Level 2 — RecallStatistics & Probability — Intermediate

Statistics & Probability — Intermediate

40 marksprintable — key stays hidden on paper

Level 2 — Recall & Standard Problems

Time: 30 minutes Total Marks: 40

Answer all questions. Show working where required. Give probabilities as fractions or decimals to 3 s.f.

Q1. The marks of 8 students are: $12, 15, 15, 18, 20, 22, 25, 25$ . (a) Find the mean. (2) (b) Find the median. (1) (c) State the mode. (1)

Q2. The following grouped data show the time (minutes) taken by 40 people to complete a task.

Time (min)	0–10	10–20	20–30	30–40	40–50
Frequency	4	10	14	8	4

(a) Estimate the mean. (3) (b) Estimate the median using the grouped median formula. (3)

Q3. For the data set $2, 4, 4, 6, 8, 10$ : (a) Find the variance. (3) (b) Find the standard deviation. (1)

Q4. For the ordered data $3, 5, 7, 8, 10, 12, 14, 16, 18$ : (a) Find the lower quartile $Q_1$ and upper quartile $Q_3$ . (2) (b) State the interquartile range (IQR). (1)

Q5. A fair die is rolled once. Let $A$ = "even number", $B$ = "number greater than 4". (a) Find $P(A)$ and $P(B)$ . (2) (b) Find $P(A \cup B)$ . (2)

Q6. State the three Kolmogorov axioms of probability. (3)

Q7. Events $A$ and $B$ are independent with $P(A)=0.6$ , $P(B)=0.5$ . (a) Find $P(A \cap B)$ . (1) (b) Find $P(A \cup B)$ . (2)

Q8. In a class, $P(\text{plays football})=0.7$ , $P(\text{plays football and cricket})=0.28$ . Given a student plays football, find the probability they also play cricket. (2)

Q9. (a) Evaluate $^{7}P_{3}$ . (1) (b) Evaluate $^{7}C_{3}$ . (1) (c) Find the general term $T_{r+1}$ in the expansion of $(x+2)^{6}$ and hence the coefficient of $x^{4}$ . (3)

Q10. A biased coin has $P(\text{head})=0.3$ . It is tossed 5 times. Let $X$ be the number of heads. (a) Write down the distribution of $X$ . (1) (b) Find $P(X=2)$ . (2) (c) State the mean and variance of $X$ . (2)

Answer keyMark scheme & solutions

Q1. (a) Mean $=\dfrac{12+15+15+18+20+22+25+25}{8}=\dfrac{152}{8}=19$ . (2) — sum (1), divide (1). (b) $n=8$ even; median = average of 4th and 5th values $=\dfrac{18+20}{2}=19$ . (1) (c) Mode $=15$ and $25$ (both appear twice) — bimodal. (1)

Q2. Midpoints: 5, 15, 25, 35, 45. (a) $\sum fx = 4(5)+10(15)+14(25)+8(35)+4(45)=20+150+350+280+180=980$ . Mean $=980/40=24.5$ min. (3) — midpoints (1), $\sum fx$ (1), divide (1). (b) $N/2=20$ . Median class = 20–30 (cf reaches 28 there; before it cf=14). $\text{Median}=L+\dfrac{N/2-CF}{f}\times h = 20+\dfrac{20-14}{14}\times10 = 20+\dfrac{60}{14}=24.29$ min. (3) — locate class (1), formula (1), answer (1).

Q3. Mean $=\dfrac{2+4+4+6+8+10}{6}=\dfrac{34}{6}=5.6667$ . Deviations squared: $(2-5.667)^2=13.44$ , $(4-5.667)^2=2.778$ (×2), $(6-5.667)^2=0.111$ , $(8-5.667)^2=5.444$ , $(10-5.667)^2=18.78$ . $\sum=13.44+2.778+2.778+0.111+5.444+18.78=43.33$ . (a) Variance $=43.33/6=7.222$ . (3) — mean (1), squared devs (1), divide (1). (b) SD $=\sqrt{7.222}=2.687$ . (1)

Q4. $n=9$ , median is 5th value = 10. (a) Lower half (below median): $3,5,7,8$ → $Q_1=\dfrac{5+7}{2}=6$ . Upper half: $12,14,16,18$ → $Q_3=\dfrac{14+16}{2}=15$ . (2) (b) IQR $=Q_3-Q_1=15-6=9$ . (1)

Q5. (a) $A=\{2,4,6\}$ , $P(A)=3/6=1/2$ ; $B=\{5,6\}$ , $P(B)=2/6=1/3$ . (2) (b) $A\cap B=\{6\}$ , $P=1/6$ . $P(A\cup B)=1/2+1/3-1/6=4/6=2/3$ . (2)

Q6. (3) (1 each)

For any event $A$ , $P(A)\ge 0$ (non-negativity).
$P(S)=1$ (sample space has probability 1).
For mutually exclusive events $A,B$ : $P(A\cup B)=P(A)+P(B)$ (countable additivity).

Q7. (a) $P(A\cap B)=P(A)P(B)=0.6\times0.5=0.3$ . (1) (b) $P(A\cup B)=0.6+0.5-0.3=0.8$ . (2)

Q8. $P(C\mid F)=\dfrac{P(C\cap F)}{P(F)}=\dfrac{0.28}{0.7}=0.4$ . (2) — formula (1), answer (1).

Q9. (a) $^{7}P_{3}=7\times6\times5=210$ . (1) (b) $^{7}C_{3}=\dfrac{210}{6}=35$ . (1) (c) $T_{r+1}={6\choose r}x^{6-r}2^{r}$ . For $x^4$ : $6-r=4\Rightarrow r=2$ . Coefficient $={6\choose 2}2^{2}=15\times4=60$ . (3) — general term (1), $r=2$ (1), coefficient (1).

Q10. (a) $X\sim B(5,\,0.3)$ . (1) (b) $P(X=2)={5\choose2}(0.3)^2(0.7)^3=10\times0.09\times0.343=0.3087$ . (2) (c) Mean $=np=5\times0.3=1.5$ ; Variance $=npq=5\times0.3\times0.7=1.05$ . (2)

[
{"claim":"Q2a mean of grouped data is 24.5","code":"fx=4*5+10*15+14*25+8*35+4*45; result=(Rational(fx,40)==Rational(49,2))"},
{"claim":"Q2b grouped median is 20+60/14","code":"med=20+Rational(20-14,14)*10; result=(med==Rational(60,14)+20)"},
{"claim":"Q5b union probability is 2/3","code":"result=(Rational(1,2)+Rational(1,3)-Rational(1,6)==Rational(2,3))"},
{"claim":"Q10b P(X=2)=0.3087","code":"p=binomial(5,2)*Rational(3,10)**2*Rational(7,10)**3; result=(p==Rational(3087,10000))"},
{"claim":"Q9c coefficient of x^4 is 60","code":"result=(binomial(6,2)*2**2==60)"}
]