40% of all the trees in a certain ponderosa pine stand (N is very large) were at

Question

40% of all the trees in a certain ponderosa pine stand (N is very large) were attacked by mountain pine beetles.

a. If 100 trees are randomly selected from this stand, what is the probability that more than 50% of the trees are infested?

b. If 100 trees are randomly selected from this stand, what is the probability that more than 35 trees are infested? (hint: instead of figuring out the probability of p being in a certain range, figure out the probability of X=number infested being in a certain range)

Explanation / Answer

Normal Approximation

40% of all the trees in a certain ponderosa pine stand (N is very large) were attacked by mountain pine beetles.

Solution:

Here, we are given,

Sample size n = 100, p = 40% = 0.4

Mean = n*p = 100*0.4 = 40

Standard deviation = sqrt (n*p*q) = sqrt (100* 0.4 * 0.6)

Standard deviation = 4.898979

X = 100*50% = 100*0.5 = 50

We have to find P(X>50)

P(X>50) = 1 – P(X<50)

The z-score formula is given as

Z = ( X – mean ) / SD

Z = (50 – 40) / 4.898979

Z = 2.041241

P(X>50) = 1 – P(X<50) = 1 – P(Z<2.041241) = 1 – 0.979387 = 0.020613

Required Probability = 0.020613

Solution:

Sample size n = 100, p = 40% = 0.4

Mean = n*p = 100*0.4 = 40

Standard deviation = sqrt (n*p*q) = sqrt (100* 0.4 * 0.6)

Standard deviation = 4.898979

Here, we have to find P(X>35)

P(X>35) = 1 – P(X<35)

Z = ( 35 - 40) / 4.898979

Z = 1.020621

P(X>35) = 1 – P(X<35) = 1 – P(Z<1.020621) = 1 – 0.846283 = 0.153717

Required probability = 0.153717

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