Question 1

In this question, you will need to provide the R-codes used and the various relevant outputs from R.

• Apply R to simulate a set of 100 numbers, with a mean value of 20 and a standard deviation of 2. List out the set of numbers.
• Apply and construct a suitable graphical display of this generated set of 100 numbers to visualize the distribution shape of the data set. Hence provide appropriate comments on the distribution of the data.
• Add another 2 numbers to the set simulated in Question 1(a), such that the new set now has a mean of 20, but the range becomes 200. List out the set of numbers.
• Using R, for the new data set of 102 numbers,
  (i) Determine an estimate of the standard deviation,
  (ii) Apply and construct a histogram for the data set and comment on its distribution.

Question 2

In this question, show all working details of the steps carried out to determine the answers. Jenny has ten cards, which are labeled

• By arranging these ten cards, a “word” is formed. Example: MYDFMXGFFB
  (i) Determine the number of different “worlds” that Jenny can make with the ten cards.
  (ii) Determine the number of “words” in Question 2(a)(i) that begin and end with F, and in which the two letters M are consecutive?
• Fadli chooses three of Jenny’s ten cards at random. Compute the probability that
  (i) Fadli’s three cards carry the letters F or M.
  (ii) Jenny’s remaining seven cards have no repeated letters.

Question 3

A bag contains 8 identically sized balls, 6 black, and 2 white.

• Balls are drawn out one at a time, at random without replacement. The Wth draw is when a white ball is first drawn.
  (i) Comment on and explain why is W a random variable.
  (ii) Apply the stated model and construct the probability distribution of W
  (iii) Show that 𝐸[𝑊] = 3.
  (iv) Calculate 𝑉(𝑊).
• If each drawn ball is now replaced before another is drawn, show that 𝐸[𝑊] = 4

Question 4

• A telemarketer with a success rate of 10% wishes to be at least 95% confident of making a minimum of 2 sales each day. Determine the number of calls the telemarketer should make each day? Comment on the method used and
  show your detailed work.
• The random variable X is the number of successes in n independent trial of an
  an experiment in which the probability of success at any trial is p.(i) Show that
  
  (ii) Apply the result from Question 4(b)(i) above to determine the most probable number of successes when 𝑛 = 100, 𝑝 = 4/7.

Question 5

• A private psychology clinic has two resident clinical psychologists, Susan and Steven. For booking each day, Susan has six appointment slots and Steven has three. The average number of demands for consultation with Susan is three on any weekday, and five on any weekend, while that with Steven is two on any day of the week.
  Assuming these demands follow Poisson distributions and are independent.
  (i) Compute the probability there is at least one demand for consultation with Susan on a particular Tuesday.
  (ii) Compute the probability that there are demands for consultation with Steven not met on both Thursday and Friday of a particular week.
  (iii) Compute the probability that all available slots for consultation with either
  Susan or Steven (but not both) are filled on a certain Saturday.
• The stock of a branded product at a local shop is being replenished regularly throughout the day. Demands from customers are independent and each customer is allowed to purchase only one copy of the product. Demands from the customers occur singly and at a rate of 2 per 30 minutes intervals.