Which of the following statements is true with respect to the sampling distribution of a proportion?

A. The mean of the sampling distribution will equal the population proportion.





B. An increase in sample size will result in a reduction in the size of the standard deviation.





C. As long as the sample size is sufficiently large, the sampling distribution will be appromixately noraml.





D. All of the above are trueWhich of the following statements is true with respect to the sampling distribution of a proportion?
D, all the above.








A and B are covered by the following:


Let X1, X2, ... , Xn be a random sample from a population with mean 渭 and variance 蟽^2





So, here we have the following:





E(Xbar)


= E(1/n 鈭?Xi)


= 1/n * E(鈭慩i)





expectation is a linear operator so we can take the sum out side of the argurement





= 1/n * 鈭?E(Xi)





there are n terms in the sum and the E(Xi) is the same for all i





= 1/n * nE(Xi)


= E(Xi)





E(Xbar) = 渭





--





Var(Xbar)


= Var(1/n * 鈭慩i)


= 1/n^2 Var(鈭慩i)





because the samples are independent we can take the sum out side of the Var argument. if the samples where not independent then this would not be possible.





= 1/n^2 * 鈭慥ar(Xi)





there are n terms in the sum and the Var(Xi) is the same for all i





Var(Xbar) = 1/n * Var(X)





and Standard Deviation of Xbar is 蟽/sqrt(n)





This shows that the mean of the sample mean is the population mean and as n, the sample size, increases the variance/standard deviation decreases.





C is covered by the central limit theorem.





Let X1, X2, ... , Xn be a simple random sample from a population with mean 渭 and variance 蟽虏.





Let Xbar be the sample mean = 1/n * 鈭慩i


Let Sn be the sum of sample observations: Sn = 鈭慩i





then, if n is sufficiently large:





Xbar has the normal distribution with mean 渭 and variance 蟽虏 / n


Xbar ~ Normal(渭 , 蟽虏 / n)





Sn has the normal distribution with mean n渭 and variance n蟽虏


Sn ~ Normal(n渭 , n蟽虏)





The great thing is that it does not matter what the under lying distribution is, the central limit theorem holds. It was proven by Markov using continuing fractions.





if the sample comes from a uniform distribution the sufficient sample size is as small as 12


if the sample comes from an exponential distribution the sufficient sample size could be several hundred to several thousand.





if the data comes from a normal distribution to start with then any sample size is sufficient.


for n %26lt; 30, if the sample is from a normal distribution we use the Student t statistic to estimate the distribution. We do this because the Student t takes into account the uncertainty in the estimate for the standard deviation.


if we now the population standard deviation then we can use the z statistic from the beginning.


the value of 30 was empirically defined because at around that sample size, the quantiles of the student t are very close the quantiles of the standard normal.Which of the following statements is true with respect to the sampling distribution of a proportion?
To the best of my knowledge, C seems to be the most correct.