6 Sampling Distributions Lesson 6 2 Sampling Distributions
6 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? ” Sample Population Sample Sample ? Sample 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
What is the mean-ing of bias? Unbiased estimators PROBLEM: The dotplot 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. Statistics and Probability with Applications, 3 rd Edition 5
What is the mean-ing of bias? Unbiased estimators PROBLEM: Is either of these statistics an unbiased estimator of the population mean? Explain your reasoning. Statistic 1 appears to be unbiased, because the mean of its sampling distribution is very close to 40, the value of the population mean. Statistic 2 appears to be biased, because the mean of its sampling distribution is around 44, which is clearly greater than 40, the value of the population mean. Statistics and Probability with Applications, 3 rd Edition 6
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. n=1000 The sampling distribution of any statistic will have less variability when the sample size is larger. Statistics and Probability with Applications, 3 rd Edition 7
Can you hand me that wrench? Sampling variability • Statistics and Probability with Applications, 3 rd Edition 8
Can you hand me that wrench? Sampling variability (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? (a) The sampling distribution of the sample mean will be less variable because the sample size is larger. (b) The estimated mean daily sales of hand tools will typically be closer to the true mean daily sales of hand tools. In other words, the estimate will be more precise. Statistics and Probability with Applications, 3 rd Edition 9
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 10
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 11
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 12
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 13
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