16-14}{16-4}=2/12=0.167$.

Set D1

  1. $P(Z>1.96)=0.025$
  2. $P(-1.5<Z<0.5)=P(Z<0.5)-P(Z<-1.5)=0.6915-0.0668=0.6247$

Set D2

  1. $Z=(130-100)/15=2$. $P(Z>2)=0.0228$
  2. $z_{0.10}=1.282$. $x=50+1.282\times10=62.82$

Set D3

  1. Yes. $np=200\times0.4=80>5$, $nq=200\times0.6=120>5$. $\mu=80$, $\sigma=\sqrt{200\times0.4\times0.6}=\sqrt{48}=6.928$
  2. $P(X\le45)\approx P(Y<45.5)$ where $Y\sim N(45, \sqrt{31.5})$

Set E1

  1. $P(X\ge2)=P(2)+P(3)=0.4+0.2=0.6$
  2. $F(2)=P(X\le2)=0.1+0.3+0.4=0.8$

Set E2

  1. (i) $f(x)\ge0$ on $[-1,1]$ since $1-x^2\ge0$. (ii) $\int_{-1}^1\frac{3}{8}(1-x^2),dx=\frac{3}{8}[x-x^3/3]_{-1}^1=\frac{3}{8}\times\frac{4}{3}=1$ ✓
  2. $\int_0^2 kx,dx=k[x^2/2]_0^2=k\times2=1$ → $k=1/2$

Set E3

  1. $E(X)=1(0.2)+2(0.5)+3(0.3)=0.2+1.0+0.9=2.1$. $E(X^2)=1(0.2)+4(0.5)+9(0.3)=0.2+2.0+2.7=4.9$. $\text{Var}=4.9-2.1^2=4.9-4.41=0.49$
  2. $E(3X+2)=3(5)+2=17$. $\text{Var}(3X+2)=9\times4=36$

Set E4

  1. $\int_0^3 cx^2,dx=c[x^3/3]_0^3=c\times9=1$ → $c=1/9$
  2. $\sum_{x=1}^4 kx=k(1+2+3+4)=10k=1$ → $k=0.1$. $P(X<3)=0.1+0.2=0.3$

Set F1

  1. $P(X>3)=e^{-0.5\times3}=e^{-1.5}=0.2231$
  2. $\lambda=1/10=0.1$. $P(X<5)=1-e^{-0.1\times5}=1-e^{-0.5}=0.3935$

Set F2

  1. Memoryless: $P(X>130|X>80)=P(X>50)=e^{-50/100}=e^{-0.5}=0.6065$
  2. In 2 weeks, $\lambda=4$. $P(X=0)=e^{-4}=0.0183$

Set G1

  1. $\bar{X}\sim N(50, 12/\sqrt{36})=N(50,2)$
  2. $Z=(53-50)/2=1.5$. $P(Z>1.5)=0.0668$

Set G2

  1. $80\pm2.064\times10/\sqrt{25}=80\pm2.064\times2=80\pm4.128$ → $(75.872,84.128)$
  2. $500\pm2.576\times40/\sqrt{100}=500\pm2.576\times4=500\pm10.304$ → $(489.696,510.304)$

Set G3

  1. $n=(1.96\times12/3)^2=(23.52/3)^2=7.84^2=61.5$ → $n=62$
  2. Increases (higher confidence → larger critical value → larger $n$)

Set H1

  1. (i) z-test ($n\ge30$, use $s$). (ii) $H_0:\mu=100$, $H_1:\mu\neq100$
  2. (i) t-test ($\sigma$ unknown, $n<30$). (ii) $H_0:\mu\ge8$, $H_1:\mu<8$
  3. (i) z-test ($\sigma$ known). (ii) $H_0:\mu\le95$, $H_1:\mu>95$

Set H2

  1. $z=(28-25)/(9/\sqrt{36})=3/1.5=2.0$. $2.0>1.645$ → reject $H_0$. Sufficient evidence mean >25.
  2. $t=(48-50)/(5/\sqrt{20})=-2/1.118=-1.789$. $\vert-1.789\vert<2.093$ → fail to reject $H_0$. Insufficient evidence mean differs from 50.
  3. At $\alpha=0.05$: $0.023<0.05$ → reject $H_0$. At $\alpha=0.01$: $0.023>0.01$ → fail to reject $H_0$.

Set H3

  1. (a) $n=df+1=25$. (b) $p=0.0215<0.05$ → reject $H_0$. (c) CI $(50.23,54.77)$ does NOT contain 50 → consistent with reject.
  2. (a) Left-tailed ($H_1$: true mean is less than 100). (b) $p=0.0643>0.05$ → fail to reject. (c) $0.0643<0.10$ → reject $H_0$ at $\alpha=0.10$.

Set I1

  1. $\det(A)=1(4)-3(2)=4-6=-2$. $\det(B)=2(5)-0(1)=10$
  2. $A^{-1}=\frac{1}{-2}\begin{pmatrix}4&-3\-2&1\end{pmatrix}=\begin{pmatrix}-2&1.5\1&-0.5\end{pmatrix}$

Set I2

  1. M <- matrix(1:9, nrow = 3)
  2. S <- rbind(c(10,20,30), c(40,50,60))
rownames(S) <- c("R1","R2")
colnames(S) <- c("A","B","C")

  3. A %*% B
det(A)
solve(A)
t(B)

Set I3

  1. $X=\begin{pmatrix}1&3\2&4\end{pmatrix}$, $Y=\begin{pmatrix}2&0\0&2\end{pmatrix}=2I$. $X%*%Y=2X=\begin{pmatrix}2&6\4&8\end{pmatrix}$
  2. $M=\begin{pmatrix}1&2&3\4&5&6\end{pmatrix}$, $t(M)=\begin{pmatrix}1&4\2&5\3&6\end{pmatrix}$
  3. $A=\begin{pmatrix}4&2\3&1\end{pmatrix}$. $\det(A)=4(1)-2(3)=4-6=-2$. $A^{-1}=\frac{1}{-2}\begin{pmatrix}1&-2\-3&4\end{pmatrix}=\begin{pmatrix}-0.5&1\1.5&-2\end{pmatrix}$

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