Precept 6 Problem Set

Problem 1

Suppose 65% SML201 students like World Coffee better than Hoagie Haven. We selected a random sample of 20 SML201 students, and asked them which they prefer. What is the probability that more than 18 students said “World Coffee”? Write R code to compute the actual probability.

Hint:

We solved a very similar problem in class on April 2 with coin tosses. Each student answer can be thought of as being like a coin toss. Decide whether you want “Heads” to represent HH or WC, and then solve the problem using pbinom, just like we did in class.

Solution

This is like asking about the probability of a coin’s coming up heads 18 times or more out of 20 when the probability of the coin’s coming up heads is 65%:

1 - pbinom(q = 18, size = 20, prob = 0.65)

## [1] 0.00213312

The answer is 0.2%.

Here’s another way to compute the answer:

sum(dbinom(x = c(19, 20), size = 20, prob = 0.65))

## [1] 0.00213312

Problem 2

In class, we saw several ways to compute the cumulative probability for the binomial distribution: we used pbinom; we summed up the outputs of dbinom; we also generated a large sample using rbinom, and then computed the proportion of the generated numbers that was under a certain threshold.

Part 2(a)

Write a function named MyPbinom1, which works just like pbinom. You may use dbinom but not rbinom in the function you write.

MyPbinom1 <- function(q, size, prob){
  return(sum(dbinom(x = 0:q, size = size, prob = prob )))
}

MyPbinom1(q = 2, size = 10, prob = 0.45)

## [1] 0.09955965

pbinom(q = 2, size = 10, prob = 0.45)

## [1] 0.09955965

Part 2(b)

Write a function named MyPbinom2, which works just like pbinom. You may use rbinom but not dbinom in the function you write.

MyPbinom2 <- function(q, size, prob){
  sample <- rbinom(n = 10000000, size = size, prob = prob)
  return(mean(sample <= q))
}

MyPbinom2(q = 2, size = 10, prob = 0.45)

## [1] 0.0994306

pbinom(q = 2, size = 10, prob = 0.45)

## [1] 0.09955965