A recent poll shows that 76% of voters disapprove of the job that congress is doing.

Question

Find the probability that at least 136 people disapprove of congress out of a random sample of 199.

Answer 1

We've got a binomial situation here, but with so many trials, we can use the normal distribution as a handy shortcut!

1. Let n be the sample size of 199 voters,
2. p is the probability of success (in this case, "success" is "disapproval", so 0.76),
3. q is the probability of "failure" (or, "approval," so 0.24), and
4. x is the number of "successes" that we want to observe (at least 136).

Because we have a large sample size (199), we can use the normal distribution to approximate the binomial, which makes the calculations much easier.

First, check that np and nq are both at least 10 (which they are).

Next, find the mean (151.24) and standard deviation (about 6.024) of our approximating normal distribution.

1. In this case, the mean (µ) is np = 151.24, and the standard deviation (σ) is given by√(npq) = √(199 * 0.76 * 0.24) ≈ 6.024

Because we want 'at least 136', we use a continuity correction, adjusting our value to 135.5. Translating this a Z-score yields approximately -2.61.

What we are really trying to find is the probability located to the right of this -2.61 Z-score, which represents the probability mass of observing 136 or more "successes" (disapproval). From the Z-table one more time, we find this probability to be around 0.9955.

So, there! In your random sample of 199, the probability of finding at least 136 people who disapprove of Congress very high at 99.55%.

Hopefully that helps!

Answer 2

You can also use a Binomial Distribution function in Excel.

For the PC version of Excel, click in a free cell and start typing

=Binom

Double-click on the "Binom.Dist" function. Now we need to enter four numbers, each separated by a comma:

The number of successes,

The number of trials,

The probability of success,

Cumulative? Here a "0" means just give me P(x). A "1" gives me the sum of all the values of P(0) up to, and including, P(x).

So for this problem, I would start with:

=Binom.Dist(135, 199, 0.76, 1)

That answer is 0.005517 - to six decimal places.

That gives the sum of the cumulative probabilities from P(0) up to P(135). Your question, the probability of at least 136 (i.e. 136 and higher), is the complement of that answer, so if you subtract that answer from 1, you get:

1 - 0.005517 = 0.994483 - to six decimal places.