Address the questions below.the questions are complete

Which of the following criteria is a necessary and sufcient condition for the sarnple variance, defined as 32 =[ZXf2nf2], of a set of n observations front a population to be an unbiased estimator of the population variance? a 32 will always be unbiased. B S 2 will be unbiased if and only if the observations are independent. C S 2 will be unbiased if and only if the observations are normally distributed. D S2 will be unbiased if and only if the observations are independent and normally distributed. [3] A random sample from a normal distribution gives the following values: 68.8, 79.4, 63.2, 67.1, 69.4, 78.2, 80.4, 69.6 (i) Obtain a 95% confidence interval for the population mean, if the population standard deviation is: (a) unknown (b) 6. [5] (ii) Obtain a 90% confidence interval for the population standard deviation. [3] [Total 81Example 6.3.1. A large car hire firm has kept records of the cars it owned in 1997-98. All its cars are sold when they become exactly 4 years old, unless they are previously damaged in an accident or suffer mechanical failure, in which events they are disposed of. The firm also buys and sells cars for other reasons at any age under 4 years. Let a = accidental damage, and 8 = mechanical failure. no. of cars in fleet on 0% = no. of cars Ap = no. of cars curtate age damaged in an accident which break down of car, r 1.1.1997 1.1.1998 1.1.1999 at age r l.b. at age r l.b. 560 502 486 23 11 506 492 500 19 15 09 19 H O 451 511 480 15 28 309 389 372 20 56 Estimate the dependent probabilities of exit by (a) accident and (b) breakdown, at each age c. Estimate also the central rates of decrement by these modes of decrement at each age r