A particular random sample of n observations can be used to calculate a sample mean. we can determine the characteristics of the distribution of means derived by other samples of the size n taken from the same populations without taking additional samples. this distribution is called the sampling distribution of the . it is centered on and has standard deviation equal to . the theorem that postulates that the distribution will tend to be normal is called the .

a. Means; b. The Mean; c. ; d. Central Limit Theorem.  

Step-by-step explanation:

When we want to know something about a population, we can obtain a representative sample from the population. From this sample, we can infer the characteristics of that population. Roughly speaking, all the methods used for doing this are gathered in what is known as Inferential Statistics.  

Well, there are probabilistic distributions and one of the most used is the normal distribution. We can also construct a kind of distribution called the sampling distribution of the means.  

To infer the mean of a population and how far are all these samples from this mean, we can take several samples of size n from the same population. From each sample, we take its mean. The distribution of these means forms the sampling distribution of the means.

As we already know, the normal distribution is centered on the mean and has a standard deviation. These two parameters determine this distribution.

Thanks to the Central Limit Theorem, the sampling distribution of the means follows a normal distribution, and, therefore, this distribution is always symmetrical around the mean. By the way, the expected value of the mean of the sampling distribution of the means is the mean of the population .

To be more precise, this Central Limit Theorem tells us that when the value for n (sample size) for these samples is n = 30 or more, the distribution of these samples tends to be normally distributed with a standard deviation equals to (called standard error), in the case for an infinite population, where is the population standard deviation. This is true whatever the distribution of the population is.  

Thus, the distribution of the corresponding means of the samples of the size n taken from the same population is called the sampling distribution of the means. This distribution is centered on the mean and has a standard deviation equal to (for an infinite population), and the theorem that tells us that these distributions tend to be normal is called the Central Limit Theorem.