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M130: Introduction to Probability and Statistics
Tutor Marked Assignment
Cut-Off Date: Dec 7th, 2013 Total Marks: 60
The TMA covers only chapters 1, 2, 3 and 4. It consists of six questions for a total of 60 marks. Please solve each question in the space provided. You should give the details of your solutions and not just the final results.
Q−1: [2+3+3+1+1 Marks] Below are the earnings, for a particular week, of 15 staff (including the owner) working in a small business:
Weekly earnings (£) 280 370 305 285 480 1,260 210 340 280 290 315 325 370 360 280
a) Calculate the mean, median and mode.
b) Calculate the lower and upper quartiles, and find the inter-quartile range.
c) Calculate the standard deviation of the earnings.
d) Find the range of the earnings.
e) Why might the range be an unhelpful measure of spread for these particular data?
Q-2: [4+3+3Marks]
a) You are considering 10 different colleges. Before you decide to apply to the colleges, you want to visit some or all of them. In how many orders you can visit:
i- 6 of the colleges
ii- All ten colleges
b) How many permutations are there of the letters in the word BANANA?
c) A committee of 3 juniors and 3 seniors is to be formed from the members of a club having 18 juniors and 15 seniors. How many different committees are possible?
Q-3: [3+3+1+3Marks] Suppose that city records produced the following probability data on a driver being an accident on the last day of a Memorial Day Weekend.
Accident (A) No accident (A’) Total
Rain (R) 0.025 0.335 0.360
No rain (R’) 0.015 0.625 0.640
Total 0.040 0.960 1.00
a) Find the probability of accident rain or no rain.
b) Find the probability of rain, accident or no accident.
c) Find the probability of an accident and rain.
d) Find the probability of an accident, given rain.
Q-4: [2+2+3+3Marks] A manufacturer has 3 machines , make 20%, 30% and 50%, respectively, of the products. It is known from past experience that 2%, 3% and 4% of the products made by each machine respectively are defective. If a finished product is randomly selected:
a) What is the probability that it is defective?
b) What is the probability that it is not defective?
c) If the product is defective, what is the probability it was made by machine ?
d) If the product is not defective, what is the probability that it was made by machine ?
Q-5: [2+2+3+2+3Marks] A coin is tossed three times. Let X be a random variable representing the number of heads in three tosses of a coin. The coin is unbalanced and a head has a 60% chance of occurrence.
a) List the possible elements of the sample space S for the three tosses of the coin and to each sample point assign a value x of X.
b) Find the probability distribution and the cumulative distribution of the random variable X.
c) Find
d) Find the mean of the random variable X.
e) Calculate the variance and standard deviation of the random variable X.
Q-6: [2+2+2+3+3Marks] The density function for a random variable X is given by where k is a constant
a) Find k.
b) Find
c) Find p(x
