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HUDM4122 Probability and Statistical Inference

HUDM4122 Probability and Statistical Inference. April 1, 2015. Questions from last class?. Continuing from last class…. Another way to think about Confidence Intervals. For a 95% Confidence Interval 5% of the time , the true value will be outside the confidence interval.

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HUDM4122 Probability and Statistical Inference

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  1. HUDM4122Probability and Statistical Inference April 1, 2015

  2. Questions from last class?

  3. Continuing from last class…

  4. Another way to think aboutConfidence Intervals • For a 95% Confidence Interval • 5% of the time , the true value will be outside the confidence interval

  5. Another way to think aboutConfidence Intervals • For a 95% Confidence Interval • 5% of the time , the true value will be outside the confidence interval • We can write this proportion of 5% as a • If a is 0.05, then we have a 95% Confidence Interval

  6. So, for a 95% Confidence Interval • 95% Confidence Interval • a is 0.05, • Cumulative Normal Probability is from 0.025 to 0.975 • Meaning that Confidence Intervals bounds are -1.96 SE and +1.96 SE • These values are called and • = • = + 1.96

  7. Why is it ? • Because to get a 95% confidence interval • = 0.05 • 0.975 – 0.025 = 0.95 • 0.025 = /2 • 0.975 = 1-(/2)

  8. So, formally • For given a • The confidence intervals are • Where is just your standard error • And we can use s (sample standard deviation) for whenever the sample is sufficiently large

  9. Regarding this whole “sufficiently large” thing • MBB say “sufficiently large” is when N>30 • That’s probably reasonable • We’ll come back to this issue in a couple weeks when we discuss the difference between Z statistical tests and t statistical tests

  10. Questions? Comments?

  11. For a 99% Confidence Interval • 99% Confidence Interval • a is 0.01, • Cumulative Normal Probability is from 0.005 to 0.995 • Look in your table to get 0.005 and 0.995 • That’s -2.57 and +2.57 • Meaning that Confidence Intervals bounds are -2.57 SE and +2.57 SE • = • = + 2.57

  12. You try it: 90% Confidence Interval • What are the bounds?

  13. You try it: 90% Confidence Interval • 90% Confidence Interval • a is 0.10, • Cumulative Normal Probability is from 0.05 to 0.95 • Look in your table to get 0.05 and 0.95 • That’s -1.64 and +1.64 • Meaning that Confidence Intervals bounds are -1.64 SE and +1.64 SE

  14. You try it: 90% Confidence Interval • 90% Confidence Interval • a is 0.10, • Cumulative Normal Probability is from 0.05 to 0.95 • Look in your table to get 0.05 and 0.95 • That’s -1.64 and +1.64 • = • = + 1.64

  15. Comments? Questions?

  16. Note • The smaller your a • The larger your % Confidence Interval • In other words, the bigger your interval • The more certain you are • And the smaller your interval • The less certain you are

  17. What you can adjust • You can adjust • Your level of certainty • Your sample size • You can’t adjust • The population mean • The population standard deviation

  18. So if you want to be more certain • Get a bigger sample size

  19. So if you want to be more certain • Get a bigger sample size • Which is not always easy 

  20. On to today’s class…

  21. Today • Chapter 8.6-8.7 in Mendenhall, Beaver, & Beaver • Estimating Differences Between Means and Proportions

  22. Often… • We don’t just want to estimate one mean or proportion • We want to estimate the difference between two means or proportions

  23. Examples • You conduct a randomized experiment on the effectiveness of ASSISTments versus Dreambox. Which one has higher learning gains? • You test out two medicines in a randomized experiment. Which one leads to a higher proportion of patients surviving? • You test the SAT scores of students who take honors classes versus regular classes. Which group has higher SAT scores?

  24. Formally • Once assigned and treated differently, these become two populations • Population 1 • Population 2

  25. Each has their own set of statistics

  26. What is the difference between means?

  27. What is the difference between means? • We know already that each group’s mean can be treated as a single value, a point estimate • Or as a distribution

  28. What is the difference between means? • So it stands to reason that the difference between group means • Can be treated as a single value, a point estimate • Or as a distribution

  29. What is the difference between means? • Point estimate of the difference • Is the difference between the two point estimates of the mean E.g. our best estimate of is

  30. We can call this value • The mean difference between groups • Or the most likely value for the difference between groups

  31. Example • Students using ASSISTments gain 30 points pre-post. Students using Dreambox gain 5 points pre-post. What is the mean difference in learning gains? • 30-5 = 25 points

  32. You try it • You test the SAT scores of students who take honors classes versus regular classes. Honors students average 650. Regular students average 480. What is the mean difference in SAT?

  33. You try it • You test the SAT scores of students who take honors classes versus regular classes. Honors students average 650. Regular students average 480. What is the mean difference in SAT? • 170 points

  34. The Standard Error • For the difference between groups • For sufficiently large samples

  35. Example • Students using ASSISTments gain 30 points pre-post, with a standard deviation of 10 points. Students using Dreambox gain 5 points pre-post, with a standard deviation of 8 points. There are 30 students in each condition. What is the standard error for the mean difference in learning gains? • =

  36. Example • Students using ASSISTments gain 30 points pre-post, with a standard deviation of 10 points. Students using Dreambox gain 5 points pre-post, with a standard deviation of 8 points. There are 30 students in each condition. What is the standard error for the mean difference in learning gains? • = = 2.34

  37. You try it • You test the SAT scores of students who take honors classes versus regular classes. In your school, honors students average 650, with standard deviation 40. Regular students average 480, with standard deviation 20. Your school has 40 honors students and 200 regular students. What is the standard error for the mean difference in SAT?

  38. = • You test the SAT scores of students who take honors classes versus regular classes. In your school, honors students average 650, with standard deviation 40. Regular students average 480, with standard deviation 20. Your school has 40 honors students and 200 regular students. What is the standard error for the mean difference in SAT?

  39. = = • You test the SAT scores of students who take honors classes versus regular classes. In your school, honors students average 650, with standard deviation 40. Regular students average 480, with standard deviation 20. Your school has 40 honors students and 200 regular students. What is the standard error for the mean difference in SAT?

  40. = = 6.5 • You test the SAT scores of students who take honors classes versus regular classes. In your school, honors students average 650, with standard deviation 40. Regular students average 480, with standard deviation 20. Your school has 40 honors students and 200 regular students. What is the standard error for the mean difference in SAT?

  41. By Central Limit Theorem • The sampling distribution of • Is approximately normal when • Both n1 and n2 are > 30

  42. We’ll talk about cases with smaller data sets • Later in the semester

  43. Questions? Comments?

  44. Since it is normally distributed • You can compute the 95% Confidence Interval as we discussed in the last class

  45. Example • Students using ASSISTments gain 30 points pre-post, with a standard deviation of 10 points. Students using Dreambox gain 5 points pre-post, with a standard deviation of 8 points. There are 30 students in each condition. What is the standard error for the mean difference in learning gains? • 2.34

  46. Example • Students using ASSISTments gain 30 points pre-post, with a standard deviation of 10 points. Students using Dreambox gain 5 points pre-post, with a standard deviation of 8 points. There are 30 students in each condition. What is the standard error for the mean difference in learning gains? • 2.34 • 95% CI = [25-(1.96)(2.34), 25+(1.96)(2.34)]

  47. Example • Students using ASSISTments gain 30 points pre-post, with a standard deviation of 10 points. Students using Dreambox gain 5 points pre-post, with a standard deviation of 8 points. There are 30 students in each condition. What is the standard error for the mean difference in learning gains? • 2.34 • 95% CI = 20.41, 29.59

  48. You try it • You test the SAT scores of students who take honors classes versus regular classes. In your school, honors students average 650, with standard deviation 40. Regular students average 480, with standard deviation 20. Your school has 40 honors students and 200 regular students. What is the standard error for the mean difference in SAT?

  49. Estimating the Difference Between Two Proportions • Coming from a binomial distribution

  50. What is the difference between means?

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