EXAMPLE 19 Areas under a Normal Curve Are Probabilities

Suppose that 60% of adults agree with this statement: “Most people who want to get ahead can make it if they’re willing to work hard.” All adults form a population, with population proportion . Interview an SRS of 1500 people from this population and find the proportion of the sample who agree with the statement. We know that if we take many such samples, the statistic will vary from sample to sample according to a normal distribution, with

The 68—95—99.7 rule now gives probabilities for the value of from a single SRS. The probability is 0.95 that lies between 0.574 and 0.626 (within 2 standard deviations of the mean). Figure 8.20 shows this probability as an area under the normal density curve.

All that is new is the language of probability. “Probability is 0.95” is shorthand for “95% of the time in a very large number of samples.”