The quality control department of a precision engineering firm monitors the manufacturing of high-tensile bolts. Historical data indicates that the mean length of the bolts is 145.0 mm145.0\text{ mm}145.0 mm with a standard deviation of 0.8 mm0.8\text{ mm}0.8 mm.
Following a recalibration of the cutting machines, a technician suspects that the mean length of the bolts has changed. To investigate this suspicion, a random sample of 256256256 bolts is measured.
The total length of these 256256256 bolts is found to be 37148.16 mm37148.16\text{ mm}37148.16 mm.
Assuming the lengths are normally distributed, test the technician's suspicion at the 5%5\%5% significance level.
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.