Exam May 2014, questions PDF

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PAPER CODE NO. MATH 162

EXAMINER: Dr. Kai Liu, TEL.NO. 44759 DEPARTMENT: Mathematical Sciences

MAY 2014 EXAMINATIONS

Introduction to Statistics

Time allowed: Two and a half hours

INSTRUCTIONS TO CANDIDATES: Answer all questions in Section A. The marks for the best three answers in Section B will be used in the assessment.

Paper Code MATH 162

Page 1 of 8

CONTINUED

SECTION A 1. A research laboratory has a small amount of a new material to analyse. They perform 10 measurements of its melting point, obtaining the following values (in degrees Celsius): 853.8, 853.9, 853.3, 853.8, 854.0, 853.7, 853.8, 853.6, 853.9, 853.8. Find (i) the median; (ii) the interquartile range;

[2 marks] [3 marks]

(iii) the mean; [2 marks] (iv) State, giving reasons, whether any of these data could be regarded as an outlier. [1 marks] 2. Assume that E, F and G are three events satisfying: E and F are independent; F and G are independent; E and G are exhaustive. Express the following probabilities in terms of P (E), P (F ) and P (G): (i) (ii) (iii) (iv)

P (E ∩ F ); P (F ∪ G); ¯ P (G).

[1 marks] [2 marks] [2 marks]

P (E ∪ (F ∩ G)).

[3 marks]

3. Packets of washing powder are filled automatically. It has been found in the past that 3% are underweight. Assuming that the weights of packets are mutually independent, write down an expression for the probability that there are x underweight packets in a sample of size 8. [2 marks] An inspector takes a random sample of 8 packets. What is the expected number of packets that are underweight? [2 marks] Also calculate the probability that: (i) none of the packets is underweight; (ii) more than one of the packets are underweight.

[2 marks] [2 marks]

4. A sample of a radioactive material emits alpha particles. The number X of alpha particles emitted per second follows a Poisson distribution with mean 5. (i) Write down an expression for P (X = x). (ii) Evaluate P (X = 2). (iii) Evaluate P (X ≥ 2).

[3 marks] [1 marks] [2 marks]

(iv) Evaluate P (X 6= 2).

[2 marks]

5. A random variable X is normally distributed with mean 7 and variance 4. Calculate: (i) P (X > 7); (ii) P (X < 10);

[1 marks] [1 marks]

(iii) P (4 < X < 10); (iv) P (3 < X < 4.6);

[2 marks] [2 marks]

(v) The value of a such that P (X > a) = 0.3085.

[2 marks]

SECTION B 6. (a) The diagram below shows a network of unreliable components. For each component X, the event that X is working has probability P (X), independent of whether or not any other component is working. Calculate the network reliability (that is, the probability that there is a path from start to finish which contains working components only) given that P (A) = 0.7,

P (B) = 0.9,

P (G) = 0.8,

P (H) = 0.9

P (J) = 0.8,

P (M) = 0.9,

P (N ) = 0.8,

P (Q) = 0.9. [10 marks]

(b) A college has 5000 students of whom 1500 study Arts subjects and 3500 study Science subjects. Of the Arts students, 60% are female, and of the Science students 20% are female. (i) What is the probability that a randomly selected student will be female? [5 marks] (ii) If a randomly selected student is female, what is the probability that she is an Arts student; what is the probability that she is a Science student? [5 marks]

7. (a) It has been found that 5% of items made on a particular machine are “defective” and will be scrapped. An inspector randomly selects a sample of 76 items. Letting random variable X be the number of defectives in the sample, evaluate P (X = 3). [3 marks] A further 10% of the items produced are nondefective but will need modification before they can be sold. Calculate the probability that: (i) the sample contains 3 defectives and 2 nondefectives requiring modification? [6 marks] (ii) there will be 2 nondefectives requiring modification given that the sample contains 3 defectives. [3 marks] (b) Approximate the Binomial distribution for the random variable X of part (a) by a Normal distribution, and hence determine (approximately) the probability that the number of defectives will lie in the ranges 3 ≤ X ≤ 6. [6 marks] State, giving reasons, whether you would expect a Poisson approximation to the distribution of X to lead to a more or less accurate value for P (3 ≤ X ≤ 6). [2 marks]

8. (a) Under which two conditions does a sequence of numbers p0 , p1 , · · · describe a valid discrete probability distribution? Under what two conditions does a function f (·) : R → R describe a valid density function of a continuous random variable? [4 marks] (b) For this question you may use without proof the result that: Z ∞ 0

λxn e−λx dx = n!/λn ,

n = 0, 1, · · · , λ > 0.

The random variable Y has probability density function f (y) = λe−λy if 0 ≤ y < ∞ and f (y) = 0 if y < 0. (i) Obtain the mean and variance of Y in terms of λ. (ii) If the variance of Y is equal to 4.0, find the value of λ.

[8 marks] [2 marks]

(iii) For λ = 0.5, determine P (2 < Y < 8).

[6 marks]

9. (a) It is observed that flowers of the species Flora vulgaris may be blue, red, white or yellow. A geneticist believes that blue, red, white and yellow flowers should occur in the ratios 1 : 3 : 6 : 10. He collects a random sample of 120 flowers and finds the following numbers of flowers of each of the four possible colours. Blue Red White Yellow 10 25 33 52 Use a χ test to decide whether the geneticist’s theory is consistent with the observed results. [10 marks] (b) It is suspected that a respiratory disease might be caused by contact with cats. A random sample of 1000 people is found to have 300 sufferers of the disease. The ownership of cat(s), other pet(s) or no pets is given below. 2

Cats Other pets No pets Sufferers Nonsuffers

51 99

147 403

102 198

Test at the 5% significance level the hypothesis that the incidence of the respiratory disease is independent of whether a person owns a cat, other pet or no pet. [10 marks] 10. (a) Given a random sample of n observations from a normal distribution with mean µ and variance σ 2 , derive the mean and variance of the sample ¯ [8 marks] mean, X. (b) The following are the systolic blood pressures (mm Hg) of a random sample of 30 patients undergoing drug therapy for hypertension: 183 111 176 152 158 200 178 124 130 157 159 134 210 120 148 163 187 168 144 190 147 200 188 190 178 165 164 152 173 150 and the sample mean x¯ = 163.3. (i) It is believed that the systolic blood pressure in healthy people is normally distributed with mean 125 (mm Hg) and standard deviation 15 (mm Hg). Also, it is known that the standard deviation of the systolic blood pressure of patients undergoing drug therapy for hypertension is 15 (mm Hg). Give a 95% confidence interval for the mean systolic blood pressure of patients undergoing drug therapy for hypertension. Comment on whether there appears to be evidence to suppose that patients with drug therapy for hypertension have a higher systolic blood pressure than healthy people. [6 marks] (ii) Find the systolic blood pressure, x, such that the probability that the average systolic blood pressure for a random sample of 30 healthy people is greater than x is 0.1. [6 marks]...