Question

A researcher believes that about 74% of the seeds planted with the aid of a new chemical fertilizer will germinate. He chooses a random sample 135 of seeds and plants them with the aid of the fertilizer. Assuming his belief to be true, approximate the probability that at least 102 of the 135 seeds will germinate. Use the normal approximation to the binomial with a correction for continuity.

Answer 1

Solution:

Given in the question

Probability of seeds planted with the aid of new chemiclal = 0.74

No. Of sample = 135

By using the normal approximation to binomial distribution

Mean = n*p = 0.74*135 = 99.9

Standard deviation = sqrt(p*(1-p)) = sqrt(n*p*(1-p) = sqrt(0.74*(1-0.74)*135) = 5.1

We need to calculate probability that at least 102 of the 135 seeds will germinate

P(X>=102) = ?

Here we will use normal distribution table. We will calculate Z score which can be calculated as

Z-score = (X-)/ = (102-99.9)/5.1 = 0.412

From Z table we found p-value

P(X>=102) = 1 -P(X<102) = 1- 0.6598 = 0.3402

So there is 34.02% probability that at least 102 of the 135 seeds.