In a quality control audit of high-precision titanium bolts, the length of a bolt is found to have a mean of 12.4012.4012.40 mm and a standard deviation of 0.150.150.15 mm. The lengths of 95%95\%95% of these bolts are observed to fall between 12.1012.1012.10 mm and 12.7012.7012.70 mm.
Comment on whether a normal distribution may be suitable to model the length of a high-precision titanium bolt in this audit.
You may assume that the length of a high-precision bolt may be modelled by a normal distribution with mean 12.4012.4012.40 mm and standard deviation 0.150.150.15 mm.
(b) (i) Find the probability that the length of a randomly selected bolt is exactly 12.5012.5012.50 mm.
(b) (ii) Find the probability that the length of a randomly selected bolt is between 12.2212.2212.22 mm and 12.5812.5812.58 mm.
(b) (iii) Two bolts are chosen at random. Calculate the probability that both of their lengths are between 12.2212.2212.22 mm and 12.5812.5812.58 mm.
The summarised data for the lengths, yyy mm, of a random sample of 404040 standard-grade bolts is given below:
∑y=492and∑(y−yˉ)2=1.56 \sum y = 492 \quad \text{and} \quad \sum(y - \bar{y})^2 = 1.56 ∑y=492and∑(y−yˉ)2=1.56Use this data to calculate estimates of the mean and standard deviation of the lengths of standard-grade bolts.
Using your answers from part (c), compare the lengths of high-precision bolts and standard-grade bolts.
616 exam-style questions on Edexcel A Level Maths The Normal Distribution, covering 3.1 The Normal Distribution, 3.2 Finding Probabilities for Normal Distributions, 3.3 The Inverse Normal Distribution Function, 3.4 The Standard Normal Distribution, 3.5 Finding the mean and standard deviation, 3.6 Approximating a Binomial Distribution, and 3.7 Hypothesis Testing with the Normal Distribution. Each one has a worked solution and a mark scheme showing where the marks go.