Chapters 8 to 10. Twenty questions, one answer each.
These are hard on purpose. Most take more than one step, and every wrong option is the number you get by making one specific mistake — dividing by $n$ instead of $n-1$, forgetting a square root, using $\sigma$ where $\sigma/\sqrt{n}$ belongs, inverting a conditional.
So if you pick a wrong answer, the key tells you which mistake you made, which is more use than the letter.
1. $X \sim \mathrm{Bin}(6, 0.2)$. Then $\mathrm{P}(X \ge 2)$ is closest to
(A) 0.246 (B) 0.262 (C) 0.655 (D) 0.345
2. $X \sim \mathrm{Bin}(20, 0.4)$. Its standard deviation is
(A) 4.80 (B) 2.19 (C) 8.00 (D) 3.10
3. A binomial has $n = 50$, $p = 0.08$. The normal approximation is
(A) appropriate, because $n > 30$ (B) appropriate, because $n(1-p) > 5$ (C) inappropriate, because $np < 5$ (D) inappropriate, because $p < 0.5$
4. Twelve items are drawn without replacement from a batch of 30, and defectives counted. The binomial is
(A) not appropriate, because $p$ changes between draws (B) appropriate, because the draws are random (C) appropriate, because there are two outcomes (D) not appropriate, because $n$ is too small
5. $X \sim N(500, 80^2)$. The value below which 25\% of the distribution falls is closest to
(A) 420 (B) 480 (C) 446 (D) 554
6. For that same distribution, $\mathrm{P}(420 < X < 620)$ is closest to
(A) 0.841 (B) 0.775 (C) 0.933 (D) 0.593
7. Waiting time is exponential with mean 8 minutes. $\mathrm{P}(X > 10)$ is closest to
(A) 0.287 (B) 0.125 (C) 0.713 (D) 0.800
8. For that same exponential, the standard deviation is
(A) $\sqrt{8}$ (B) 64 (C) 4 (D) 8
9. $X$ is uniform on $[10, 50]$. Its standard deviation is closest to
(A) 20.00 (B) 11.55 (C) 13.33 (D) 6.67
10. For a continuous variable, which is NOT true?
(A) $\mathrm{P}(X = a)$ is a small positive number (B) $\mathrm{P}(X = a) = 0$ (C) $\mathrm{P}(X < a) = \mathrm{P}(X \le a)$ (D) the density can exceed 1
11. $t_{0.05,\,12}$ compared with $t_{0.05,\,30}$ is
(A) smaller (B) equal (C) larger only if $n < 10$ (D) larger
12. A $\chi^2$ distribution with 1 degree of freedom is
(A) symmetric about 1 (B) bell-shaped (C) right-skewed and bounded below by zero (D) bounded above by 1
13. $F_{0.05,\,8,\,20}$ and $F_{0.05,\,20,\,8}$ are
(A) different values (B) the same value (C) reciprocals (D) the same only at $\alpha = 0.05$
14. In Stata, ttail(20, 2.086) returns approximately
(A) 0.95 (B) 2.086 (C) 0.05 (D) 0.025
15. Which is the inverse of Stata’s invchi2(10, 0.95)?
(A) invchi2(10, 0.05) (B) chi2(10, 0.95) (C) chi2tail(10, 18.307) (D) chi2tail(10, 0.95)
16. A standardised binomial approaches the normal as $n$ grows because
(A) the binomial formula simplifies (B) a binomial is a sum of independent Bernoulli trials (C) $p$ approaches 0.5 (D) the variance approaches 1
17. $X \sim N(\mu, \sigma^2)$. The distribution of $(X - \mu)/\sigma$ is
(A) $N(\mu, 1)$ (B) $N(0, \sigma)$ (C) $N(0, 1)$ (D) $t$ with $n-1$ df
18. For a $t$ distribution, as $\nu \to \infty$ the critical value $t_{0.025,\nu}$ approaches
(A) 1.645 (B) 1.960 (C) 2.000 (D) 0
19. Which requires exactly one parameter?
(A) the exponential (B) the normal (C) the binomial (D) the uniform
20. A density curve reaches a height of 2.5 at its peak. This
(A) is impossible, since probabilities cannot exceed 1 (B) means the variable is discrete (C) means the curve is mis-scaled (D) is possible, since only the area must equal 1
1. D — $1 - \mathrm{P}(0) - \mathrm{P}(1)$. 0.655 is $\mathrm{P}(X \le 1)$, the complement — taken without subtracting from 1.
2. B — $\sqrt{20(0.4)(0.6)} = \sqrt{4.8}$. 4.80 is the variance, left un-rooted.
3. C — $np = 4$. Both $np$ and $n(1-p)$ must clear 5; here only one does.
4. A — 12 of 30 is a large fraction of the batch, so the probability shifts materially as items are removed.
5. C — $z = -0.6745$, so $500 - 0.6745(80)$. 554 uses $+0.6745$ and finds the 75th percentile instead.
6. B — $z = -1.00$ and $z = +1.50$, so $0.9332 - 0.1587$. 0.933 forgets to subtract the lower tail.
7. A — $e^{-10/8}$. 0.713 is $\mathrm{P}(X < 10)$, the complement.
8. D — For an exponential, $\sigma = \mu$ — unusual, and worth remembering.
9. B — $(b-a)/\sqrt{12} = 40/3.4641$. 20.00 is half the range, which is not the standard deviation of anything here.
10. A — Exactly zero, not merely small — which is why strict and weak inequalities agree.
11. D — Fewer degrees of freedom means fatter tails, so the cutoff sits further out.
12. C — It is a single squared standard normal, so it cannot be negative and its upper tail stretches.
13. A — $F$ is a ratio, so swapping the degrees of freedom asks a different question — 3.33 against 2.45 in this case.
14. D — ttail is cutoff-to-area and gives the upper tail. 0.05 would be the two-tailed total.
15. C — invchi2 takes an area and returns a cutoff; chi2tail takes a cutoff and returns an area.
16. B — Sums of many independent things go normal — the CLT, applied to a count.
17. C — Standardising always produces mean 0 and standard deviation 1. It becomes $t$ only when $\sigma$ is replaced by an estimate.
18. B — The normal’s two-tailed 5\% value. 1.645 is the one-tailed figure.
19. A — The others each take two: $(\mu,\sigma)$, $(n,p)$ and $(a,b)$.
20. D — A uniform on $[0, 0.4]$ has height 2.5 throughout.