Normal Distributions 2/27/12

1. Normal Distributions • 2/27/12 • Normal Distribution • Central Limit Theorem • Normal distributions for confidence intervals • Normal distributions for p-values • Standard Normal • Corresponding Sections: 5.1, 5.2

2. Exam 1 Grades

3. Bootstrap and Randomization Distributions Correlation: Malevolent uniforms Slope :Restaurant tips All bell-shaped distributions! What do you notice? Mean :Body Temperatures Diff means: Finger taps Proportion : Owners/dogs Mean : Atlanta commutes

4. Normal Distribution • The symmetric, bell-shaped curve we have seen for almost all of our bootstrap and randomization distributions is called a normal distribution

5. Central Limit Theorem! For a sufficiently large sample size, the distribution of sample statistics for a mean or a proportion is normal http://onlinestatbook.com/stat_sim/sampling_dist/index.html

6. Central Limit Theorem • The central limit theorem holds for ANY original distribution, although “sufficiently large sample size” varies • The more skewed the original distribution is (the farther from normal), the larger the sample size has to be for the CLT to work

7. Central Limit Theorem • For distributions of a quantitative variable that are not very skewed and without large outliers, n ≥ 30 is usually sufficient to use the CLT • For distributions of a categorical variable, counts of at least 10 within each category is usually sufficient to use the CLT

8. Normal Distribution • The normal distribution is fully characterized by it’s mean and standard deviation

9. Normal Distribution

10. Bootstrap Distributions If a bootstrap distribution is approximately normally distributed, we can write it as N(parameter, sd) N(statistic, sd) N(parameter, se) N(statistic, se) sd = standard deviation of variable se = standard error = standard deviation of statistic

11. Confidence Intervals If the bootstrap distribution is normal: To find a P% confidence interval , we just need to find the middle P% of the distribution N(statistic, SE)

12. Best Picture What proportion of visitors to www.naplesnews.comthought The Artist should win best picture?

13. Best Picture www.lock5stat.com/statkey

14. Area under a Curve • The area under the curve of a normal distribution is equal to the proportion of the distribution falling within that range • Knowing just the mean and standard deviation of a normal distribution allows you to calculate areas in the tails and percentiles http://davidmlane.com/hyperstat/z_table.html

15. Best Picture http://davidmlane.com/hyperstat/z_table.html

16. Best Picture

17. Confidence Intervals For a normal sampling distribution, we can also use the formula to give a 95% confidence interval.

18. Confidence Intervals For normal bootstrap distributions, the formula gives a 95% confidence interval. How would you use the N(0,1) normal distribution to find the appropriate multiplier for other levels of confidence?

19. Confidence Intervals For a P% confidence interval, use where P% of a N(0,1) distribution is between –z* and z*

20. Confidence Intervals 95% -z* z*

21. Confidence Intervals Find z* for a 99% confidence interval. http://davidmlane.com/hyperstat/z_table.html z* = 2.576

22. News Sources “A new national survey shows that the majority (64%) of American adults use at least three different types of media every week to get news and information about their local community” The standard error for this statistic is 1% Find a 99% confidence interval for the true proportion. Source: http://pewresearch.org/databank/dailynumber/?NumberID=1331

23. News Sources

24. Confidence Interval Formula From N(0,1) From original data From bootstrap distribution

25. First Born Children • Are first born children actually smarter? • Based on data from last semester’s class survey, we’ll test whether first born children score significantly higher on the SAT • From a randomization distribution, we find SE = 37

26. First Born Children What normal distribution should we use to find the p-value? N(30.26, 37) N(37, 30.26) N(0, 37) N(0, 30.26)

27. Hypothesis Testing

28. p-values If the randomization distribution is normal: To calculate a p-value, we just need to find the area in the appropriate tail(s) beyond the observed statistic of the distribution N(null value, SE)

29. First Born Children N(0, 37) http://davidmlane.com/hyperstat/z_table.html p-value = 0.207

30. First Born Children

31. Standard Normal • Sometimes, it is easier to just use one normal distribution to do inference • The standard normal distribution is the normal distribution with mean 0 and standard deviation 1

32. Standardized Test Statistic • The standardized test statistic is the number of standard errors a statistic is from the null value • The standardized test statistic (also called a z-statistic) is compared to N(0,1)

33. p-value Find the standardized test statistic: The p-value is the area in the tail(s) beyond z for a standard normal distribution

34. First Born Children Find the standardized test statistic

35. First Born Children Find the area in the tail(s) beyond z for a standard normal distribution p-value = 0.207

36. z-statistic • Calculating the number of standard errors a statistic is from the null value allows us to assess extremity on a common scale

37. Formula for p-values From original data From H0 From randomization distribution Compare z to N(0,1) for p-value

38. Standard Error • Wouldn’t it be nice if we could compute the standard error without doing thousands of simulations? • We can!!! • Or rather, we’ll be able to on Wednesday!