A theorem that allows us to use the normal probability distribution to approximate the sampling distribution of sample means and sample proportions whenever the sample size is large is known as the:____________.
a. approximation theorem
b. normal probability theorem
c. central limit theorem
d. central normality theorem

A theorem simply means a statement that can be demonstrated to be true by accepted mathematical operations and arguments. A proof is the process of showing a theorem to be correct.

From the options given:

a) Approximation Theorem: Approximation theory simply means how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced.

b) Normal Probability Theorem: In probability theorem, independent random variables are added, their properly normalized sum tends toward a normal distribution even if the original variables themselves are not normally distributed.

c) Central Limit Theorem: The central limit theorem means that the distribution of sample means approximates a normal distribution as the sample size gets larger.

d) Central normality theorem: The central normality theorem states that if you have a population with mean (μ) and standard deviation and take sufficiently large random samples from the population with replacement , then the distribution of the sample means will be approximately normally distributed.