Which theorem states that as samples become large (above about 30) the sampling distribution will take the shape of a normal distribution, with the t-distribution for small samples, and the standard error of the mean equals s/√N?

Which theorem states that as samples become large (above about 30) the sampling distribution will take the shape of a normal distribution, with the t-distribution for small samples, and the standard error of the mean equals s/√N?

• A. Law of Large Numbers
• B. Student's t-distribution
• C. Central limit theorem
• D. Chebyshev's inequality

Answer: (C)

The main idea is how the distribution of a sample mean behaves as you collect more data. The central limit theorem says that as the sample size grows, the distribution of the sample mean becomes approximately normal, even if the underlying population isn’t normal, provided the population has finite mean and variance. When the population standard deviation is unknown and you estimate it with the sample standard deviation, the standardized mean follows a t distribution for small samples; as the sample size increases, this t distribution approaches the normal distribution. The standard error of the mean is estimated as s/√n, reflecting the precision of the mean estimate that improves with more data. In short, the CLT explains why big samples give a normal-shaped sampling distribution for the mean, why small samples involve the t distribution, and why the standard error is s/√n.

The main idea is how the distribution of a sample mean behaves as you collect more data. The central limit theorem says that as the sample size grows, the distribution of the sample mean becomes approximately normal, even if the underlying population isn’t normal, provided the population has finite mean and variance. When the population standard deviation is unknown and you estimate it with the sample standard deviation, the standardized mean follows a t distribution for small samples; as the sample size increases, this t distribution approaches the normal distribution. The standard error of the mean is estimated as s/√n, reflecting the precision of the mean estimate that improves with more data. In short, the CLT explains why big samples give a normal-shaped sampling distribution for the mean, why small samples involve the t distribution, and why the standard error is s/√n.