The proportion of eligible voters in the next election who will vote for the incumbent is assumed to be 53.2%.

Question

What is the probability that in a random sample of 490 voters, less than 49.6% say they will vote for the incumbent?

Answer 1

In this case, our mean is .532 (the population proportion they gave us). Our standard deviation can be found using the formula: square root [p(1-p)/n] where p= .532 and n = 490. Plugging all of that in, we get a standard deviation of .0225.

Now we have all of the information that we need to find our answer. We can solve two different ways, through calculating the z-score and finding the probability on the standard normal table, or by using the Normalcdf function on a graphing calculator. I’ll explain both ways just in case!

1) z-score method

We can calculate the z-score by using the formula

(x - mean)/standard deviation.

In in this case, that would be .496 - .532 / .0225 = -1.6

Then, we locate the z-score of -1.6 on the Standard Normal Table, and the probability is .0548 or 5.48%

2) Calculator

Access the Normalcdf function of your calculator (for Ti-84 plus it is under 2nd cars). For the lower limit, use a large negative number like -100000000. For the upper limit, use .496. For the mean, use .532 and for the standard deviation, use .0225. Enter all of that in and you should get .0548