1 / 38

Normal Distributions 2/27/12

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. Exam 1 Grades. Bootstrap and Randomization Distributions.

krysta
Download Presentation

Normal Distributions 2/27/12

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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!

More Related