F71SM STATISTICAL METHODS
Tutorial on Section 6 SAMPLING DISTRIBUTIONS, CLT, t and F DISTRI- BUTIONS
Substantial quantities of data show that the sizes of claims which arise under policies of a certain type can be modelled by a normal distribution with mean m = £3600 and standard deviation s = £800. Consider a random sample of 16 such claim amounts. Find the probability that the sample mean claim amount exceeds £3700. [0.3085]
(b) Substantial quantities of data show that the sizes of claims which arise under policies of a certain type can be modelled by a distribution with mean m = £1200 and stan- dard deviation s = £$150. Consider a random sample of 100 such claim amounts. Find the approximate probability that the total of the claim amounts is less than £121,500. [0.8413]
One hundred random samples, each of 10 children, are taken independently from a large population consisting of an equal number of boys and girls. Find the probabilities that:
(a) a sample contains at least 6 boys [0.3770], and
(b) at least 40 of the samples contain at least 6 boys. [0.3557]
(a) Private cars pass a checkpoint at an average rate of 3 per minute and the number which pass in a given time interval can be modelled by a Poisson r.v. Find the probability that more than 190 private cars pass the checkpoint in an hour. [0.2177]
(b) The number of claims that arise in a year under a policy of a particular type follows a Poisson distribution with mean 0.2. Find the probability that a total of 180 or fewer claims arise in a year under a group of 1000 independent such policies. [0.0838]
Let X1,X2,…,X10 be a random sample of a N(μ,σ2) r.v. with sample mean X ̄ and variance S2. Find
(a) P X − μ < σ/2 [0.8858]
(b) P X − μ < S/2 [0.852]
(c) P (S2 < 2σ2) [0.9648]
A random sample of 16 observations is drawn from a N(μ,122) distribution. Indepen- dently, a random sample of 25 observations is drawn from a N(μ,202) distribution. Let X ̄ and Y ̄ denote the respective sample means, and let SX2 and SY2 denote the respective sample variances. Find
(a) P |X ̄ − Y ̄ | < 5 [0.6826] (b) P (SX2 > 200) [0.143]
(c) P (SY2 > 6.356SX2 ) [0.05]