Sampling Distributions

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2 Sampling Error As you may remember from the first lecture, samples provide incomplete information about the population In particular, a statistic (e.g., M, s) computed on any particular sample drawn from a population will tend to differ from the population parameter (e.g., µ, σ) it represents. This discrepancy between a sample statistic (e.g., M) and its corresponding population parameter (e.g., μ) results from sampling error Recall that sampling error is the variability of a statistic from sample to sample due to chance

3 The Sampling Distribution of the Mean Sampling Distributions Due to sampling error, the statistics computed from sample to sample drawn from a fixed population vary. The sampling distribution of a statistic describes this variation For example, the sampling distribution of the mean describes how the mean (M) computed for a sample of size n varies from one sample to the next when the underlying population distribution remains the same.

4 Example 1: Six-Sided Dice Let s consider a fair 6-sided die The population of interest is the set of all rolls. Since there is no limit to the number of times you can roll a die, the size of the population N is potentially infinite A sample is a set of n rolls There are 6 possible discrete outcomes {1,2,3,4,5,6}, each of which is equally likely, therefore the mean population outcome is 3.5 Let s consider the mean outcome for a sample of n rolls

5 Example 1: Six-Sided Dice Theoretical (analytical) distribution for single roll of a fair 6-sided die μ = 3.5 σ = 1.71

6 Example 1: Six-Sided Dice Let s try it. Roll your die once and enter the resulting score using your clicker

7 Example 1: Six-Sided Dice Now, let s compute the sample mean (M) for a sample of size 2 1. Roll your die twice 2. Compute the mean of the two rolls and round to the nearest whole number 3. Enter the resulting score using your clicker

8 Example 1: Six-Sided Dice Now, let s compute the sample mean (M) for a sample of size 4 1. Roll your die four times 2. Compute the mean of the 4 rolls and round to the nearest whole number 3. Enter the resulting score using your clicker

9 Example 1: Six-Sided Dice Finally, let s compute the sample mean (M) for a sample of size 8 (This is the last one, I promise) 1. Roll your die eight times 2. Compute the mean of the 8 rolls and round to the nearest whole number 3. Enter the resulting score using your clicker

10 Example 1: Six-Sided Dice

11 Example 1: Six-Sided Dice

12 Example 1: Six-Sided Dice

13 Example 1: Six-Sided Dice

14 Example 1: Six-Sided Dice

15 Central Limit Theorem Thanks to the central limit theorem we can compute the sampling distribution of the mean without having to actually draw samples and compute sample means. Central limit theorem: Given a population with mean µ and standard deviation σ, the sampling distribution of the mean (i.e., the distribution of sample means) will itself have a mean of µ and a standard deviation (standard error) of / n Furthermore, whatever the distribution of the parent population, this sampling distribution will approach the normal distribution as the sample size (n) increases.

16 Central Limit Theorem Conceptually, we can break up the theorem into three parts: 1. The mean (µ M ) of a population of sample means (M) is equal to the mean (µ) of the underlying population of scores 2. The standard deviation (σ M ) of a population of sample means (called the standard error) becomes smaller as the size of the sample (n, the number of scores used to compute M) increases 3. Whatever the distribution of the underlying scores, the sampling distribution of the mean will approach the normal distribution as the sample size (n) increases

17 The Mean & Variance as Estimators Sampling Distributions The expected value of a statistic is its long-range average over repeated sampling Notated as E(). For example the expected value of s is denoted E(s) In statistics, bias is a property of a statistic whose expected value does not equal the parameter it represents M is an unbiased estimator of μ (i.e., E(M) = μ) s 2 is an unbiased estimator of σ 2 (i.e., E(s 2 ) = σ 2 ) In statistics, minimum variance (or efficiency) is a property of a statistic whose long-range variability is as small as possible M is a minimum variance estimator of μ s 2 is not a minimum variance estimator of σ 2

18 Standard Error Just as the standard deviation (σ)of a population of scores provides a measure of the average distance between an individual score (x) and the population mean (µ), the standard error (σ M ) provides a measure of the average distance between the sample mean (M) and the population mean (µ). M n

19 Example 2: Exam Scores Student Score

20 Example 2: Exam Scores Sample 1 M 74.2 Student Score (M)

21 Sampling Distribution of the Mean Sampling Distributions Sample 1 M 74.2 Sample 2 M 73.4 Student Score Student Score (M)

22 Example 2: Exam Scores Sample 1 M 74.2 Sample 2 M 73.4 Student Score Student Score Sample 3 M 67.6 Student Score (M)

23 Example 2: Exam Scores Sample 1 M 74.2 Student Score Remember: 70.0, samples (with N=5) x x Sample 2 M 73.4 Sample 3 M 67.6 Student Score Student Score M M (M)

24 p(m) Sampling Distributions Sample Size & Standard Error n 1 M 7 n 4 M n 16 M (M)

25 The Sampling Distributions of s 2 and s Sampling Distributions Sample 1 M 74.2 Sample 2 M 73.4 Student Score Student Score s s 6.76 s s n 5 Sample 3 M 67.6 Student Score s s 3.44

26 Standardizing the Sampling Distribution Sampling Distributions Because the distribution of sample means tends to be normal, the z-score value obtained for a sample mean can be used with the unit normal table to obtain probabilities. The procedures for computing z-scores and finding probabilities for sample means are essentially the same as we used for individual scores However, when computing z-scores for sample means, you must remember to consider the sample size (n) and use the standard error (σ M ) instead of the standard deviation (σ)

27 Standardizing the Sampling Distribution Sampling Distributions Remember, to compute z-scores for individual values: z x x To compute z-statistics for sample means: z M M M M n, because M n

28 Sample problems: For a population of exam scores with μ = 70 and σ = 10: What is the probability of obtaining M between 65 and 75 for a sample of 4 scores? What is the probability of obtaining M > 75 for a sample of 16 scores?

29 z Upper-Tail Probabilities