Sickness and absence records were kept on 30 employees in a company over a 91-day period. These data are tabulated below: Number of employees absent 0 DO US W L Number of days 44 19 10 7 Calculate the sample mean and standard deviation of the number of employees absent per day. [3] 2 A set of claim amounts (f) is given below: 192 136 253 138 87 112 221 176 336 203 159 55 308 165 254 Present these data graphically using a boxplot. [3] 3 The probability that a car accident is due to faulty brakes is 0.02, the probability that a car accident is correctly attributed to faulty brakes is 0.95, and the probability that a car accident is incorrectly attributed to faulty brakes is 0.01. Calculate the probability that a car accident which is attributed to faulty brakes was due to faulty brakes. [3] 4 The probability that information held on an individual record on a company's database is incorrect (e.g. information is out-of-date) is 0.13. Calculate the probability that in a random sample of 200 records at most 20 contain incorrect information. [3]5 Consider a random sample of size n from a normal distribution Mu,o-) and let $2 denote the sample variance. (i) State the sampling distribution for (n-1)5- , and specify an approximate sampling distribution for this expression when n is large. [2] (ii) For n = 101 calculate an approximate value for the probability that S' exceeds o' by more than a factor of 10%, i.e. P(S' > 1.1 67). [1] [Total 3] 6 Calculate the maximum possible width of a symmetrical two-sided 95% confidence interval for the proportion of a population who possess a particular characteristic, based on the corresponding information in a random sample of size 1600 from the population. [3]7 A group of 500 insurance policies gave rise to a total of 83 claims during the last year. Assuming a Poisson model for the occurrence of claims, calculate an approximate 95% confidence interval for 2, the claim rate per policy per year. [3] 8 The ratio of the standard deviation to the mean of a random variable is called the coefficient of variation. For each of the following distributions, decide whether increasing the mean of the random variable increases, decreases, or has no effect on the value of the coefficient of variation: (a) Poisson with mean 2 (b) exponential with mean u (c) chi-square with n degrees of freedom [6] 9 An insurance company specifies in its advertising literature that 75% of all small claims (less than (200) on household policies are fully settled within one month. As part of an internal audit a random sample of 200 small claims is examined. (i) It is found that 146 of these small claims were fully settled within one month. Obtain an approximate one-sided 99% confidence interval for the true percentage of all small claims which are fully settled within one month. [3] (ii) The mean and standard deviation of the individual claim amounts were calculated as Y = f1 12.41 and $ = $51.62 respectively. Obtain an approximate two-sided 99% confidence interval for the population mean of all small claim amounts. [2]
