Chapter 7 – Sampling DistributionsIn Chapter 6 we spent time defining population distributions, which is how the individuals in a population are distributed.In statistics, particularly inferential statistics, we are usually dealing with a sample and its statistics, so we want to describe how those statistics are distributed (e.g., the distribution of the sample mean or sample proportion).Definition: the probability distribution of a sample statistic is called a sampling distribution. We use the sampling distribution to draw inference about the population parameters (the things that define the probability distributions) using the sample data at hand.The sampling distribution of a sample proportion ^p is defined by its mean and standard error. If we are dealing with binary data, a random sample of size n will yield a sample proportion of successes ^p, our sample statistic.The mean of ^p is p, the population proportion. (μ^p= p)The standard error of ^p is √p(1− p)n where n is the sample size and p is the population proportion (s . e.{^p}=√p(1− p)n¿It turns out that if n is sufficiently large (i.e., np and n(1-p) are both at least 15), the distribution of ^p is approximately normal.That is, for large enough sample sizes, the sampling distribution of the sample proportion is approximately normal with mean p and standard deviation √p(1− p)n. We can therefore find probabilities for ^p using the standard normal distribution if we collect a big enough sample.Example: You sample 100 Skittles randomly and get 5 red candies.1. What is the mean of the sampling distribution?2. What is the standard error?3. What is the sample proportion?4. What is the probability of getting a value lower than your answer to (3)?1.μ^p= p=0.20 (the true proportion of red Skittles in the population)2.s . e.{^p}=√p(1− p)n=√(.2) (.8)100=0.043.^p=xn=5100=0.054.Pr[^p ≤0.05]=Pr[^p− ps .e .{^p}≤0.05−.20.04]=Pr[Z ≤−3.75]´¿0.00 (between 0.0000317 and 0.000233)With a quantitative r.v., we describe the sample using its mean, ´x. The sampling distribution of ´x is defined by its mean and standard error:The mean of ´x is µ , the mean of the probability distribution (μ´x=μ)The standard error of ´x is σ√n, where n is the sample size and σ is the population standarddeviation (s.e.´{x }=σ√n)This is true for any quantitative r.v.Interesting results:1. If the r.v., comes from a normal distribution, the sampling distribution of the mean will also be normal.2. If the sample size is sufficiently large, the sampling distribution of the standardized values (i.e., Z =´X−μσ /√n¿ will be standard normal, regardless of the distribution of the r.v.(2) results from the Central Limit Theorem.Example: A random sample of 8 female huskies is selected to pull a sled for the Iditarod. Their average weight is 44 lbs.1. What is the mean of the sampling distribution?2. What is the standard error of the sampling distribution?3. What is the sample statistic?4. What is the probability of obtaining a heavier team?1.μ´x=μX=42 lbs2.s . e.´X=σ√n=0.67√8=0.237lbs3.´x=44 lbs4.Pr[´X >44 lbs]=Pr [´X −μσ√n>44−42.67√8]=Pr[Z >8.439]=1−Pr[Z ≤ 8.439]´¿0.00Remember my earlier claim that sample size plays an important role in statistical inference. Whatif I had only sampled 2 huskies? What would happen to my standard error? To my resulting