Motivation for why we do this
• The main objective of inferential statistics is to utilize sample data to make population inferences.
• One such case involves using the sample average 𝑋̅ to estimate the population mean μ.
• However, different samples typically yield different sample averages: some are much smaller than μ, some are much larger than μ, and some are fairly close to μ.
Why need 𝑋̅ to make inferences about μ
• prior to sampling, the sample average 𝑋̅ is itself a random variable and therefore, 𝑋̅ follows some probability distribution; in order to use 𝑋̅ to draw inferences about μ, it is necessary to understand the distribution of 𝑋̅.
Law of large numbers
Even with this small population, we can see that as the size of the sample grows, its sample mean will (in general) tend closer and closer to the value of the population mean (with some “ups and downs”).
sampling distribution of the mean = sampling distribution
• Each one of these samples is equally likely,
— this is the definition of Simple Random Sampling.
• Table defines a discrete random variable.
= sampling distribution of the mean
• will always be the case, that the average of all sample averages equals the population average.
— 𝜇𝑋̅ = 𝜇
When we consider larger populations, a second trend emerges in the sampling
distribution of the mean.
• The “standard deviation of the sampling distribution of the means” is also called the Standard Error of the Mean.
- As the sample size increases, the shape of the distribution increasingly
approximates the normal distribution. - As the sample size increases, the standard deviation of the sampling distribution decreases.
Central Limit Theorem
• Given any population with mean 𝜇 and standard deviation 𝜎 and a sample of size n,
— the distribution of the samples means will (as n increases) approximate a normal distribution with mean 𝜇𝑋̅= 𝜇 and standard deviation 𝜎𝑋̅ = 𝜎 ⁄ √𝑛.
• we can assume that if the sample size is large enough (and we will say n ≥ 30) the sampling distribution will be “close enough” to the normal distribution for us to use the normal tables to find probabilities.
• does not depend on the shape of the original distribution, Even if the original population is very skewed, or bimodal, or completely bizarre, the sampling distribution will eventually approximate a normal curve.
Sample Distribution
Vs
Sampling Distribution
Sample Distribution: The distribution of the values in one specific sample.
Sampling Distribution: The distribution of all the possible sample means for a given sample size n
