Stats Definition and Theory Handout
2017 Specimen Stats Comments
 5(i) The “2!” in your working rearrange the couple among themselves. But we have 6 different couples, all of whom can rearrange themselves so we should have $2! \times 2! \times \cdots \times 2! = (2!)^6$. This gives us the final answer of $6! \times 2^6 = 46080$.
5(ii),(iii) We’d discuss them together during class. 
7(i) The two assumptions we usually use are (1) constant probability and (2) each event being independent. For most of the time having a fixed number of trials is usually clear in the question, so we do not need an assumption there.
6(ii) Good
6(iii) Well done on observing that 2 being the mode means that $P(X=2) > P(X=3)$. However, there is also the other side where $P(X=2) > P(X=1)$. Try this side out to get the final answer of $\frac{2}{7} < p < \frac{3}{7}$. 
8(ii) Rather than expanding for tough looking algebra you may want to consider using graphs to straightaway jump from $\frac{6(3y+1)}{(4+y)(3+y)(2+y)}=\frac{7}{20}$ to the answer $y=2$.
8 Well done otherwise for the rest of the question.  9: Well done except 9(iii) seems to be missing
 10 Well done. We’d discuss 10(v) together in class
2018 Stats Comments
 5(i) Good. The best answer will elaborate more on what “random” means: that the probability of a fan being chosen is the same for each fan and that the fans are independently selected.
5(ii) Correct up until the line $\sigma / \sqrt{43} > 468.1267609$. Careless mistake in the next step (should have multiplied but went to divide instead). Also, the final answer asks for variance, so we need to square the final result to get $\sigma^2 > 9\,420\,000$.  6,7: Update me after you’ve attempted them from our discussions/the videos I uploaded. We can discuss this during class too. Q7(i),(ii) calculations are good.

8(i)(ii): Good
8(iii) Give it a try again based on what I sent over text. See if you can get the answer of $g(n)=22n^2+78n+36$. 
11(i): We’d discuss in class
11(ii)(v) Good
2019 Stats Comments
 6(i) Good.
6(ii) (Theory: We want our sample to be random. That is, each member of the population must have equal probability of being selected. And each selection should be independent of each other).
He could obtain a sample of 50 supporters to interview. The sample should be random (by selecting 11, 12, 13 and 14 supporters from the clubs in Division One, Two, Three and Four respectively). Taking a random sample ensures that prevents bias in our results.
6(iii) Multiplying is the correct approach. The answer is $7.24 \times 10^{18}$. 
7(i),(ii): Good
7(iii). Notice that there is a new random variable about the number of days in which there are “at least 7 faulty mugs”. So we have $Y \sim B(5,0.10187)$. and the answer is $P(Y \leq 2) = 0.991$.
7(iv) When there is 1 faulty mug, probability should be $(10.08)(0.08)\times 2!$ (you missed out the 2! to account for the arrangement of faulty vs nonfaulty mugs). Similarly we missed out the 2! for case 3 for 1 faulty saucer. Make the change and see if you can get the answer of 0.0689 (no need to expand. use the GC graph when equations get complicated). 
8(i)Good
8(ii)(a) Since both cases are identical (neither is orange, there is no need for 2!. Or, we can use the identical formula to get 2!/2! which is 1). For cases like one orange, one nonorange we will need the 2!.
8(ii)(b) You missed out the case of one yellow dog and a nonyellow nondog. See if adding that will get us the answer of $\frac{21}{110}$.
We can also combine the 3 cases you have into 1 case since the question doesn’t care about horse, rider, nor birds. So the 3 cases can be combined as “one nonyellow dog and a yellow nondog” for $\frac{10}{56} \times \frac{7}{55} \times 2!$.
8(iii) Send me what you have when you’re done/or if you want hints/a discussion 
9(i) Good.
9(ii): Circled the mistake in whatsapp text.
9(iii): Since we no longer know if the population is normally distributed, in order to carry out our test we need to ensure that CLT can be used so that $\overline{X}$ is normally distributed approximately. Hence we need to take a large sample size $(n \geq 30)$ vs the original test that only took a sample size of 8. 
11(i)(iii) Good.
11(iv) Send me what you have when you’re done/or if you want hints/a discussion
Recommended Schedule
 Thursday: 2016 P1 Q9,10,11
 Friday: 2016 P2 Q1,2,3
 Saturday: 2017 P1 Full paper
 Sunday: 2017 P2 Q14
 Monday: Breather/buffer
 Tuesday: 2017 P2 Q510
HW 23rd September
TYS
Stats Practice
 Stats – NYJC 2019 Skip Q4,8.
 Stats – SAJC 2019 Skip Q4,9.
 Stats – TMJC 2019 Skip Q8.
 Answers: Look at them only after you complete each paper. I can help you mark as well if you send your working to me.
 Pure Math Revision (optional)
 Functions and graphs: Q10 3_Functions_2020
 Integration: Q4 Stats – TMJC 2019
 Parametric: P2Q3 JPJC 2019 Prelims
 Vectors: P1Q3 JPJC 2019 Prelims

Sigma Notation: Q5b, 6 T1 Ineq, Eq, APGP, Sigma
There is a typo in 6c) instead of $\displaystyle \sum_{r=5}^20 \ldots$ it should be $\displaystyle \sum_{r=5}^{20} \ldots$
Prelim Papers
CJC 2019 Prelims
JPJC 2019 Prelims
Integration, volumes, DE discussion questions
Google Docs Link
Integration, volume, DE selected questions – Google Docs
2019 CJC Prelims
CJC 2019 Prelims
Recommended questions before next class: P1 Q2,4,5,6,7,8,9. P2 Q1,2,3,4,5(i),(ii),(iii),(v).
HW questions on vectors/hypothesis testing
HW questions on complex numbers
HW questions on transformations and Maclaurin series
HW 25th26th July
HW 9th12th July
5th July
HW 29th June5th July
HW 5th12th June
 Calculus module 4: Q5 (DE)
 Complex module 5: Q3
 Probability and p and c module 6a: Q45
 PnC, Integration
HW 29th May
 Calculus module 4: Q5 (DE)
 Complex module 5: Q13
 Probability and p and c module 6a: everything
Plan for 2027th May
 Normal Q3aii, 5iii, 5iv
 Integration Q1b,4i
 Binomial + DRV
HW Plan
 10th17th May: Module 2 Graphs Q2,3a(ii),3b,5,7. Module 6c Normal distribution Q15
 17th24th May: Module 4 Calculus. Module 6b DRV, Binomial distribution
 24th31st May: Module 5 Complex numbers. Module 6a PnC, Probability
 31st May – June NYJC Prelims
Plan for 10th May
 Review any questions from last week/school/tutorial
 Selected TYS questions (stats)
 ListMF26
HW 6th10th May
 3 normal and sampling questions: Normal and sampling worksheet
Plan for 3rd May
 Review any questions from last week/school/tutorial
 Start on sampling
 Discuss holiday revision schedule
HW for 26th April3rd May:
 Q3,6,7 S5_Normal
 Binomial: HW_Binomial2
Plan for 26th April:
 Review any questions from last week/school/tutorial
 Past prelim normal questions: S5_Normal
HW for 22nd26th April:
 Normal Basic Questions Q1b, 2, 7: Normal Basic Questions
 Binomial: HW_Binomial2
HW for 20th22nd April:
Plan for 18th April:
 Review any questions from last week/school/tutorial
 Binomial: discussions on mode and using formula: Q6bi, 10 S4_Binomial_2020
 Start discussion on Normal: Normal Summary Notes
 Basic questions: Normal Basic Questions
 Past prelim normal questions: S5_Normal
Plan for 12th April:
 Review any questions from last week/school/tutorial
 Start discussion on Binomial: DRV Binomial notes
 Basic binomial questions: Binomial Basic Questions
 Past prelim binomial questions: S4_Binomial_2020
Zoom meeting link
Meeting ID: 417 847 7601
Password: 117964
HW 5th12th April:
Plan for 12th April:
 Review any questions you have
 Discuss binomial distribution
 Linear combinations of DRV
Google Drive I will be storing files in for this period of time: Google Drive link
Plan for 5th April:
 Review any questions you have
 Work on and discuss Probability Worksheet: Q17, 1215 together.
 Summary notes on probability: Probability Summary Notes
 Start on the new topic: DRV.
 Basic concepts of DRV
 Pdf (table/function)
 $E(X)$, $Var(X)$ and related formulas.
 Linear combinations of DRV: e.g. $2X+3$, $X5Y$.
 Past prelims worksheet on DRV: DRV Worksheet
 Selected questions for DRV: Q17.
For the probability worksheet, the questions with asterisks (**) are from the next topic of DRV. Q2c is a tough question. I’ve explained the technique in tackling these sort of questions in two TYS examples in the following youtube videos. You can either look at those videos or ask me to explain to you directly.