Download Pointers -‐‑Section 8.1

Pointers -­‐‑Section 8.1 Goal: To understand the concepts of a sampling distribution of sample means. Definitions A sampling distribution of a statistic is a probability distribution for all possible values for that statistic for a sample size of size n.* (This is a generic definition. Remember a statistic is a number that describes the sample.) A sampling distribution for the sample mean, 𝑥 , is the probability distribution for all possible values of the random variable 𝑥 computed from a sample of size n from a population with mean 𝜇 and standard deviation 𝜎 .* (This is the same definition applied to a specific statistic – the sample mean.) WHAT DOES THAT EVEN MEAN??? REVIEW. (Assume you are using Numeric Data.) • Population o The population is the entire group to be studied. o The shape of the distribution for the population may be symmetric (bell-­‐‑shaped), uniform, skewed left, skewed right, bimodal, etc. The shape of the distribution for a population is a FIXED shape. (There is only ONE population.) o The mean of a population is represented by 𝝁. There is ONLY ONE population mean. The population mean is a FIXED quantity. (The mean is the “center” or “representative” value.) o The standard deviation of a population is represented by 𝝈. There is ONLY ONE population standard deviation. The standard deviation of a population is a FIXED quantity. (The standard deviation describes the spread from the mean.) • Sample of size n o A sample is a subset of the population. If the sample size is n, then it contains “n” objects from the population. o There are many different samples in a population. Every sample will have its own unique shape – symmetric (bell-­‐‑shape), uniform, skewed left, skewed, right, bimodal etc. The shape of the distribution for an individual sample will vary from sample to sample and it may be the same or different than the population. o The mean of a sample is represented by 𝒙. Every sample will have its own sample mean, or 𝑥. The sample means, or 𝑥 ’s, will VARY from sample to sample. o The standard deviation of a sample is represented by s. Every sample will have its own sample standard deviation, or s. The sample standard deviations, or s’s, will VARY from sample to sample. Developed by Sharleen McCarroll in conjunction with the textbook Interactive Statistics: Informed Decisions Using Data, by Michael Sullivan III & George Woodbury, 2016. th
Reference: Sullivan III, Michael, Fundamentals of Statistics: Informed Decisions Using Data, 4 ed. Pearson, 2014. NEW. • Sampling Distribution of the Sample mean. o A sampling distribution is created by taking multiple samples of size n from the same population. In fact, a sampling distribution is created by taking every possible sample of size n, from a population. (Note: All of the samples in a sampling distribution have the same sample size. A sampling distribution might consist of all possible samples of size 5, 15, 35, etc.) o Remember: Every sample in a sampling distribution has its own sample mean, or 𝑥 . (Some 𝑥’s might be the same, some might be different.) o A sampling distribution of the sample mean is created by using all of the sample means, or 𝑥 ’s, from your samples to form another, separate distribution. à Essentially, you can think of a sampling distribution of the sample mean as “a bunch of 𝑥 ’s”, or a bunch of sample means. o The mean for the sampling distribution of the sample mean is denoted: 𝜇( . (Think of this as the mean of means. It is the mean of all of the 𝑥 ’s.) o The standard deviation for the sampling distribution of sample mean is denoted 𝜎( . (Think of this as the standard deviation of sample means. It is the standard deviation of the 𝑥’s.) PICTURE Example: Find the sampling distribution of the sample mean from the population: { 1, 2, 3}. Here we will use n = 2. Step 1: Find all possible samples of size n = 2 from the population. (There are 9 “different” samples because the ordering is important). Step 2: Find the sample mean for each sample. Step 3: Use the sample means (or 𝑥 ’s) to create a NEW distribution. This is the sampling distribution of sample means. You can then find the mean and standard deviation of the “new” distribution of sample means. The Central Limit Theorem Why is the Central Limit Theorem (CLT) Important? The Central Limit Theorem says that as we draw more and more samples from our population, the sampling distribution will have a normal (symmetric) distribution. In fact, as n gets larger, the sampling distribution will have an approximately normal distribution, regardless of the population it came from. This is an amazing concept in Statistics because even if our population does not have a symmetric distribution, or if it is an unknown distribution, if our sample size is large enough, the sampling distribution will have an approximate normal distribution. Why is a normal distribution important? If our sampling distribution is normal, then all of the properties that we learned in Chapter 7 will apply to the sampling distribution as well. (NOTE: The concept of the normal distribution will continue to be utilized throughout the rest of this course. Chapter 9 – 14.) Central Limit Theorem for 𝒙-­‐‑ CLT You can assume that the sampling distribution for the sample mean will have a normal distribution if: a) The sampling distribution comes from a population that has a normal distribution. -­‐‑OR -­‐‑ b) The sample size is sufficiently large. (We will use 𝑛 ≥ 30). How to calculate the mean & standard deviation for a sampling distribution of the sample mean Suppose that a simple random sample of size n is drawn from a population, where the population mean is 𝜇 , the population standard deviation is 𝜎 , and 𝑛 ≤ 0.05𝑁. (That is to say that your sample size is no more than 5% of the population size.) • The mean of the sampling distribution for the same mean is: 𝜇( = 𝜇 • The standard deviation of the sampling distribution for the sample mean is: 𝜎
𝜎( =
𝑛
NOTE: This is also referred to as the standard error for the mean.