Next we're asked to describe the probability

distribution of number of uninsured Americans who plan

to get health insurance through a government health

insurance exchange among a random sample of 100.

Once again, p is 0.56, this time n is 100. This seems like a larger sample size.

So let's see if it's actually large enough to yield a nearly normal distribution.

The rule was that we need at

least 10 expected successes and 10 expected failures.

Expected number of successes is 100 times 0.56 which is 56, which is

indeed greater than 10 and the expected number of failures is 100 times

0.44 which is equal to 44 which is also greater than 10.

So we do know that the shape of the distribution will be nearly normal.

Normal distributions have two parameters, mean and standard deviation.

So to fully describe the distribution, we need to

calculate these parameters, which we know can be estimated

by the binomial, mean, and standard deviation.

Mean, in other words the expected number of successes, is 56.

We already calculated this.

And the standard deviation can be calculated as the square root of n times

p times 1 minus p. So that's the square root of 100 times

0.56 times 0.44 roughly 4.96. So this binomial distribution's

shape actually follows a normal distribution

with mean 56 and standard deviation 4.96.

Lastly, let's consider the following question.

What is the probability that at least 60 out of a random sample of

100 uninsured Americans plan to get health

insurance through a government health insurance exchange?

Once again, we'll present a variety of ways of solving

this, though you can really just pick one and stick with it.

One approach is, once again, to use the applet.

So let's take a look at how we could solve this question there.

Once again the distribution is binomial.

This time our number of trials is 100, so we're going to slide over our n to 100.

Probability of success is 0.56. The observation of interest

is 60 successes.

So we're going to slide our cutoff value to 60, and we're looking for

not just exactly 60 successes, but 60 or more successes.

So we want to find the upper tail area and the

bound we're interested in is greater than or equal to.

Here's the shaded area of interest, the probability comes out to be 24.1% chance.