Question

Two friends are opening a coffee shop. As they write their business plan, they research the amount of debt similar businesses can have in the first two years of opening. It is known that 72% of coffee shops have a debt of over $50,000 within the first two years of opening. If a random sample of 36 coffee shops is obtained, what is the probability that more than half of them had debt of over $50,000 within the first two years of opening?

Answer 1

This is Binomial distribution with normal approximation application.

n = 36, p = 0.72, q = 0.28

mean = n*p = 36*0.72 = 25.92

SD = n * p*g= V36 * 0.72 * 0.28 ~ 2.694

More than half so more than 36/2 = 18

Applying normal approximation

P(x > 18.5) (by continuity correction)

= P(Z > 18.5 – 25.92 2.694

= P(Z > -2.75)

= 0.9970

Hence 0.9970 is the probability that more than half of them had debt of over $50000 within the first two years of opening.

Please comment if any doubt. Thank you