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2 ANSWER ALL QUESTIONS Question 1: (25 Marks) A random sample of 35 homeowners was taken from the village Penville and their ages were recorded (i) Illustrate the results in a stem and leaf diagram. (2 Marks) (ii) On the graph paper provided, draw a box plot to represent the data. Show clearly any outliers. (6 Marks) (b) (i) State and prove the Bayes Theorem. (8 Marks) Page 2 of 7
3 (ii) In a bolt factory, machines A, B and C manufacture respectively 25%, 35% and 40% of the total. Of their output respectively 5, 4 and 2 per cent are defective bolts. A bolt is drawn at random from the output and is found to be defective. What is the probability that it was manufactured by machine B? (c) Find the probability that in a group of 18 people (that includes no twins) at least two have the same birthday that is, will have been born on the same day of the same month but not necessarily in the same year. (5 Marks) Page 3 of 7
4 Question 2: (25 Marks) A bakery has 7 different types of items, with lots of each type available. How can 12 baked goods be selected? (2 Marks) (b) Consider the toss of two coins: a 5p piece and a 10p piece. Let A be the event of the 5p coin showing head, B be the event of the 10p coin showing head and C be the event of them being either both heads or tails. Discuss the independence of A, B and C. (6 Marks) (c) Suppose that all the cards in a deck of n different cards are placed in a row, and the cards in another similar deck are then shuffled and placed in a row on top of the cards of the first deck. Find the probability that there will be at least one match between the corresponding cards from the two decks. (10 Marks) (d) A and B throw alternatively a pair of dice. A wins if he obtains a sum of 6 before B throws 7 and B wins if he throws 7 before A throws 6. Find their respective chances of winning, if A begins. (7 Marks) Page 4 of 7
5 Question 3: (25 Marks) The following table shows the number of plants having certain characters: Flatness of leaf Flat leaves Curled leaves Total White Flowers Colour of flower Red Flowers Total At 10% level of significance, test whether the flower colour is independent of flatness of leaf. (7 Marks) (b) A random variable has a normal distribution with mean µ and standard deviation 3. The null hypothesis µ = 20 is to be tested against the alternative hypothesis µ > 20 using a random sample of size 25. It is decided that the null hypothesis will be rejected if the sample mean is greater than (i) Find the probability of making a type I error. (3 Marks) (ii) Find the probability of making a type II error, when µ = 21. (2 Marks) Page 5 of 7
6 (c) A company produces small jars of coffee. It is known that the weights of the jars are normally distributed with standard deviation 4.8g. The manager is to take a random sample. He wishes to ensure that there is at least a 95% probability that the estimate of the population mean is within 1.25g of its true value. Compute the minimum sample size required. (d) A sample of 6 persons in an office respectively revealed an average daily smoking of 10, 12, 8, 9, 16 and 5 cigarettes. Find a 90% confidence interval for the average level of smoking in the whole office. (9 Marks) Page 6 of 7
7 Question 4: (25 Marks) Suppose that X is a random variable having a Poisson distribution. (i) State the probability mass function of X. (1 Marks) (ii) Derive the mean and the variance of the Poisson distribution. (13 Marks) (iii) Given that the third central moment and fourth central moment are m and 3m 2 + m respectively. Find the coefficient of skewness and the coefficient of kurtosis of the Poisson distribution. (b) The number of defects per metre in a roll of cloth has a Poisson distribution with mean Find the probability that (i) The total number of defects in a randomly chosen 6 metres length of cloth is more than 2. (3 Marks) (ii) A tailor buys 300 metres of cloth. Using a suitable approximation, calculate the probability that the tailor s cloth will contain less than 90 defects. ***END OF PAPER*** Page 7 of 7
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