If it has an emergency locator, what is the probability that it will not be discovered? (ii) If it does not have an emergency locator, what is the probability that it will be discovered?
[ marks]Define and give the example of: (i) Random variable, (ii) Independent Events.
[3 marks]Two fair six-sided dice are tossed independently. Let Mbe the maximum of the two tosses. What is the probability mass function (pmf) of M?
[4 marks]Seventy percent of the light aircraft that disappear while in flight in a certain country are subsequently discovered. Of the aircraft that are discovered, 60% have an emergency locator, whereas 90% of the aircraft not discovered do not have such a locator. Suppose a light aircraft has disappeared.
[7 marks]Fit a straight line to the following data. Also, estimate the value of y at x=72. x: 65 66 67 67 68 69 71 73 y: 67 68 64 68 72 70 69 70
[7 marks]State the probability function of Exponential and Gamma distribution.
[3 marks]Adice is thrown 264 times with the following results. Show that the dice is biased. [Use 𝜒2 = 11.07 for 5 degree of freedom]. 0.05 No. appeared on dice 1 2 3 4 5 Frequency 40 32 28 58 54 52
[6 marks]Fit the second degree parabola using the least square method to the following data: x: 1 2 3 4 y: 5 12 26 60 97 Also, estimate y at x=6.
[5 marks]Acar hire firm has two cars, which it hires out day by day. The number of demands for a car on each day is distributed as a Poisson distribution with mean of 1.5. Calculate the proportion of days on which (i) neither car is used, (ii) some demand is refused, (iii) only one car is used.
[4 marks]State the properties of the Normal Distribution
[3 marks]If a random variable has a Poisson distribution such that P(X=1)=P(X=2), find
[4 marks]the mean of the distribution, (ii) P(X=5), (iii) P(X>1), and (iv) P(1<X<4).
[ marks]Define Binomial Distribution. Aparticular telephone number is used to receive both voice calls and fax messages. Suppose that 25% of the incoming calls involve fax messages, and consider a sample of 10 incoming calls. What is the probability that (i)At most 3 (ii) Exactly 3 (iii) At least 3 (iv) More than 3, of the calls involve a fax message?
[7 marks]The mean and variance of a binomial distribution are 4 and 2. Find P(X ≥ 2). 1/3
[3 marks]Ten objects are chosen at random from a large population and their weights are found to be in grams:61,63,64,65,68,69,69,70,71,71. Discuss the suggestion that the mean is 65 g. [Use t = 2.262 at v = 9]. 0.05
[3 marks]Explain the term related to testing of hypothesis: (i) Type I Error, (ii) Type II Error, (iii) Level of Significance.
[3 marks]Acoin was tossed 960 times and returned heads 183 times. Test the hypothesis that the coin is unbiased. Use 5% level of significance. [use Z = 1.96]. 0.05
[4 marks]Two types of batteries are tested for their length of life and the following data are obtained: No. of samples Mean Life in hours Variance Type A 9 600 121 Type B 8 640 144 Is there a significant difference in the two means? [Use t = 2.132] 0.05,15
[7 marks]The means of simple samples of sizes 1000 and 2000 are 67.5 cm and 68 cm respectively. Can the samples be regarded as drawn from the same population of standard deviation 2 cm. [use Z = 1.96] 0.05
[4 marks]Two random samples are drawn from two populations and the following results were obtained: Sample I 21 24 25 26 27 Sample II 22 27 28 30 31 36 Find the variances of the two samples and test whether the two populations have the same variances.[Use F (5,4) = 6.26.] 0.05
The probability distribution of a random variable Xis given below. Find (i) E(X), (ii) V(X) X: -2 -1 0 1 P(x=X) 0.2 0.1 0.3 0.3 0.1
[2 marks]The following are the lines of regression 9y = x+288 and 4y = x +38. Estimate y when x = 99 and x when y = 30. Also, find the means of x and y.
[4 marks]Ten competitors in a test are ranked by three judges in the following order: Rank by First Judge: 6 10 2 9 8 1 5 3 4 Rank by Second Judge: 5 4 10 1 9 3 8 7 2 Rank by Third Judge: 4 8 2 10 7 5 9 1 3 Use the method of rank correlation to gauge which pairs of judges has nearest common approach.
[6 marks]For a group of 10 items, Σx = 452,Σx2 = 24270,and mode = 43.7. Find Karl Pearson’s coefficient of Skewness. 2/3
[3 marks]Find the correlation coefficient for the following data: X: -3 -2 -1 0 1 2 Y: 9 4 1 0.5 1 4 9
[3 marks]Calculate the regression coefficients and find the two lines of regression for the following data: x: 57 58 59 59 60 61 62 64 y: 67 68 65 68 72 72 69 71 Find the value of y when x=65. 3/3
[7 marks]Define Standard normal variate. The lifetime of a certain kind of batteries has a mean life of 400 hours and the standard deviation as 45 hours. Assuming the distribution of lifetime to be normal. Find The percentage of batteries with lifetime (i) at least 490 hours, (ii) between 385 and 490 hours. Also, find the minimum life of the best 5% of batteries. [Use: P(0< z < 2) = 0.4772, P(0 < z < 0.33) = 0.1293 and P(0< z < 1.65) = 0.45]
[7 marks]