A team of engineers tests the operational lifespan of a new emergency beacon under sub-zero temperatures. A random sample of 12 beacons is activated, and the time until failure, ttt hours, is recorded. The data are summarized as follows:
∑t=504,∑t2=21450 \sum t = 504, \quad \sum t^2 = 21450 ∑t=504,∑t2=21450You may assume that the lifespans are normally distributed.
Calculate a 98% confidence interval for: (i) the mean lifespan of the beacons, (ii) the variance of the lifespan of the beacons.
Beacons that fail in less than 40 hours are designated as "short-life". Using the relevant confidence limits from part (a), determine the lowest estimate for the proportion of beacons that are short-life.
390 exam-style questions on OCR (MEI) A Level Maths 2.4 Probability Distributions, covering 2.4.1 Recognise binomial situations, 2.4.2 Probability of success p, 2.4.3 Calculate binomial probabilities, 2.4.4 Mean of the binomial distribution, 2.4.5 Expected frequencies for binomial, 2.4.6 Probability functions and discrete random variables, 2.4.7 Numerical probabilities for a simple distribution, 2.4.8 Normal distribution as a model (A-level only), 2.4.9 Shape of the Normal curve (A-level only), 2.4.10 Linear transformation and standardising (A-level only), 2.4.11 Symmetry and inflection of Normal curve (A-level only), 2.4.12 Calculate probabilities from a Normal distribution (A-level only), 2.4.13 Model with probability distributions, and 2.4 Probability Distributions. Each one has a worked solution and a mark scheme showing where the marks go.