A large number of insects are expected to be attracted to a certain variety of r

Question

A large number of insects are expected to be attracted to a certain variety of rose plant. A commercial insecticide is advertised as being 99% effective. Suppose 2000 insects infest a rose garden where insecticide has been applied, and let X=number of surviving insects.

(A)   What probability distribution might provide a reasonable model for this?

(B)   Write down an expression for the probability that fewer than 100 insects survive (do not evaluate)

(C)   Approximate the expression in part (B).

I see the answer for this else where but can anyone do problem (C)??

Explanation / Answer

Part a

The normal probability distribution might provide a reasonable model for the given scenario.

Part b

We have to find P(X<100)

Part c

We are given

Insecticide is 99% effective.

This means about 99% of insects are killed.

This means about 1% of insects are not killed or they are surviving.

We are given

Total number of insects = n = 2000

Probability of insects surviving = p = 1% = 0.01, q = 1 – p = 1 – 0.01 = 0.99

n*p = 2000*0.01 = 20

n*q = 2000*0.99 = 1980

n*p > 10 and n*q > 10

We can apply normal approximation.

Mean = n*p = 2000*0.01 = 20

SD = sqrt(n*p*q) = sqrt(2000*0.01*0.99) = 4.449719

We have to find P(X<100)

By subtracting continuity correction 0.5, we have to find P(X<99.5)

Z = (X – mean) / SD

Z = (99.5 - 20) / 4.449719

Z = 17.8663

P(Z<17.8663) = 1.0000

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