Question

RANDOM VARIABLES AND DISTRIBUTIONS Central limit theorem: Sample sum A small business owner estimates his mean daily profit as $969 with a standard deviation of $128. His shop is open 103 days a year. What is the probability that his annual profit will not exceed $100,000? Carry your intermediate computations to at least four decimal places. Report your result to at least three decimal places. xml?

Answer 1

Let X denote the daily profit earned by the business. Using Central limit theorem, we know,

$\sum X_{i}\sim N(n\mu,n\sigma^{2})\sim N(103(969),103(128)^{2})\sim N(99807,1687552)$

Required probability =

100000 – 99807. PAΣX, S 100000) = P(Z< 100 1687552

= P(Z < 0.1486) = 0.5591