6. (a) A manufacturer obtains thermostats from a supplier in batches of 500. Five batches are examined, and found to contain 20,21,25,35 and 29 defective thermostats. Assume that X the number of defectives in a batch can be described by a Binomial distribution,X Bin(n,),and estimate the proportion π ofdefective items in a batch by the maximum likelihood estimator 1 where n500 and m-5 Noting that approximately Show that and hence show that an approximate 95% confidence interval for π is Compute a 95% confidence interval for the proportion ofdefective items in a batch, given that-,-2. The supplier claims that on average, less than 3% of a batch will be defective- are they meeting this target? [11 points] (b) An engincering firm has been awarded a contract to maintain a fleet of light aircraft. A condition of the contract is that on average there should be no more than 3 major breakdowns per year. Afler the first year of operation of the contract, there have been 5 major breakdowns. Perform a statistical hypothesis test to determine whether the firm has met the conditions of the contract. 9 points 6. (a) A manufacturer obtains thermostats from a supplier in batches of 500. Five batches are examined, and found to contain 20,21,25,35 and 29 defective thermostats. Assume that X the number of defectives in a batch can be described by a Binomial distribution,X Bin(n,),and estimate the proportion π ofdefective items in a batch by the maximum likelihood estimator 1 where n500 and m-5 Noting that approximately Show that and hence show that an approximate 95% confidence interval for π is Compute a 95% confidence interval for the proportion ofdefective items in a batch, given that-,-2. The supplier claims that on average, less than 3% of a batch will be defective- are they meeting this target? [11 points] (b) An engincering firm has been awarded a contract to maintain a fleet of light aircraft. A condition of the contract is that on average there should be no more than 3 major breakdowns per year. Afler the first year of operation of the contract, there have been 5 major breakdowns. Perform a statistical hypothesis test to determine whether the firm has met the conditions of the contract. 9 points