A simple random sample of size n = 1,160 is obtained from a population whose size is N-1,700,000 and
whose population proportion with a specified characteristic is p=0.24. Describe the sampling distribution of
P.
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The sampling distribution of P (the sample proportion) will be approximately normally distributed. This is due to the Central Limit Theorem, which states that the distribution of sample means (or proportions) will be approximately normal if the sample size is large enough, regardless of the shape of the population distribution.
The mean (expected value) of the sampling distribution of P will be equal to the population proportion p, which is 0.24 in this case.
The standard deviation of the sampling distribution of P (also known as the standard error) can be calculated using the formula sqrt[(p(1-p))/n], where p is the population proportion and n is the sample size. In this case, the standard error would be sqrt[(0.24(1-0.24))/1160] = 0.013.
So, the sampling distribution of P is approximately normal with a mean of 0.24 and a standard deviation of 0.013. This means that if we were to take many samples of size 1160 from this population and calculate the sample proportion for each, we would expect the distribution of those sample proportions to be approximately normal with a mean of 0.24 and a standard deviation of 0.013.
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