Question

In a survey, an Institute discovered that 40% of people who responded couldn’t remember any of the laws by the First Amendment. You decide to build a distribution for how many respondents could not recall any of the laws. You take on a random sample of 10 Americans.

A. What are the assumptions of a binomial distribution? Does the example match the assumptions?

B. What is the probability that the sample has exactly n successes, for n=1,2,3…10?

C. Plot the probabilities that were calculated in B.

D. Find the probability that the sample has at least 5 successes.

E. Find the probability that the sample has at most 3 successes.

Answer 1

A. Define X= number of respondents not being able to recall any of the laws out of 10 Americans.

p= proportion of   people who responded but couldn’t remember any of the laws

Here the number of trials (i.e. the number of respondents chosen) is fixed, i.e. k=10

Each trial has two possibilities: either can recall or not

The trials are identical and independent in nature

The probability of success (i.e. not being able to recall) remains constant, i.e. p=0.4

Thus all the assumptions are satisfied and hence X~Binomial(10,.40)

B.

with p=.40.

C.

D. P(The sample has at least 5 successes)=

E. P(The sample has at most 3 successes.)

=P(X<=3)=P(X=0)+P(X=1)+P(X=2)+P(X=3)=0.006046618+..+.214991 =0.3822806