According to data from the 2010 United States Census, 11.4% of all housing units in the United States were vacant. Suppose Maria, a researcher, takes a random sample of 200 housing units in the United States and finds that 15 are vacant. Let ^ p represent the sample proportion of housing units that were vacant. What are the mean and standard deviation of the sampling distribution of ^ p

The mean of the sampling distribution of ^ p is 0.075 = 7.5% and the standard deviation is 0.0186 = 1.86%

Step-by-step explanation:

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean and standard deviation , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean and standard deviation .

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

For the distribution of the sampling proportions of a proportion p in a sample of size n, we have that the mean is and the standard deviation is

In this problem, we have that:

So

Mean 0.075

Standard deviation

The mean of the sampling distribution of ^ p is 0.075 = 7.5% and the standard deviation is 0.0186 = 1.86%

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