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SECTION A
- Survival times ( in days from diagnosis) are listed below for 20 patients with chronic granulocytic leukemia:
16, 21, 22, 6, 40, 2, 10, 26, 14, 13, 15, 35, 2, 8, 4, 6, 19, 4, 14, 18.
(a) Display the above data as a stem-leaf plot. [4 marks] (b) Describe the shape of the distribution. [2 marks] (c) Calculate the median survival time. For this data, would you prefer the median or the mean as a measure of location? Explain your answer. [2 marks]
- A continuous random variable has probability density function given by
for some constant
(a) Show that the value of the constant is. [2 marks] (b) State the probability of [1 mark] (c) Calculate [2 marks] (d) Calculate the variance of [3 marks]
- A roulette wheel is divided into six sectors of unequal area, marked with the numbers 1, 2, 3, 4, 5, and 6. The wheel is spun and is the random variable ‘the number on which the wheel stops’. The probability distribution of is as follows:
1 2 3 4 5 6 1/16 3/16 1/4 3/16 1/
(a) Find the value of and the mean of [2 marks] (b) Find the standard deviation of [3 marks] (c) Calculate the conditional probability [3 marks]
- State conditions under which the binomial distribution, B(n,p), can be approximated by a normal distribution. [1 mark] At a particular hospital, records show that each day, on average, only 80% of people keep their appointment at the outpatients’ clinic. Denote by the number of people keep their appointment on a day when 200 appointments have been booked. Assuming that follows the binomial distribution, B(200, 0.80).
(a) Verify that the binomial distribution can be approximated by a normal distribution. [1 mark] (b) Find the probability that more than 170 patients keep their appointments. [2 marks] (c) Calculate the probability that the number of people keep their appointment, , lies in the range [4 marks]
SECTION B
(a) Events and are such that and.
(i) Explain why the events and are not independent.
(ii) Find the value of
[5 marks]
(b) Two events and are such that
, , calculate the probability that
(i) both events occur, (ii) only one of the events occurs, (iii) neither event occurs. [7 marks]
(c ) A system consists of six components as illustrated below. Components 2 and 3 are connected in parallel, so that subsystem works if either 2 or 3 works. Since components 4 and 5 are connected in series, that subsystem works if both 4 and 5 work. Components work independently of one another and the probability any given component works is 0.9. Calculate the probability that the system works from START to FINISH.
[8 marks]
4 5
2
3 START 1 6 FINISH
- A certain type of flashlight is sold with the five batteries included. A random sample of 100 flashlights is obtained, and the number of defective batteries in each is determined, resulting in the following data:
Number of defective, 0 1 2 3 4 5 Frequency 1 6 14 33 31 15
It is suggested that the above data might be expected to follow a binomial distribution.
(a) On the assumptions that the actual number of defectives, , follows a binomial distribution with parameters and , estimate. [4 marks]
(b) Calculate the binomial probabilities for where is a random variable following the binomial distribution with and success probability equal to the value estimated in part (a) above. [6 marks]
(c) Use a goodness of fit test at the 5% level to assess whether the binomial distribution is appropriate for the given data. [10 marks]
(a) A study of the relationship between facility conditions at gasoline stations and aggressiveness in the pricing of gasoline reports the accompanying data based on a sample of stations.
Observed pricing policy Aggressive Neutral Nonaggressive Total
Condition
Substandard 24 15 17 56 Standard (^52 73 80) 205 Modern 58 86 36 180 Total 134 174 133 441
Test at the 5% significance level the hypothesis that there is no association between the facility condition and the pricing policy. [10 marks]
(b) An unmanned monitoring system uses high-tech video equipment and microprocessors to detect intruders. A prototype system has been developed and is in use outdoors at a weapons munitions plant. The system is designed to detect intruders with a probability of 0.90. However, the design engineers expect this probability to vary with weather condition. The system automatically records the weather condition each time an intruder is detected. Based on a series of controlled tests, in which an intruder was released at the plant under various weather conditions, the following information is available: Given the intruder was, in fact, detected by the system, the weather was clear 75% of the time, cloudy 20% of the time, and raining 5% of the time. When the system failed to detect the intruder, 60% of the days were clear, 30% cloudy, and 10% rainy. Use this information to find the probability of detecting an intruder, given rainy weather conditions.(Assuming that an intruder has been released at the plant.)
[10 marks]
(a) For a random variable which is Poisson distributed with parameter , show that the
expectation is given by.
Show also that the variance of is equal to. [11 marks]
(b) The school photocopier breaks down, on average, eight times during the school week
(Monday to Friday). Assuming that the number of breakdowns in any period can be
modelled by a Poisson distribution, find the probability that it breaks down
(i) five times in a given week,
(ii) once on Monday,
(iii) eight times in a fortnight.
[9 marks]
