MTH219: Fundamentals of Statistics and Probability 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 s

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