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This is the Past Exam of Mathematics which includes Determine, Represents, Angle, Measured, Formula, Measured in Degrees, Value, Exact Value, Formula, Numerically etc. Key important points are: Data, Response Times, Integrated Circuits, Interquartile Range, Patien, Sample, Treated, Particular Disease, Split, Other Group
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MATH 161 — Jan. 2010
EXAMINER: Dr A.Piunovskiy, Extension 44737
Time allowed: Two and a half hours
Answer all of Section A and THREE questions from Section B. Only the best three answers from Section B will be taken into account. Normal and Chi-squared tables are provided at the end of the paper.
4.6, 4.0, 3.7, 4.1, 4.1, 5.6, 4.5, 5.1, 6.0, 3.4, 5.1, 4.6, 3.7, 4.2, 4.6, 4.7, 4.1, 3.7, 3.4, 3.
(i) Display the above data as a stem-and-leaf plot. Describe the shape of the distribution. [4 marks]
(ii) Calculate the median and the interquartile range of the above data. [4 marks]
Response to treatment Drug Yes No A 30 42 B 18 54
Test the hypothesis that there is no association between the drug given and the response to treatment. [8 marks]
(ii) Show that, for any two events A and B,
P (A ∩ B) ≤ min{P (A), P (B)}.
[4 marks]
(i) Let X be the (random) number of cars passing that point in one minute interval. What is the reasonable probability distribution of X? Calculate E[X]. [5 marks] (ii) Due to impending roadworks it is estimated that congestion will occur closer to the city centre if more than 5 cars pass the point in any one minute. What is the probability of congestion occurring during any one minute interval? [8 marks] (iii) Calculate probability that there will be no congestion during a 20 minutes time interval. [7 marks]
17 13 18 19 17 21 29 22 16 28 21 15 26 23 24 20 8 17 17 21 32 18 25 22 (i) Calculate the arithmetic average ¯x (the sample mean) and the sample variance s^2 using formula
s^2 =
∑ x^2 i − (
∑ xi)^2 /n n − 1
[5 marks] (ii) If the assemble time X is normally distributed, use your calculations to estimate the probability that X > 30. Here you can assume that ¯x and s^2 coincide with the population mean and variance. [6 marks]
(iii) Calculate a 90% confidence interval for the exact value of the population mean μ. Here and below, you can assume that V ar = σ^2 = s^2 is known. [6 marks] (iv) Suppose instead you were asked to calculate a 99% confidence interval. Without carrying out any further calculations, say whether the 99% confidence interval will be wider or narrower than the interval calculated in part (iii) above, explaining your reasoning. [3 marks]
(ii) A group of students form a football team. In their first 90 matches they score the following numbers of goals.
Goals 0 1 2 3 4 Number of matches 18 30 27 9 6
Test at the 5% level the hypothesis that these data are drawn from a Poisson distribution with mean 1; [6 marks] a Poisson distribution with mean determined from the data. [8 marks]
(i) If you buy a toy, what is the probability that it is defective? [6 marks] (ii) If your toy is defective what is the probability that you bought an au- thentic product of Acme Gadgets Inc.? (Only in this case you are able to get a replacement.) [7 marks]
(iii) If you get a nondefective toy (possibly after replacement) you loose noth- ing, but if you get a defective pirate version (which will not be replaced) you simply loose the price of the toy, say £30. How much do you loose on average each time you buy this toy? [7 marks]
Paper Code MATH 161 Page 8 of 8 END
