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 A Level Maths 2.4 Statistical Distributions, covering 2.4.1 Discrete probability distributions, 2.4.2 Binomial distribution as a model, 2.4.3 Calculating binomial probabilities, 2.4.4 Mean and variance of the binomial (A-level only), 2.4.5 Normal distribution as a model (A-level only), 2.4.6 Probabilities using the normal distribution (A-level only), 2.4.7 Links to histograms, mean and standard deviation (A-level only), and 2.4.8 Selecting an appropriate distribution (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.