# Sampling Distributions Lesson 6 2 Sampling Distributions Center

Sampling Distributions Lesson 6. 2 Sampling Distributions: Center and Variability Statistics and Probability with Applications, 3 rd Edition Starnes & Tabor Bedford Freeman Worth Publishers

Sampling Distributions: Center and Variability Learning Targets After this lesson, you should be able to: ü Determine if a statistic is an unbiased estimator of a population parameter. ü Describe the relationship between sample size and the variability of a statistic. Statistics and Probability with Applications, 3 rd Edition 2

Sampling Distributions: Center and Variability When using a statistic to estimate a parameter, different samples from a population will produce different values of the statistic. This basic fact is called sampling variability: the value of a statistic varies in repeated random sampling. To make sense of sampling variability, we ask, “What would happen if we took many samples? ” Statistics and Probability with Applications, 3 rd Edition 3

Sampling Distributions: Center and Variability Unbiased Estimator A statistic used to estimate a parameter is an unbiased estimator if the mean of its sampling distribution is equal to the value of the parameter being estimated. The design of a statistical study shows bias if it would consistently underestimate or consistently overestimate the value you want to know when you repeat the study many times. Caution: Don’t trust an estimate that comes from a biased sampling method. Statistics and Probability with Applications, 3 rd Edition 4

Unbiased estimators The doplot below displays simulated sampling distributions of two statistics that can be used to estimate the mean of a population distribution. The simulated sampling distributions are based on 1000 SRSs of size n = 5, and the population mean μ = 40. The mean of each distribution is indicated by a blue line segment. Is either of these statistics an unbiased estimator of the population mean? Explain your reasoning. Statistics and Probability with Applications, 3 rd Edition 5

Sampling Distributions: Center and Variability To get a trustworthy estimate of an unknown population parameter, start by using a statistic that’s an unbiased estimator. This ensures that you won’t tend to overestimate or underestimate. Unfortunately, using an unbiased estimator doesn’t guarantee that the value of your statistic will be close to the actual parameter value. The sampling distribution of any statistic will have less variability when the sample size is larger. Statistics and Probability with Applications, 3 rd Edition 6

Sampling variability A local auto parts store has records on the daily sales of hand tools (in dollars) over the last several years. To estimate the average daily sales, a manager selects 10 days and finds the sample mean daily sales. Here is a simulated sampling distribution of , the sample mean daily sales of hand tools (in dollars) for 1000 samples of size n = 10. • (a) What would happen to the sampling distribution of the sample mean if the sample size were n = 30 instead? Justify. • (b) What is the practical consequence of this change in sample size? Statistics and Probability with Applications, 3 rd Edition 7

Sampling Distributions: Center and Variability We can think of the true value of the population parameter as the bull’seye on a target and of the sample statistic as an arrow fired at the target. Both bias and variability describe what happens when we take many shots at the target. Bias means that our aim is off and we consistently miss the bull’s-eye in the same direction. Our sample values do not center on the population value. High variability means that repeated shots are widely scattered on the target. In other words, repeated samples do not give very similar results. Statistics and Probability with Applications, 3 rd Edition 8

LESSON APP 6. 2 How many tanks does the enemy have? During World War II, the Allies captured many German tanks. Each tank had a serial number on it. Allied commanders wanted to know how many tanks the Germans had so that they could allocate their forces appropriately. They sent the serial numbers of the captured tanks to a group of mathematicians in Washington, D. C. , and asked for an estimate of the total number of German tanks N. Here are simulated sampling distributions for three statistics that the mathematicians considered, using samples of size n =7. The blue line marks N, the total number of German tanks. The shorter red line segments mark the mean of each simulated sampling distribution. Statistics and Probability with Applications, 3 rd Edition 9

LESSON APP 6. 2 How many tanks does the enemy have? 1. Do any of these statistics appear to be unbiased? Justify. 2. Which of these statistics do you think is best? Explain your reasoning. 3. Explain how the Allies could get a more precise estimate of the number of German tanks using the statistic you chose in Question 2. Statistics and Probability with Applications, 3 rd Edition 10

Sampling Distributions: Center and Variability Learning Targets After this lesson, you should be able to: ü Determine if a statistic is an unbiased estimator of a population parameter. ü Describe the relationship between sample size and the variability of a statistic. Statistics and Probability with Applications, 3 rd Edition 11

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