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.
