Calculation of the probability of an error?

Question

Suppose we want to estimate the average speed of cars travelling on a highway, and we want to be able to assert with probability 0,99 that the error of our estimate will be at most 3 miles per hour. How large a sample will we need if it can be assumed that $\sigma=7,1$ miles per hour?

Answer 1

The standard formula in such base cases, if we assume a normal distribution, is to look at the following statistic

$$Z=\frac{X-\mu}{\sigma/\sqrt{n}} \sim N(0,1) \; .$$

You want to find a 0.99 two-sided confidence interval with error of at most 3 miles per hour. This implies that

$$z_{0.995}\frac{\sigma}{\sqrt{n}}\leq3$$

or

$$n\geq \left(\frac{z_{0.995}\sigma}{3}\right)^2 = \left(\frac{2.576 \cdot 7.1}{3}\right)^2 = 37.16\; .$$

So, you need a sample of at least 38 cars.