Supplier Confidence [Tesla Statistics Interview Question]

A 3rd-party widget supplier for a key-component of the Tesla Model S manufactures widgets whose length follows a $N(\mu, 25mm) distribution. You take a sample of the widgets and measure their length.

How many widgets do you need to sample, to have a 95% confidence interval for $\mu$ with width 1mm?

The 95% confidence interval for the mean is:

$ x \pm z_0.025 * \frac{\sigma}{\sqrt(n)}

This has width $2 * z_0.025 * \frac{\sigma}{\sqrt(n)}$

Setting the width equal to 1, and substituting $z_0.025 = 1.96$ and $\sigma=5$:

$2 * 1.96 * \frac{5}{\sqrt(n)} = 1 \Rightarrow \sqrt(n) = 19.6$

Thus, $n = (19.6)^{2} = 384$