1) a. A random variable X has the following probability distribution: X O XA 5 XA 10 % A 15 X A 20 X A 25 X A P(X = x) 0.1 2m 0.2 0.1 0.04 0.07 a Find the value of m. (3 Marks] b. What is the probability that X is fewer than 10? (3 Marks] C. What is the probability that X is at least (15 x A)? (4 Marks) d. Find the mean/expected value E(x) and standard deviation of the given probability distribution. (10 Marks) e . Find E(-4x + 3) and V(6x - 7) (6 Marks) 2) A calculator manufacturing factory noticed that 4% are defective. If 10 calculators are selected at random, what is the of getting a. Exactly 4 are defective. (4 Marks] b. At least 3 are defective. (10 Marks) 3) A Dell laptop company produces hard discs whose life follows a normal distribution, with mean (120 + A) months and standard deviation (25 + A) months. If we choose a hard disc at random what is the probability that its lifetime will be a. Less than 90 months? (4 Marks] b. More than 130 months? (6 Marks] c. Between 70 and 100 months? (10 Marks)4) a. A group of doctors on a particular hospital identified that they attend an average of (15 + A) patients per hour. Assume the standard deviation is 4. A random sample of 36 hours was chosen. Find the 95% confidence interval. (14 Marks) 13 b. Find the 90% confidence interval for the variance if a study of (9 + A) students found the 6.5 years as standard deviation of their ages. Assume the variable is normally distributed. (12 Marks) 5) A sample of 400 male students is found to have a mean height (55 + A) inches. Can it be reasonably regarded as a sample from a large population with mean height 66 inches and standard deviation 1.30 inches? Test at 5% level of significance. (14 Marks)