1. Copyright (c) Bani Mallick1 Stat 651 Lecture 5

2. Copyright (c) Bani Mallick2 Topics in Lecture #5 Confidence intervals for a population mean  when the population standard deviation  is known. Properties of confidence intervals: what things make them longer and shorter. Sample size calculation for a population mean  when the population standard deviation  is known : a simple illustration of a method.

3. Copyright (c) Bani Mallick3 Book Sections Covered in Lecture #5 Chapter 5.1 Chapter 5.2 Chapter 5.3

4. Copyright (c) Bani Mallick4 Lecture 4 Review: Pr(X < c) for Normal Populations Compute the z-score Look up value in Table 1

5. Copyright (c) Bani Mallick5 Lecture 4 Review: Pr(X > c) for Normal Populations Compute the z-score Look up the value for z in Table 1 Subtract this value from 1.0

6. Copyright (c) Bani Mallick6 Lecture 4 Review: Inference The sample mean is a random variable Its own “population” mean is  It’s standard deviation is Note how the standard deviation of the sample mean becomes smaller as the sample size becomes larger More data = more precision!!!!!

7. Copyright (c) Bani Mallick7 Lecture 4 Review: Central Limit Theorem The sample mean is a random variable Its own “population” mean is  It’s standard deviation is In “large enough” samples, the sample mean is very nearly normally distributed, i.e., has a bell--shaped histogram

8. Copyright (c) Bani Mallick8 Confidence Interval for a Population Mean A considerable part of basic statistics is to make inferences about the population mean  It is impossible to know the value of  exactly. This is a key factoid: why do I say this with such certainty?

9. Copyright (c) Bani Mallick9 Confidence Interval for a Population Mean A considerable part of basic statistics is to make inferences about the population mean  It is impossible to know the value of  exactly. Because (almost) every sample will give you a unique sample mean, and that sample mean will not equal the population mean.

10. Copyright (c) Bani Mallick10 Confidence Interval for a Population Mean What we can do is to construct an interval of possible values for the population mean . The interval is determined by how much “confidence” we want in saying that the population mean  is in the interval. The interval is always of the form

11. Copyright (c) Bani Mallick11 Confidence Interval for a Population Mean The confidence factor is determined by how much confidence we want in concluding that the population mean is actually in the interval Which interval has higher confidence of including the population mean? -100 to -50 OR -150 to 0

12. Copyright (c) Bani Mallick12 Confidence Interval for a Population Mean: Formal Method The first method assumes that the population standard deviation  is known. Suppose we want to be 95% confident that our interval includes the population mean , i.e., the probability is 95% that the population mean  is in the interval. Here is the interval:

13. Copyright (c) Bani Mallick13 WOMEN’S INTERVIEW SURVEY OF HEALTH (WISH)  computed the reported mean caloric intake at the start of the study, and the mean reported caloric intake at the end My random variable X was the change (difference) My hypothesis is that the population mean of X is < 0. In other words, I think women report less calories the more they are asked about their diet (Hawthorne Effect).

14. Copyright (c) Bani Mallick14 WISH: Change in Caloric Intake Does it look like a big change? Note that the scale of the box plot is -3000 to 2000

15. Copyright (c) Bani Mallick15 WISH The sample size is n = 271 The sample mean change = -180 I am going to pretend that the population standard deviation is  = 600.

16. Copyright (c) Bani Mallick16 WISH: Change in Reported Caloric Intake n = 271,  = 600, = -180 95% CI = -251 to -109

17. Copyright (c) Bani Mallick17 Review  = 600, n = 271, = -180 Then, with 95% probability, true population mean change is in the interval from -251 to - 109 The chance is 95% that the population mean change is between 251 and 109 calories lower Is there a Hawthorne effect?

18. Copyright (c) Bani Mallick18 Confidence Intervals You can construct a confidence interval for the population mean with any level of confidence. Generally, people report the 95% CI, but sometimes they report the 90% and 99% confidence intervals. This is easy to do via a formula, and even easier to do via SPSS.

19. Copyright (c) Bani Mallick19 Confidence Interval for a Population Mean  when  is Known Want 90%, 95% and 99% chance of interval including . 90% 95% 99%

20. Copyright (c) Bani Mallick20 Confidence Intervals There is a general formula given on page 200 If you want a (1-  )100% confidence interval for the population mean  when the population s.d.  is known, use the formula The term z  is the value in Table 1 that gives probability 1 -  /2.  = 0.10, z  = 1.645:  = 0.05, z  = 1.96,  = 0.01, z  = 2.58

21. Copyright (c) Bani Mallick21 WISH The sample size is n = 271 The sample mean change = -180 I am going to pretend that the population standard deviation is  = 600. I want a 99% confidence interval: z  = 2.58

22. Copyright (c) Bani Mallick22 WISH: Change in Reported Caloric Intake n = 271,  = 600, = -180 99% CI = -274 to -86

23. Copyright (c) Bani Mallick23 WISH: Change in Reported Caloric Intake 99% CI = -274 to -86 The chance is 99% that the population mean change in reported caloric intake is between 274 and 86 calories The chance is less than 1% that there is no change in the population mean.

24. Copyright (c) Bani Mallick24 WISH: Change in Reported Caloric Intake 99% CI = -274 to -86 95% CI = -251 to -109 Note that the 99% CI is longer than the 95% CI. This is clear(!): the more confidence you want, the longer the CI has to be. Put another way, the less willing you are to be wrong, the more conservative your claims.

25. Copyright (c) Bani Mallick25 Effect of Sample Size 95% CI = -251 to -109 with n = 271 If n = 1000, the 95% CI would be from -217 to -143 Note how the CI gets shorter in length as the sample size gets larger. This is a general fact: the larger the sample size the shorter the CI.

26. Copyright (c) Bani Mallick26 Effect of Population Standard Deviation 95% CI = -251 to -109 with  = 600 If  = 2000, the 95% CI would be from -418 to +58 Note how the CI gets longer in length as the population standard deviation  gets larger. This is a general fact: the larger the population standard deviation  the longer the CI.

27. Copyright (c) Bani Mallick27 Using SPSS to Construct CI SPSS actually assumes that the population standard deviation is unknown: we will consider this case later. Its default is a 95% CI You can easily change to any level of confidence SPSS demo using Wish Data

28. Copyright (c) Bani Mallick28 Sample Size Determination In general, this is a relatively complex issue, depending very heavily on the experiment. I will show you a simple calculation in the special case that the population standard deviation  is known. Of course,  is not known in practice, and more complex methods are required, but this will give you a feel for the process.

29. Copyright (c) Bani Mallick29 Sample Size Determination The usual answer to “what sample size should I take” is “what can you afford”. Remember, more precision with larger sample sizes Less precision with smaller sample sizes

30. Copyright (c) Bani Mallick30 Sample Size Determination The length of a confidence interval is 2 x confidence factor Thus, our 95% CI for WISH was -251 to -109, so that the length was 142 calories What if I wanted the length to be 100 calories? Then the CI would have to be

31. Copyright (c) Bani Mallick31 Sample Size Determination The length of the CI is If I want the length of a confidence interval to be 2 x E then I have to set Now I do some algebra

32. Copyright (c) Bani Mallick32 Sample Size Determination I want the length of a confidence interval to be 2 x E then the sample size I need is

33. Copyright (c) Bani Mallick33 Sample Size Determination Consider WISH, where = 600. Suppose I want the confidence interval length of 95% CI to be 2xE = 100 E = 50, z  = 1.96

34. Copyright (c) Bani Mallick34 Sample Size Determination Consider WISH, where = 600. Suppose I want the confidence interval length of 95% CI to be 2xE = 60 E = 30, z  = 1.96

35. Copyright (c) Bani Mallick35 Sample Size Determination 95% confidence Length = 100, E = 50, n = 553 Length = 60, E = 30, n = 1,557 General fact: the more precise you want to be (shorter CI), the larger the sample size you will need.

36. Copyright (c) Bani Mallick36 Sample Size Determination General fact: the larger the population standard deviation, the larger the sample size you will need to have a CI of length 2xE

37. Copyright (c) Bani Mallick37 Reactiver Oxygen Species (ROS) Data Rats fed with Fish oil enhanced diets Response is the change in ROS for an animal when the cells are exposed to butyrate

38. Copyright (c) Bani Mallick38 ROS Data

39. Copyright (c) Bani Mallick39 ROS Data Sample mean = 3.21 Sample size is n = 20 Pretend  = 3.33 Then = 0.74 95% interval for population mean change is [3.21 - 0.74 * 1.96, 3.21 + 0.74 * 1.96] = [1.76, 4.66]: Does butyrate increase ROS? How certain are we?

40. Copyright (c) Bani Mallick40 ROS Data  = 3.33, n = 20 95% interval for population mean change is [1.76, 4.66] The length of the CI is 2xE = 2.90 What sample size would I need to make the length of the CI = 1.00? Here 2xE = 1.00, E = 0.50, and