\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
In a quality control process, the proportion of defective items is \(p=0.02\). We examine a batch of \(n=400\) independent items. Let \(\overline{X}_{400}\) be the proportion of defective items in the sample.
  1. Identify the mean \(\mu\) and standard deviation \(\sigma\) of the Bernoulli variable modeling the state of one item.
    \(\mu=\)
    and \(\sigma=\)
  2. Deduce the expectation \(E(\overline{X}_{400})\) and the standard deviation \(\sigma(\overline{X}_{400})\) of the sample proportion.
    \(E(\overline{X}_{400})=\)
    and \(\sigma(\overline{X}_{400})=\)