Chapter 6: Sampling Distributions

McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions Where We’ve Been The objective of most statistical analyses is inference. Sample statistics (mean, standard deviation) can be used to make decisions. Probability distributions can be used to construct models of populations. McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions

McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions Where We’re Going Develop the notion that sample statistic is a random variable with a probability distribution. Define a sampling distribution for a sample statistic. Link the sampling distribution of the sample mean to the normal distribution. McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions

McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions In practice, sample statistics are used to estimate population parameters. A parameter is a numerical descriptive measure of a population. Its value is almost always unknown. A sample statistic is a numerical descriptive measure of a sample. It can be calculated from the observations. McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions

McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions Parameter Statistic Mean µ  Variance 2 s2 Standard Deviation  s Binomial proportion p McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions

6.1: The Concept of a Sampling Distribution Since we could draw many different samples from a population, the sample statistic used to estimate the population parameter is itself a random variable. The probability distribution of a sample statistic calculated from a sample of n measurements is called the sampling distribution of the statistic. McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions

6.1: The Concept of a Sampling Distribution Imagine a very small population consisting of the elements 1, 2, and 3. Its mean is μ = 2 and std dev σ = 1. Below are the possible samples that could be drawn, along with the means of the samples and the mean of the means. n = 1  1 2 3 n = 2  1, 2 1.5 1, 3 2 2, 3 2.5 n = 3  1, 2, 3 2 McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions

6.1: The Concept of a Sampling Distribution Imagine a very small population consisting of the elements 1, 2, 3, and 4. Its mean is μ = 2.5 and std dev σ = 1.29. Below are the possible samples that could be drawn, along with the means of the samples and the mean of the means. n = 1  1 2 3 4 n = 2  1, 2 1.5 1, 3 2 1,4 2.5 2, 3 2,4 3 3,4 3.5 n = 3  1,2,3 2 1,2,4 7/3 1,3,4 8/3 2,3,4 3 n = 4  1, 2, 3,4 2.5 McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions 8

6.1: The Concept of a Sampling Distribution µ To estimate …should we use …  ? the median … or … McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions

6.1: The Concept of a Sampling Distribution µ To estimate …should we use …  ? the median … or … The answer can be Yes for either! (Depending on the distribution of the random variable, e.g., for symmetric distributions.) McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions

McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions 6.2: Properties of Sampling Distributions: Unbiasedness and Minimum Variance A point estimator is a single number based on sample data that can be used as an estimator of the population parameter  µ p s2 σ2 McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions

McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions 6.2: Properties of Sampling Distributions: Unbiasedness and Minimum Variance If the sampling distribution of a sample statistic has a mean equal to the population parameter, the statistic is said to be an unbiased estimate of the parameter. McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions

McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions 6.2: Properties of Sampling Distributions: Unbiasedness and Minimum Variance If two alternative sample statistics are both unbiased, the one with the smaller standard deviation is preferred. Here, μA = μB, but σA < σ B, so A is preferred. McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions

Properties of the Sampling Distribution of  6.3: The Sampling Distribution of  and the Central Limit Theorem (CLT) Properties of the Sampling Distribution of  The mean of the sampling distribution equals the mean of the population The standard deviation (or standard error) of the sample mean equals the population standard deviation (of the population) divided by the square root of n McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions

6.3: The Sampling Distribution of  and the CLT Here’s our small population again, this time with the standard deviations of the Its mean is μ = 2 and std dev σ = 1. sample means. Notice the mean of the sample means in each case equals the population mean and the standard error decreases as n increases. n = 1  1 2 3 n = 2  1, 2 1.5 1, 3 2 2, 3 2.5 n = 3  1, 2, 3 2 McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions

6.1: The Concept of a Sampling Distribution Imagine a very small population consisting of the elements 1, 2, 3, and 4. Its mean is μ = 2.5 and std dev σ = 1.29. Below are the possible samples that could be drawn, along with the means of the samples and the mean of the means. n = 1  1 2 3 4 n = 2  1, 2 1.5 1, 3 2 1,4 2.5 2, 3 2,4 3 3,4 3.5 n = 3  1,2,3 2 1,2,4 7/3 1,3,4 8/3 2,3,4 3 n = 4  1, 2, 3,4 2.5 McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions 16

6.3: The Sampling Distribution of  and the CLT If a random sample of n observations is drawn from a normally distributed population, the sampling distribution of  will be normally distributed McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions

6.3: The Sampling Distribution of  and the CLT The Central Limit Theorem The sampling distribution of , based on a random sample of n observations, will be approximately normal with µ = µ and  = /√n . The larger the sample size, the better the sampling distribution will approximate the normal distribution. McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions

6.3: The Sampling Distribution of  and the CLT Suppose existing houses for sale average 2200 square meter in size, with a standard deviation of 250 m2. What is the probability that a randomly selected sample of 36 houses will average at least 2300 m2 ? McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions

6.3: The Sampling Distribution of  and the CLT Suppose existing houses for sale average 2200 square meter in size, with a standard deviation of 250 m2. What is the probability that a randomly selected sample of 36 houses will average at least 2300 m2 ? McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions

6.3: The Sampling Distribution of  and the CLT Suppose existing houses for sale average 2200 square meter in size, with a standard deviation of 250 m2. Assuming the areas of the house are normal, what is the probability that a randomly selected house will have at least 2300 m2 ? McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions

6.3: The Sampling Distribution of  and the CLT Suppose existing houses for sale average 2200 square meter in size, with a standard deviation of 250 m2. Assuming the areas of the house are normal, what is the probability that a randomly selected house will have at least 2300 m2 ? McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions

6.3: The Sampling Distribution of  and the CLT Suppose existing houses for sale average 2200 square meter in size, with a standard deviation of 250 m2. (1) Assuming the areas of the house are normal, what is the probability that a randomly selected house will have at least 2300 m2 ? (2) What is the probability that a randomly selected sample of 36 houses will average at least 2300 m2 ? Notice that in (1) we assumed normality of the areas of the houses, but in (2) we do not need to assume this, because n=36>30 so by CLT the average of 36 houses is approximately normal. McClave: Statistics, 11th ed. Chapter 6: Sampling Distributions