Please give step by step calculation

GET GENUINE OFFICE Your license isn't genuine, and you may be a victim of software counterfeiting Aword interuption and keep your files safe with genuine Office today Last genuine Office Learn more Question 1 (a) A number is selected at random from the following set of integers; 5 = { 3. 4. 5, 8. 12, 15, 18, 19, 23, 24. 25, 28, 29. 31, 36] Let A. B and C denote the events (i) a prime number (ii) divisible by 4 and (lii) greater than 15 respectively. Find the probability of_(1) A (i) InC (ill) AUC [TV) BIC Are B and C Independent events? (a] A candidate is selected for Interview of management trainee position by 2 companies. For the first company there are 4 candidates to be Interviewed and for the second there are 5. What is the chance of his success in (() both companies (li] at least one company? (Assume that all the candidates have the same chance of being selected), (b) Three ladies, Aileen, Barbers and Cathy pack biscuits in a factory, From the lot allocated to them, Aileen pack: 40%%, Barbara 35% and Cathy the remainder. The probability that Cathy breaks some biscuits in a packet is 0.02. The respective probabilities for Aileen and Barbara are 0.025 and 0.015 (1) What is the probability that a packet selected at random from a day's production contains some broken biscuits? Given that a selected packed contains some broken biscuits, what is the probability that it was packed by Barbara? Question 2 (=) A Geiger counter reaches an average of 3 counts per second in the vicinity of some radioactive material. What is the probability that there will be exactly 4 counts in one second? Find the probability that in a 10- second period, the count is between 27 and 32 inclusive (use a normal approximation] (b) An electrical device consists of 10 separate parts connected in such a manner that it will work only if all of the parts operate successfully. (1) if the probability of successful operation for each part is 0.95, what is the probability the device will work? What will this probability become if the device works provided that at least a of the ten parts operate successfully? (c) A manufacturer of machine parts claims that on the average, 10% of his parts are defective. A purchaser requires 120 such parts but because there will be defective items he places an order of 140. if the manufacture's claim is valid, what is the probability that the purchaser will receive at least 120 good parts? (Use normal approximation). (d) The marks of 1200 students obtained in an examination have a normal distribution with a mean value of 56 and standard deviation 8. The pass mark of the examination is 40. What is the probability that = given student scored 70 marks or more? Estimate the number of students who scored between 50 and 60 marks. Estimate the number of students who passed the examination.1. A normal distribution has a mean value of 50 and standard deviation 14.5. A random sample of size 100 gave a sample mean of 52.6. Test the given null hypothesis on the population mean against the alternative that the mean has increased (Use a=0.05) 2. Experience has shown that the scores obtained in a particular test are normally distributed with a mean score of 58 and standard deviation 8. The test Is taken by a random sample of 64 students and the mean score obtained is 56.6 Can we say the students have not performed as well as expected? (Use a=0.05) 3. From past experience, a given normal distribution has a mean value of 420 cm and standard deviation 12 cm. A sample of size 100 is selected at random from the distribution. The sample mean obtained is 423 cm. Can we say that the population mean has changed? (Use a=0.05). 4. A researcher obtained a random sample of 25 measurements (in cm) from a normal distribution with mean p and variance of The following values are obtained; En = 340 Ex = 5506 (i) Find the unbiased estimate of p and of (ii) Work out the 95% confidence interval for the population mean. What is the interpretation of the interval obtained? (iii) Use the interval obtained in (il) to test the hypothesis Hosp = 16 sgainst Hitp # 16. (iv) Test the following hypothesis Ho: p = 16 against Hj:p 1) (iii) Pr(X = 2). (d) Find the mean value and variance of X 6. The pdf of a continuous variable X is, [(x) = k(+2x); 0 67) = 0.95 What is the minimum value of n required? Question 4 (a] Twelve readings of the resistance in ohms of a piece of wire gave the following results; 151 149 1.54 154 1.48 150 1.46 149 156 1.45 1.58 If the wire is pure silver, the meen resistance is 1.50 ohms, If the wire is impure, the resistance increases, Test the hypothesis that the wire is pure silver (use a=0.05), (b) A manufacturer claims that the average life of his electric light bulbs is 80 days of continuous lighting. A random sample of 64 bulbs is tested and the lighting duration x (in days) at the end of the experiment is noted. The summarized values obtained are as follows: I x = 5216 )(x - 7) ' = 2709. Test the following hypotheses on the given population mean. (1) Ho: u = 80 against H, : u > 80 (il] Ho: p = 80 against H, : p = 80. (c) Find 95% confidence interval for the population mean in (b). Use the interval to test the hypothesis in (b]ill. (d) Two firms conducted a research in a constituency before an election on the proportion of voters expected to support candidate X on the actual election. Firm A found that out of the 50 voters selected at random, 18 will vote for the candidate while firm B found that of the 30 voters 9 will vote for the candidate. Test whether the proportion of voters favoring candidate X in the two firms can be taken to be the same [Use a= 0.05)