Question

An astronomer is interested in measuring in light years, the distance from his observatory to a distant star. If the astronomer believes that the values of the measurements are independent and identically distributed random variables with common mean d (the actual distance) and a common variance of 4 (light years), how many measurements must he make to be 95% sure that his estimated distance is accurate to within + - 0.5 light years? (Note: for 95% confidence, Phi(Z) >=0.975, which means that Z = 1.96) (Note: for 99% confidence, Phi(Z) >=0.995, which means that Z = 2.575)

Answer 1

Solution :

standard deviation = $\sigma$ = $\sqrt{}$ 4 = 2

Z_$\alpha$/2 = Z_0.025 = 1.96  

sample size = n = [Z_$\alpha$/2* $\sigma$ / E] ²

n = [1.96 * 2 / 0.5]²

n = 61.46

Sample size = n = 62