Question

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

Answer 1

E(T) = 7l mn Var(#) = m2 mn 2 a hence π =- 7m nm π-π tn and nm nm ence 六(1-f tn Now, P 六(1-f tn where, 1-2γ= 0.95 or, γ = 0.975 Hence approximate 95% C. I. for π is nm nm

Here, π (20 21 + 25 35 + 29)/(5 * 500) = 0.052 Approximate 95% C. I. for is (0.0431, 0.0609) Since 0.03 is less than lower boundary of proportion of defectives so there is insufficient evidence to conclude that on average, less than 3% of a batch will be de fective. Hence they do not meet the target (b) Let X = no. of breakdowns per year ~ Poisson(A) Null hypothesis, Ho :A 3 vs. Alternative hypothesis, H1:A>3 p-value-P(X > 51-3)-1-Σ 0.18470.05 r! r-0 Hence we fail to reject Ho at 5% level of significance hence the firm has met the conditions of the contact