A city currently does not have a National Football League team. Fifty-four percent of all the city’s residents are in favor of attracting an NFL team. A random sample of 1000 of the city’s residents is selected, and asked if they would want an NFL team.  What is the probability the percentage of those residents polled who are in favor of attracting an NFL team is less than 50%?

SOLUTION

For the population of all residents, p, the proportion in favor of attracting an NFL team, is equal to .54.

The sampling distribution of the sample proportion, p, for a random sample of n = 1000 residents from this population has the following properties:

The distribution of the sample proportion is approximately normal because:

The sample was randomly selected, 

Transforming p = .5 to a z-score yields

 

The figure below displays the essential information related to this problem. The normal table area corresponding to z = -2.53 is .494.  The desired probability is .500 - .494 = .006.

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