EXAMPLE 5 Using Normal percentiles*

What is the probability that the opinion poll in Example 4 will get a sample in which 52% or more say Yes? Because 0.52 is not 1, 2, or 3 standard deviations away from the mean, we can’t use the 68–95–99.7 rule. We will use Table B of percentiles of Normal distributions.

435

image
Figure 18.4: Figure 18.4 The Normal sampling distribution, Example 5. The outcome 0.52 has standard score 0.9, so Table B tells us that the area under the curve to the left of 0.52 is 0.8159.

To use Table B, first turn the outcome into a standard score by subtracting the mean of the distribution and dividing by its standard deviation:

Now look in Table B. A standard score of 0.9 is the 81.59 percentile of a Normal distribution. This means that the probability is 0.8159 that the poll gets a smaller result. By Rule C (or just the fact that the total area under the curve is 1), this leaves probability 0.1841 for outcomes with 52% or more answering Yes. Figure 18.4 shows the probabilities as areas under the Normal curve.