Answer to Question #152581 in Microeconomics for Salah yahye

Question

1. An economist in Kenya argued that 3.5% of all SMEs would file for bank loans next year. For a random sample of 100 SMEs, estimate the probability that at least three will file bank loans next year. Assuming that the economist’s prediction is correct.

2. A politician believes that 25% of all macroeconomists in senior positions would strongly support a proposal he wishes to advance. Suppose that this belief is correct and that five senior macroeconomists are approached at random. What is the probability that at least one of the five would strongly support the proposal?

Answer 1

1. n = 100

P(file loans) = 3.5%

= 0.035

P( at least 3) = P(X≥3) = 1 – P(X<3) = 1 – P(X≤2)

= 1 – [P(X=0) + P(X=1) + P(X=2)]

$P(X=x) = C^n_xp^x(1-p)^{n-x}$

$P(X≥3) = 1 – (0.0284 + 0.1029+0.1847) \\ = 1-0.316 \\ = 0.684$

The probability that at least three will file bank loans next year is 0.684.

2. n = 5

p = 0.25

$P(X=x) = C^n_xp^x(1-p)^{n-x}$

Compute the probability that at least one of the five senators will support the tax proposal.

$P(X≥1) = 1 - P(X<1) \\ = 1 - P(X=0) \\ = 1 - (C^5_0 \times 0.25^0 \times 0.75^{5}) \\ = 1 -0.2373 \\ = 0.7627$

The probability that at least one of the five senators will support the tax proposal is 0.7627.