The Sampling Distribution

1. The Sampling Distribution Introduction to Hypothesis Testing and Interval Estimation

2. Outline • Distinctions • Sampling Distribution • The Central Limit Theorem • Confidence Intervals

3. Random Sampling

4. Key things to keep in mind • Population- what we want to talk about • Sample- what we have with our data • Sampling distribution- the means by which we will go from our sample to the population

5. Sampling Distribution • Sampling distributions concern any statistic we can come up with. Examples: • Measures of Central Tendency • Measures of Variability • Measures of Relationship • Ratios • Sample != sampling distribution • Recall also that sampling distributions can be theoretical (used in most studies) or empirical (seeing wider use via bootstrapping). • It is the properties of the sampling distribution

6. Central Limit Theorem (CLT) • Suppose X is • random • mean  • standard deviation • not necessarily normal

7. Terms Concerning Sampling Distribution of the Mean • Standard Error of the mean: • Is just the standard deviation of the sampling distribution. • i.e. it is a particular standard deviation • Sampling error • The sample cannot be fully representative of the population • As such, there is variability due to chance • We could have a thousand sample means and none of them equal exactly the population mean. However…

8. CLT (continued) • Properties of the sampling distribution of the mean • random • has a mean of  • has a standard error • Distributed approximately normal for large samples • Normal for all samples if the variable X is normal

9. The Central Limit Theorem • For any population of scores, regardless of form, the sampling distribution of the mean will approach a normal distribution as the sample size (N) gets larger. • This of course begs the question of what is ‘large enough’ • Furthermore, the sampling distribution of the mean will have a mean equal to µ (the population mean), and a standard deviation equal to

10. Central Limit Theorem • With the mean, we can use sample data and the normal curve to reach conclusions about the population of interest • We of course desire large, random samples in order to do • Non-random selection can result in under-selection or over-selection of subsections of the population. • e.g. carry out a telephone opinion poll • http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/index.html

11. In summary: sample means • are random • are normally distributed for large sample sizes • distribution has mean  • distribution has standard error (standard deviation)

12. Confidence intervals • Draw a sample, gives us a mean • is our best guess at µ • For most samples will be close to µ • is a ‘point’ estimate • However, we can also give a range or interval estimate that takes into account the uncertainty involved in that estimate • Using the normal distribution

13. Confidence interval equation Where = sample mean Z = z value from normal curve = standard error of the mean

14. 95% confidence interval • Let’s say we want a 95% confidence interval. • Obtain1 the ‘critical’ z-score for p =.025 • 2.5% above +z, and 2.5% below -z • p = .025 then z = 1.96 • When the population standard deviation is not known, we use the t critical value instead

15. Confidence interval example • Randomly selected a group of 10 of you folks with a mean score of 89 (s = 6) on the midterm. • What guess can we make as to the true mean of the class?

16. 89 + 2.26* • 89 + 2.26(1.90) • (89 - 4.294) < < (89 + 4.294) • 84.71 < < 93.294 • This seems pretty wide; it essentially covers a full letter grade. Why do you think that is?

17. Important: what a confidence interval means • A 95% confidence interval means that: 95% of the confidence intervals calculated on repeated sampling of the same population will contain µ • Note that the population value does not vary i.e. it’s not a 95% chance that it falls in that specific interval1 • In other words, the CI attempts to capture the true population mean, but we would have a different interval estimate for each sample drawn • http://www.ruf.rice.edu/~lane/stat_sim/conf_interval/index.html • In R library(animation) conf.int(.95)

18. Question to think about • How does one know if the confidence interval calculated actually contains the true population mean?