Question

Assume a binomial probability distribution with n=45n=45 and π=0.30π=0.30 . Compute the following: (Round all your z values to 2 decimal places.)

1. The mean and standard deviation of the random variable. (Round your "σ" to 4 decimal places and mean to 1 decimal place.)

2. The probability that X is 16 or more. (Use the rounded values found above. Round your answer to 4 decimal places.)

3. The probability that X is 10 or less. (Use the rounded values found above. Round your answer to 4 decimal places.)

Answer 1

Solution:

Given: X follows a binomial probability distribution with n=45 and π=0.30

Part a) Compute the mean and standard deviation of the random variable

Mean is:

Standard Deviation is:

Part b) Find the probability that X is 16 or more

Since sample size n is large , we will use Normal approximation to Binomial to find above probability and hence we need to use continuity correction.

That is we add or subtract 0.5 in x value

Here we have subtract 0.5 from 16, so that 16 will not exclude from the range of x values.

Thus we get:

Find z score for x = 15.5

Thus we get:

Look in z table for z = 0.6 and 0.05 and find area.

Thus from z table we get:
P( Z< 0.65) = 0.7422

Thus

Part c) Find the probability that X is 10 or less

That is:

( We add 0.5 , since X is less than or equal to 10, so as to include 10 in the range, we need to add 0.5 )

Find z score for x= 10.5

Look in z table for z = -0.9 and 0.08 and find area.

Thus from z table we get:
P( Z < -0.98) =0.1635

Thus