1. A manufacturer of skateboard wheels purchases ball bearing from their supplier in 2 kg and 4kg containers. The quality of the ball hearings in n 2 kg containers and m 4kg containers were assessed to ensure that they meet the necessary specication for man- ufacturing wheels. Let the independent random variables X1, . . . 1X\" and 1/1.... ,Ym denote the number of faulty ball hearings in the 2 kg and 4kg containers respectively, where X1- N Pois[2)i) [for f = 1.... ,n) and Y;- N PoisHA) (for j = l, . . . , m). The follow- ing two estimators have been proposed to estimate the population rate parameter A for the number of faulty ball bearings per kilogram: 3']: + J? 17 J? + 17 and T = . 4 s 2 6 ' where X : $22121 X1: and "l? : % 23:1 l3- denote the sample mean estimator for the 2kg and 4 kg containers respectively. Show that both estimators are unbiased in estimating J. and derive an expression for their standard error. Under what circumstances [i.e. the relationship between H and m) will make the estimator T1 more efficient than estimator T2 in estimating A? 2. A newspaper journalist is investigating 'BEtterCai', a high-street coffee chain. The jour- nalist has received a number of health concerns by BetterCaf's customers where they believe that the decaffeinated coffee contains a higher concentration of caffeine than what the chain claims. On their website: BetterCaf states that the expected caffeine concentration in their decaffeinated coffee is 1-5 Mg);K ml. To investigate, the journalist pur- chases a decaffeinated coffee at 8 BetterCaf cafes and serid a sample of each to a lab for analysis. The lab report states that the average caffeine content across the samples is 21.4 pg} ml with a standard deviation of 9.3 pg,E ml. The journalist approaches you to interpret the lab report into whether there is any evidence to substantiate the claims made by BetterCaf's customers. Perform an appropriate hypothesis test at the 5% signicance level. Clearly state your hypotheses. all denitions and any assumptions you have made in your investigation