Lecture 14- Binomial Distribution

Question 1

What two parameters define a binomial distribution?

Answer 1

• The fixed number of trials (n)

- The constant probability of success on each trial (π)

Question 2

What are the three conditions for a binomial distribution?

Answer 2

• Outcome is binary (success or failure)
• n independent trials
• The probability of success (π) is the same, for all trials

Question 3

A report suggests that 75% of New Zealand children under 18 live with
both parents. A random sample of 20 NZ children is selected, and X is the
binomial random variable for the number of these 20 who live with both
parents.
1. Define the parameters of the distribution of X.
2. In the sample of 20, how many children might we expect to find are
living with both parents?
3. Find Pr(X = 15).
4. Find the probability that 10 or fewer live with both parents i.e.
Pr(X ≤ 10).
5. A random sample of 20 NZ children had only 10 living with both
parents. Does this result provide any evidence to support the claim
that 75% of NZ children live with both parents?

Answer 3

Answers on Lecture 14 slides/ notes

Question 4

What command in r can you use to find individual binomial probabilities?

Answer 4

dbinom(x=15,size=20,prob=0.75)

change the values depending on problem

Question 5

What command in r can use to find the sum of all binomial probabilities less than or equal to a value?

Answer 5

pbinom(q=10,size=20,prob=0.75)

change values depending on problem

Question 6

Suppose the standard drug for treating a tumour halves the tumour size in
30% of patients.
Suppose we treat 7 randomly selected patients with a new, experimental
drug, and observe how many have their tumour size halved (X).
1. List the conditions for X to be Binomial.
2. Write down the probability that three of the patients have their
tumour size halved.
Pr(X = 3)
3. Find the probability that three or more of the patients have their
tumour size halved.
Pr(X ≥ 3)
4. In a pilot study in Auckland, three out of seven patients given a new
drug had their tumour size halved. What conclusion, if any, can be
drawn about the new drug?

Answer 6

Answers on lecture 14 slides/ notes

Question 7

How do we use a binomial distribution to estimate the population mean?
What about the variance?

Answer 7

mean= np
variance= np (1 − p)

Question 8

How do you use pbinom to determine greater than probabilities?
What do you have to be careful of?

Answer 8

• Being binary means the events are complimentary therefore can just go 1- probability
• You need to be careful of if you are calculating a greater than or equal to probability. In this case because pbinom calculates the probability of being less than or equal to a value you would need to set q to be one value below the number given.