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# QUESTION TWO (a) Assume that the probability of an individual coal miner being killed in a mine accident during a year is $\frac{1}{2400}$. Use the Poisson approximation to the Binomial to calculate the probability that in a mine employing 200 miners, there will be atleast one fatal accident in a year. (b) A package of $8 \mathrm{AA}$ batteries contains 2 batteries that are defective. A student randomly select four batteries and replaces the batteries in his calculator. i. What is the probability that all four batteries work? ii. What is the mean and variance for the number of batteries that work? 2 (c) The probability that a driver passes the written test for a driver's license is 0.75 . What is the probability that a person will fail the test on the first try and pass the test on the second try? (d) The principal of a college wants to estimate the proportion of smokers among his students. What size of a sample should be selected so as to have the proportion of smokers not to exceed by $20 \%$ with almost certainty? It is believed from previous records that the proportion of smokers was 0.55 . (e) Ten students were given intensive coaching for a month in statistics. The scores obtained in tests 1 and 7 are given below: \begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|} \hline Serial No. of students & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline Marks in $1^{\text {st }}$ test & 50 & 52 & 53 & 60 & 65 & 67 & 48 & 69 & 72 & 80 \\ \hline Marks in $7^{\text {th }}$ test & 65 & 55 & 65 & 65 & 60 & 67 & 49 & 82 & 74 & 86 \\ \hline \end{tabular} Does the scores from test 1 to test 7 show an improvement? Test at the $2 \%$ level of significance.  