University of California Characteristics of A Binomial Random Variable Questions Recall that the characteristics of a binomial random variable are: 1. The

University of California Characteristics of A Binomial Random Variable Questions Recall that the characteristics of a binomial random variable are:

1. The experiment consists of a fixed number (we usually use n) of trials.

2. The trials are independent.

3. There are only two possible outcomes on each trial, which for convenience we call “success” and “failure.”

4. The probability of a success (p), is the same for each observation.

5. If X is a binomial random variable then its mean is E[X]=?x=npE[X]=?x=np and its standard deviation is ?x=?np(1?p)?x=np(1-p) .

For the next three questions of the following examples, decide whether X is a binomial random variable using ONLY the information given. State why or why not.

Question 1: The pool of potential jurors for a murder case contains 100 persons chosen at random from the adult residents of a large city, and asked whether he or she opposes the death penalty; X is the number who say “Yes”.

No, more than two outcomes.
Yes, meets all requirements of a binomial random variable
No, probability is not the same for each person.
No, trials are not independent.

Question 2: Suppose a jury of 12 people is chosen from the above pool, and this jury hears a case and discusses the verdict; X is the number who think the defendant is guilty.

Yes, meets all requirements of a binomial random variable.
No, there is not a fixed number of trials.
No, the trials are not independent.
No, there are more than two possible outcome

Question 4: Using the information from the previous question, find the expected number of people cured, i.e., E(X)E(X)(answer in decimal form round to nearest tenth if necessary).

E(X)=E(X)= people

Question 5: Now find the standard deviation of the number of people cured X (answer in decimal form round to nearest tenth if necessary).

SD(X)=SD(X)= people

Question 6: If 31 of the 40 people were cured, would this be unusual?

No
Yes

Also recall that the standard deviation of a Poisson is the square root of the mean: E(X)=?xE(X)=?x (also commonly written as ?? ) and SD(X)=?x=??SD(X)=?x=? .

Suppose a pet clinic gets an average of 9 cats per day to be spayed, and they only have the resources (material supplies and volunteer time) to spay a maximum of 14 cats on any given day. Can we find the probability that more cats will show up on a day than they can spay? We could work with the theoretical distribution, but instead, let’s estimate it using relative frequencies in JMP. But FIRST let’s answer some theoretical questions:

Question 1: What are the theoretical expected value for the number of cats showing up at the clinic on a given day (don’t over think this)?

E(X)=E(X)= cats/day

Question 2: What is the theoretical standard deviation for the number of cats showing up at the clinic on a given day? Round to two decimal places.

SD(X)=SD(X)= cats/day