Sampling Distributions: Understanding Parameters and Statistics

1. CHAPTER 7Sampling Distributions 7.1 What Is A Sampling Distribution?

2. What Is A Sampling Distribution? • DISTINGUISH between a parameter and a statistic. • USE the sampling distribution of a statistic to EVALUATE a claim about a parameter. • DISTINGUISH among the distribution of a population, the distribution of a sample, and the sampling distribution of a statistic. • DETERMINE whether or not a statistic is an unbiased estimator of a population parameter. • DESCRIBE the relationship between sample size and the variability of a statistic.

3. Introduction The process of statistical inference involves using information from a sample to draw conclusions about a wider population. Different random samples yield different statistics. We need to be able to describe the sampling distribution of possible statistic values in order to perform statistical inference. We can think of a statistic as a random variable because it takes numerical values that describe the outcomes of the random sampling process. Population Sample Collect data from a representative Sample... Make an Inference about the Population.

4. Parameters and Statistics As we begin to use sample data to draw conclusions about a wider population, we must be clear about whether a number describes a sample or a population. A parameter is a number that describes some characteristic of the population. A statistic is a number that describes some characteristic of a sample. Remember s and p: statistics come from samples and parameters come from populations

5. Categorical Variables: proportion Parameter: Statistic: Population proportion Sample proportion Quantitative Variables: mean Population mean Sample mean Parameter: Statistic:

6. (c) On Tuesday, the bottles of Arizona Iced Tea filled in a plant were supposed to contain an average of 20 ounces of iced tea. Quality control inspectors sampled 50 bottles at random from the day’s production. These bottles contained an average of 19.6 ounces of iced tea. (d) On a New York–to–Denver flight, 8% of the 125 passengers were selected for random security screening before boarding. According to the Transportation Security Administration, 10% of passengers at this airport are chosen for random screening.

7. Sampling Variability 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?” Population Sample ? Sample Sample Sample Sample Sample Sample Sample

8. Sampling Distribution If we took every one of the possible samples of size n from a population, calculated the sample proportion for each, and graphed all of those values, we’d have a sampling distribution. The sampling distribution of a statistic is the distribution of values taken by the statistic in all possible samples of the same size from the same population. In practice, it’s difficult to take all possible samples of size n to obtain the actual sampling distribution of a statistic. Instead, we can use simulation to imitate the process of taking many, many samples.

9. Sampling Distribution vs. Population Distribution There are actually three distinct distributions involved when we sample repeatedly and measure a variable of interest. The population distribution gives the values of the variable for all the individuals in the population. The distribution of sample data shows the values of the variable for all the individuals in the sample. The sampling distribution shows the statistic values from all the possible samples of the same size from the population.

10. Describing Sampling Distributions The fact that statistics from random samples have definite sampling distributions allows us to answer the question, “How trustworthy is a statistic as an estimator of the parameter?” To get a complete answer, we consider the center, spread, and shape. Center: Biased and unbiased estimators In the chips example, we collected many samples of size 20 and calculated the sample proportion of red chips. How well does the sample proportion estimate the true proportion of red chips, p = 0.5? Note that the center of the approximate sampling distribution is close to 0.5. In fact, if we took ALL possible samples of size 20 and found the mean of those sample proportions, we’d get exactly 0.5. A statistic used to estimate a parameter is an unbiased estimator if the mean of its sampling distribution is equal to the true value of the parameter being estimated.

11. Describing Sampling Distributions Spread: Low variability is better! 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=100 n=1000 Larger samples have a clear advantage over smaller samples. They are much more likely to produce an estimate close to the true value of the parameter.

12. Describing Sampling Distributions There are general rules for describing how the spread of the sampling distribution of a statistic decreases as the sample size increases. One important and surprising fact is that the variability of a statistic in repeated sampling does not depend very much on the size of the population. Variability of a Statistic The variability of a statistic is described by the spread of its sampling distribution. This spread is determined mainly by the size of the random sample. Larger samples give smaller spreads. The spread of the sampling distribution does not depend much on the size of the population, as long as the population is at least 10 times larger than the sample.

13. Bias, Variability, and Shape We can think of the true value of the population parameter as the bull’s- eye 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. Repeated samples do not give very similar results.

14. Sample Proportions • FIND the mean and standard deviation of the sampling distribution of a sample proportion. CHECK the 10% condition before calculating the standard deviation of the sample proportions. • DETERMINE if the sampling distribution of sample proportions is approximately Normal. • If appropriate, use a Normal distribution to CALCULATE probabilities involving a sample proportion.

15. The Sampling Distribution of Consider the approximate sampling distributions generated by a simulation in which SRSs of Reese’s Pieces are drawn from a population whose proportion of orange candies is either 0.45 or 0.15. What do you notice about the shape, center, and spread of each?

16. The Sampling Distribution of What did you notice about the shape, center, and spread of each sampling distribution?

17. The Sampling Distribution of In Chapter 6, we learned that the mean and standard deviation of a binomial random variable X are As sample size increases, the spread decreases.

18. The Sampling Distribution of Sampling Distribution of a Sample Proportion As n increases, the sampling distribution becomes approximately Normal. Before you perform Normal calculations, check that the Normal condition is satisfied: np ≥ 10 and n(1 – p) ≥ 10.

19. The Sampling Distribution of

20. Sample Means • FIND the mean and standard deviation of the sampling distribution of a sample mean. CHECK the 10% condition before calculating the standard deviation of a sample mean. • EXPLAIN how the shape of the sampling distribution of a sample mean is affected by the shape of the population distribution and the sample size. • If appropriate, use a Normal distribution to CALCULATE probabilities involving sample means.

21. The Sampling Distribution of When we record quantitative variables we are interested in other statistics such as the median or mean or standard deviation. Sample means are among the most common statistics. Consider the mean household earnings for samples of size 100. Compare the population distribution on the left with the sampling distribution on the right. What do you notice about the shape, center, and spread of each?

22. The Sampling Distribution of When we choose many SRSs from a population, the sampling distribution of the sample mean is centered at the population mean µ and is less spread out than the population distribution. Here are the facts. Sampling Distribution of a Sample Mean as long as the 10% condition is satisfied: n ≤ (1/10)N.

23. Sampling From a Normal Population Sampling Distribution of a Sample Mean from a Normal Population

24. The Central Limit Theorem Most population distributions are not Normal. What is the shape of the sampling distribution of sample means when the population distribution isn’t Normal? It is a remarkable fact that as the sample size increases, the distribution of sample means changes its shape: it looks less like that of the population and more like a Normal distribution! When the sample is large enough, the distribution of sample means is very close to Normal, no matter what shape the population distribution has, as long as the population has a finite standard deviation.

25. The Central Limit Theorem Consider the strange population distribution from the Rice University sampling distribution applet. Describe the shape of the sampling distributions as n increases. What do you notice?

26. The Central Limit Theorem As the previous example illustrates, even when the population distribution is very non-Normal, the sampling distribution of the sample mean often looks approximately Normal with sample sizes as small as n = 25. Normal/Large Condition for Sample Means The central limit theorem allows us to use Normal probability calculations to answer questions about sample means from many observations even when the population distribution is not Normal.

27. The Sampling Distribution of