Executives from Six Flags, a well-known amusement park chain, had interest in constructing a Six Flags theme park in a location near Ames city limits. Experts believed that approximately 15% of the surrounding population would be interested in becoming season ticket holders. A random sample of 500 residents of Story County was collected (of the approximately 80,000 residents of Story County). Of the 500 people sampled, 124 said that they would be interested in purchasing season tickets to a Six Flags in Ames. The mean of the sampling distribution for the sample proportion when taking samples of size 500 from this population is equal to ___________.

The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean [tex]\mu = p[/tex] and standard deviation [tex]s = \sqrt{\frac{p(1-p)}{n}}[/tex]

Of the 500 people sampled, 124 said that they would be interested in purchasing season tickets to a Six Flags in Ames.

This means that [tex]p = \frac{124}{500} = 0.248[/tex]

The mean of the sampling distribution for the sample proportion when taking samples of size 500 from this population is equal to

By the Central Limit Theorem, it is equal to the sample proportion of 0.248.