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L07 Ch 07: Sampling and Sampling Distributions Population Sample Census Real Statistic Parameter Statistical Inference estimates Parameters vs. Statistics Numerical characteristics about a population are called _________ ◦ Mean, Standard Deviation, etc… Numerical characteristics about a sample are called ______________ ◦ Mean, Standard Deviation, etc... ◦ Statistics ESTIMATE parameters ◦ X-bar is a point estimate of the population mean It estimates the population mean with a single point rather than an interval ◦ s is a point estimate of s Summary of Point Estimates Obtained from a Simple Random Sample Population Parameter Parameter Value m = Population mean 990 x = Sample mean s = Population std. 80 s = Sample std. deviation for SAT score p = Population proportion wanting campus housing .72 p = Sample proportion wanting campus housing SAT score deviation for SAT score Point Estimator Point Estimate SAT score Sampling Distributions Turns out that sample statistics have their own shape! That is right, they have their own distributions. To see why: Consider the classroom a population of interest. Want to know average height but collecting census is too ―costly‖ ◦ Form SAMPLES of size three ◦ In your sample, calculate the average height in inches. ◦ Notice that before a sample is drawn, x-bar is not known. Therefore it is called a random variable ◦ Will all samples produce the same value for x-bar? ◦ How many unique samples of size 3 can I draw? ◦ Use formula for ______________ Visualize Sampling Distributions In what follows, we will: ◦ Take a sample from a population of observations ◦ From the sample, we will calculate the mean ◦ We will plot the mean on a number line ◦ We will repeat many, many times ◦ Goal: See the shape, or distribution of our sample statistic x-bar Link Sampling Distribution of X-Bar The sampling distribution (shape) of x-bar has a center and it has spread. The center of the x-bar distribution is located at the mean of the underlying __________ E( x ) = mxbar = mx Where m is mean of the population of data points your are drawing from Sampling Distribution of X-Bar Standard Deviation of X-Bar, Two cases 1. Sampling from Finite Population ◦ Where N is the population size, sx N n Nn s x n is sample size and n N 1 N 1 is the finite population correction factor. ◦ When n—the sample size—is less than 5 percent of N—the population sample size—then you x can ignore the correction factor. Infinite population Standard Deviation of a statistic is called the standard error 2. ◦ Above is the standard error of x-bar What is the shape of X-Bar Central Limit Theorem tells us the shape of x-bar If the sample size is ―big enough‖ then the shape of x-bar is that of the ________ __________________ ◦ It does not matter what the underlying Link distribution is. If the underlying distribution is normal, xbar is ____________no matter the sample size. Link Making Probability Statements about X-bar A population has a sample mean of 200 and a standard deviation of 50. Suppose a simple random sample of size 100 is selected and x-bar is used to estimate m ◦ What is the distribution of x-bar? ◦ What is the probability that the sample mean will be within + or – 5 of the population mean? Notice how this builds on chapter 6! We are going to use the formula to convert to z. Instead of dividing by the standard deviation of x, we are going to be dividing by the standard deviation of x-bar. OLD Z FORMULA New Z FORMULA Z xm sx Proportions Can we find the proportion of accounting majors in a population? Can we find the proportion of accounting majors in a sample? What would be a general name for these two quantities? If we take two samples, from the same population, are we likely to get the same value for the proportion? ◦ ___________ ◦ If we plot ____________, we get a shape of the proportion statistic The sample proportion is a ―statistic‖ and hence it has its own shape called the ―sampling distribution‖ Proportions P-bar = sample proportion The shape can be approximated by the ___________________ when ◦ np ≥ 5 number in sample times proportion of accounting majors is greater than 5 ◦ and n(1-P) ≥ 5 number in sample times non accounting majors The center of 𝑝 is at p the population proportion E ( p) p Proportion Standard Deviation of p-bar Finite Population 𝝈𝒑 = 𝒑(𝟏 − 𝒑) 𝑵 − 𝒏 𝒏 𝑵−𝟏 • If sample is ______relative to population • If n/N > .05 Infinite Population sp p (1 p ) n • If sample is ______relative to population • If n/N < .05 Proportions Why do we care about the shape of the sample proportion? ◦ Helps us to understand how close or far away we are from the POPULATION proportion. A simple random sample of size 100 is selected from a population with p = .40 ◦ ◦ ◦ ◦ What is the expected value of 𝑝? What is the standard error of 𝑝? Show the distribution of 𝑝 What does the sampling distribution of 𝑝 show? Properties of Point Estimators Let’s talk about the desirable properties of point estimators ◦ Wouldn’t it be nice if you expected your point estimate to be a correct guess? ____________ 𝐸(𝜃) = 𝜃 This is why we use the finite correction factor and why we divide by n-1 when we first introduced the sample standard deviation. ◦ Wouldn’t it be nice if your estimate did not have a lot of variation? __________ ◦ Wouldn’t it be nice if large sample sizes provided better estimates? __________
