Answer to Question #171930 in Statistics and Probability for Bikram Chaudhary

Question

A manufacturing concern wants to estimate the average amount of purchase of its product in a month by the customers. If the standard deviation is Rs. 10, find the sample size, if the maximum error is not be exceed Rs. 3 with probability of 0.99.

Answer 1

Let $\bar{X}=$ the average amount of purchase: $\bar{X}\sim N(\mu, \sigma^2/n).$

The critical value for $\alpha=0.01$ is $z_c=z_{1-\alpha/2}=2.576.$

The corresponding confidence interval is computed as shown below:

$$CI=(\bar{X}-z_c\times\dfrac{\sigma}{\sqrt{n}}, \bar{X}+z_c\times\dfrac{\sigma}{\sqrt{n}})$$

Given $\sigma=10$

$$z_c\times\dfrac{\sigma}{\sqrt{n}}\leq3$$

$$n\geq(\dfrac{z_c\times\sigma}{3})^2$$

$$n\geq(\dfrac{2.576\times10}{3})^2$$

$$n\geq74$$