代写 MXB107 assignment Statistical Models for Data

MXB107 Semester 1 2016 Assignment 2 Question Sheet Page 1 out of 2

MXB107 Semester 1 2016 Assignment 2 Question Sheet Page 1 out of 2

Queensland University of Technology

ience and Engineering Faculty

School of Mathematical Sciences

MXB107 – Statistical Models for Data: Relationship and Effects

Assignment #2 – Semester 1, 2016 – Question Sheet

Due: by 5:00pm on Friday 22nd April 2016.

Please submit your answers to this assignment in the MXB107 Assignment Box located

on the 6th Floor of O-Block. Please ensure you attach a signed Official Assignment

Cover Sheet and include your tutorial time and tutor’s name.

Please do NOT submit your answers via Assignment Minder!!!

Question 1 [4.5 marks]

Consider two continuous variables: X ~ Unif(0, 10), with f X (x) = 0.1, 0 ≤ x ≤ 10 and

Y ~ Exp(0.2), with f Y (y) = 0.2exp(–0.2y), y ≥ 0. Note that E(X) = E(Y) = 5.

a) Find Pr(X>a) and Pr(Y>a) for a = 5, 6, 8, 9.

b) Hence find Pr(X>a+1|X>a) for a = 5, 8. [Hint: The events A = (X>a+1)&(X>a) and

B = (X>a+1) are the same. (Can you see why?)] How do these probabilities compare

with Pr(X>1)?

c) Likewise, find Pr(Y>a+1|Y>a) for a = 5, 8 and compare with Pr(Y>1).

d) Comment briefly on how your results from parts (a) and (b) differ.

Question 2 [2.5 marks]

Let Z be a standard normal random variable and consider the equation

Pr{a < Z ≤ b} = 0.90.

a) If b = 2, find a. Similarly, if b = 2.5, find a.

b) Are there values of b for which the equation has no solution for a? If so, what are

they?

c) Consider all the values of b for which there is a corresponding value of a. What

choice of b do you think makes the range (a, b) shortest (i.e., makes b – a smallest)?

Question 3 [2 marks]

An established market research consultancy has decided to check its own market

profile. Out of a random sample of 100 local residents, they find that 42 have heard of

their brand name.

a) Based on this sample, construct a 99% confidence interval for the true proportion p

of local residents who have heard this brand name.

b) After constructing this interval, chief consultant feels that it is too wide to b useful and would like to shorten the interval; however, she does not want to chang the confidence level and decides instead to increase the sample size. Assuming the

observed proportion stays the same, how large a sample is needed to ensure an

interval with width of no more than 0.02?

MXB107 Semester 1 2016 Assignment 2 Question Sheet Page 2 out of 2

Question 4 [3 marks]

Consider the following inspection scenarios:

(1) identify which of 50 nominally identical items have manufacturing defects, when it

has been established that 4% of items from this production line typically have defects;

(2) inspect 50km of copper wire for flaws, when flaws have been found to occur

randomly in this type of wire at an average rate of one flaw per 25km.

Let X be the number of defective items found in scenario (1) and Y be the number of

flaws found in scenario (2). Note that E(X) = E(Y) = 2.

a) Given that X ~ Bin(50, 0.04), calculate Pr(X=0), Pr(X=1) and Pr(X>1).

b) State a realistic distribution for Y and calculate Pr(Y=0), Pr(Y=1) and Pr(Y>1).

c) Now consider another variable Z ~ Bin(500, 0.004). Based on your findings from (a)

and (b), estimate Pr(Z=0), Pr(Z=1) and Pr(Z>1) to 3 decimal places.

Question 5 [3 marks]

[Note: the context of this question was framed by last semester’s lecturer, Prof. Steve

Stern, but I liked it so much that I have only changed the numbers.] Believe it or not,

while writing this assignment, I traveled forward in time and collected a random sampl of 25 submissions for this assignment. The names on the assignments did not survive

the return time-trip, but I could still calculate the average total score on the sampled

assignments, which was 10, and their sample standard deviation, which was 1.8. On the

basis of this information:

a) Find a 95% confidence interval for the actual average score μ for the entire class.

b) Construct a 99/95 tolerance interval for your score on this assignment (i.e., an

interval that includes 99% of possible observations with 95% confidence). Discuss

any issues.

代写 MXB107 assignment Statistical Models for Data

代写 MXB107 assignment Statistical Models for Data

c) While collecting the sample, I met my future self, who told me the actual mean and

standard deviation of scores for all submitted assignments. Unfortunately, the

time-travel sickness I experienced on my return trip to the present meant I wa only able to remember the standard deviation, which was 2.0. Given this

information, how would you change the interval you calculated in part (a), if at all?

代写 MXB107 assignment Statistical Models for Data

代写 MXB107 assignment Statistical Models for Data