Question

USA Today reported that about 20% of all people in the United States are illiterate. Suppose you take eleven people at random off a city street. (a) Make a histogram showing the probability distribution of the number of illiterate people out of the eleven people in the sample.

Answer 1

Answer:

Seee explanation below.

Step-by-step explanation:

1) Previous concepts

The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".

2) Solution to the problem

Let X the random variable of interest, on this case we now that:

$X \sim Binom(n=11, p=0.2)$

The probability mass function for the Binomial distribution is given as:

$P(X)=(nCx)(p)^x (1-p)^{n-x}$

Where (nCx) means combinatory and it's given by this formula:

$nCx=\frac{n!}{(n-x)! x!}$

For this case in order to do the plot we can use the following R code:

> x <- seq(0,11,by = 1)

> y <- dbinom(x,11,0.2)

> plot(x,y,main = "Binomial distribution n=11, p=0.2")

> plot(x,y,main = "Binomial distribution n=11, p=0.2",type = "h")

And the results are represented in the figure attached.

We can see that the distribution is right skewed with the higher probabilities for values of X between 0 and 4