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 a debt of over $50,000 within the first two years of opening?

Answer 1

The answer is 50,036

Answer 2

Answer: 0.9984

Explanation:

Let p be the proportion of coffee shops have a debt of over $50,000 within the first two years of opening.

As per given , p= 72%=0.72

Sample size : n= 36

Required probability :-

`P(\hat{p}>0.50)=P(\frac{\hat{p}-p}{\sqrt{(p(1-p))/(n)}}>\frac{0.50-0.72}{\sqrt{(0.72(1-0.72))/(36)}})\\\\=P(z>-2.94)\ \ \ [\because z=\frac{\hat{p}-p}{\sqrt{(p(1-p))/(n)}}]\\\\=P(z<2.94)=0.9984\ \ \ [\text{By p-value table}]`

Hence, the probability that more than half of them had a debt of over $50,000 within the first two years of opening = 0.9984