An electronics firm produces capacitors with a nominal capacitance of 470 μF470\text{ μF}470 μF. A quality control inspector suspects that the automated assembly line is under-filling the components, resulting in a mean capacitance lower than the target. A random sample of 60 capacitors is tested, yielding a sample mean of 466.8 μF466.8\text{ μF}466.8 μF and a sample standard deviation of 8.4 μF8.4\text{ μF}8.4 μF.
Conduct a hypothesis test at the 1% significance level to determine whether there is evidence to support the inspector's suspicion. Clearly state your null and alternative hypotheses.
Construct a 95% confidence interval for the true mean capacitance μ \mu\,μ based on this sample.
Suggest what action, if any, the electronics firm should take based on the results of parts (a) and (b).
Following a calibration of the assembly line, the standard deviation is reduced to σ=4.2 μF\sigma = 4.2\text{ μF}σ=4.2 μF while the mean is μ\muμ. A researcher uses the sample mean Xˉ\bar{X}Xˉ of a new sample of size n n\,n to estimate μ\muμ.
Calculate the smallest value of n n\,n required such that P(∣Xˉ−μ∣<1.0)≥0.98P(|\bar{X} - \mu| < 1.0) \ge 0.98P(∣Xˉ−μ∣<1.0)≥0.98.
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.