Question

Dr Z decided to investigate the productivity of her plutonium mines. The standard deviation for the output is known (fat hint: use z-tables) and is 5 tons a day. The average output in a sample of a 400 mines was 66 tons. At 95% confidence level, what is the lower border of the confidence interval? (Two decimals) Chint: don't get confused here: you have been always asked for both, the lower and the upper values for the confidence interval but there is no way to set the question like that here, so I am only asking for one]

Answer 1

Given that, population standard deviation $(\sigma)$ = 5 tons a day

sample size (n) = 400 and sample mean $(\bar x)$ = 66 tons

A 95% confidence level has significance level of 0.05 and critical value is, $Z_{\alpha/2} = 1.96$

We want to find, the lower bound of the 95% confidence interval for the population mean,

$\text { Lower Bound } = \bar x - Z_{\alpha/2} *\frac {\sigma}{\sqrt n }$

$=> \text { Lower Bound } = 66- 1.96 *\frac {5}{\sqrt {400}}$

$=> \text { Lower Bound } = 66- 0.49$

$=> \text { Lower Bound } = 65.51$

Therefore, required lower border = 65.51 tons