The document summarizes key concepts from chapters 6 and 7 of a statistics textbook. Chapter 6 discusses sampling and calculating standard error for infinite and finite populations. Chapter 7 introduces estimation, including interval estimates and point estimates. It provides examples of calculating standard error and confidence intervals. The document also lists SPSS tips for t-tests.

1. Statistics 2 Dr. Ning DING IBS I.007 [email_address] You’d better use the full-screen mode to view this PPT file.

2. Table of Contents Chapter 6 Sampling - Review: Sampling and Standard Error - Calculating Standard Error-Infinite Population - Calculating Stanard Error-Finite Population Chapter 7 Introduction of Estimation - Types of Estimates - Interval Estimates SPSS Tips for t-test

3. Sampling and Sampling Distribution Population = all items chosen for study Sample = a portion chosen from the population Parameter Statistic  Greek or capital letters  Lowercase Roman letters Chapter 6 Sampling - Review: Sampling and Standard Error - Calculating Standard Error-Infinite Population - Calculating Stanard Error-Finite Population Chapter 7 Introduction of Estimation - Types of Estimates - Interval Estimates SPSS Tips for t-test

4. Sampling and Sampling Distribution Population Sample Parameter Statistic N = number μ = mean σ = standard deviation n = number X = mean SD = standard deviation Chapter 6 Sampling - Review: Sampling and Standard Error - Calculating Standard Error-Infinite Population - Calculating Stanard Error-Finite Population Chapter 7 Introduction of Estimation - Types of Estimates - Interval Estimates SPSS Tips for t-test

5. Sampling Distribution Mean Mean Mean Chapter 6 Sampling - Review: Sampling and Standard Error - Calculating Standard Error-Infinite Population - Calculating Stanard Error-Finite Population Chapter 7 Introduction of Estimation - Types of Estimates - Interval Estimates SPSS Tips for t-test

6. Standard Error = Standard deviation of the distribution of a sample statistic Larger Standard Error Smaller Standard Error Which one is better? Chapter 6 Sampling - Review: Sampling and Standard Error - Calculating Standard Error-Infinite Population - Calculating Stanard Error-Finite Population Chapter 7 Introduction of Estimation - Types of Estimates - Interval Estimates SPSS Tips for t-test

7. Standard Error Chapter 6 Sampling - Review: Sampling and Standard Error - Calculating Standard Error-Infinite Population - Calculating Stanard Error-Finite Population Chapter 7 Introduction of Estimation - Types of Estimates - Interval Estimates SPSS Tips for t-test

8. Standard Error Sample size Dispersion of sample means Standard Error Chapter 6 Sampling - Review: Sampling and Standard Error - Calculating Standard Error-Infinite Population - Calculating Stanard Error-Finite Population Chapter 7 Introduction of Estimation - Types of Estimates - Interval Estimates SPSS Tips for t-test

9. Standard Error µ = 100 σ = 25 =95 =106 =101 Population Range=80~240 Sample Range=90~120 Standard Error of mean Standard Deviation of population ____ Chapter 6 Sampling - Review: Sampling and Standard Error - Calculating Standard Error-Infinite Population - Calculating Stanard Error-Finite Population Chapter 7 Introduction of Estimation - Types of Estimates - Interval Estimates SPSS Tips for t-test

10. Calculating the Standard Error Chapter 6 Sampling - Review: Sampling and Standard Error - Calculating Standard Error-Infinite Population - Calculating Stanard Error-Finite Population Chapter 7 Introduction of Estimation - Types of Estimates - Interval Estimates SPSS Tips for t-test Sample means frequency

11. Calculating the Standard Error individual savings accounts µ= $2000 σ = $600 Sample= 100 accounts the probability that the sample mean lies betw. $1900~$2050 ? Standard Error of the mean Population standard deviation Sample size Example: Chapter 6 Sampling - Review: Sampling and Standard Error - Calculating Standard Error-Infinite Population - Calculating Stanard Error-Finite Population Chapter 7 Introduction of Estimation - Types of Estimates - Interval Estimates SPSS Tips for t-test

12. Calculating the Standard Error the probability that the sample mean lies betw. $1900~$2050 ? Sample mean Population mean Standard error of the mean 0.4525 0.2967 + = 0.7492 74.92% of our sample means lies betw. $1900~$2050 Chapter 6 Sampling - Review: Sampling and Standard Error - Calculating Standard Error-Infinite Population - Calculating Stanard Error-Finite Population Chapter 7 Introduction of Estimation - Types of Estimates - Interval Estimates SPSS Tips for t-test

13. Calculating the Standard Error 6-30 Chapter 6, No. 6-30 P.321 Known: Normal distribution, μ =375 σ =48 P=95% n = ? Chapter 6 Sampling - Review: Sampling and Standard Error - Calculating Standard Error-Infinite Population - Calculating Stanard Error-Finite Population Chapter 7 Introduction of Estimation - Types of Estimates - Interval Estimates SPSS Tips for t-test

14. Calculating the Standard Error 6-30 Chapter 6, No. 6-30 P.321 Known: Normal distribution, μ =375 σ =48 P=95% n=? Step 1: P=P z1 +P z2 =0.950 z 1 =-1.96 z 2 =1.96 370< <380 -1.96< z <1.96 Step 2: 1.96= Step 3: n=354.04 The sample size is at least 355 Chapter 6 Sampling - Review: Sampling and Standard Error - Calculating Standard Error-Infinite Population - Calculating Stanard Error-Finite Population Chapter 7 Introduction of Estimation - Types of Estimates - Interval Estimates SPSS Tips for t-test

15. Calculating the Standard Error Infinite population Finite population Chapter 6 Sampling - Review: Sampling and Standard Error - Calculating Standard Error-Infinite Population - Calculating Stanard Error-Finite Population Chapter 7 Introduction of Estimation - Types of Estimates - Interval Estimates SPSS Tips for t-test

16. The Finite Population Multiplier population size sample size Finite population multiplier F.P.M. Chapter 6 Sampling - Review: Sampling and Standard Error - Calculating Standard Error-Infinite Population - Calculating Stanard Error-Finite Population Chapter 7 Introduction of Estimation - Types of Estimates - Interval Estimates SPSS Tips for t-test

17. The Finite Population Multiplier 1) N= 20 n=5 0.888 2) N= 20 n=19 0.229 3) N= 20 n=20 0 4) N= 1000 n=20 0.99 When to use F.P.M.? If Chapter 6 Sampling - Review: Sampling and Standard Error - Calculating Standard Error-Infinite Population - Calculating Stanard Error-Finite Population Chapter 7 Introduction of Estimation - Types of Estimates - Interval Estimates SPSS Tips for t-test

18. Calculating the Standard Error SC 6-7a Chapter 6, SC No. 6-7 P.327 Known: N=125 n=64 μ =105 σ =17 =? Step 1: n/N=64/125= 0.512 >0.05 Yes, it is allowed to use F.P.M. Step 2: = = = Chapter 6 Sampling - Review: Sampling and Standard Error - Calculating Standard Error-Infinite Population - Calculating Stanard Error-Finite Population Chapter 7 Introduction of Estimation - Types of Estimates - Interval Estimates SPSS Tips for t-test

19. Calculating the Standard Error SC 6-7b Chapter 6, SC No. 6-7 P.327 Known: N=125 n=64 μ =105 σ =17 =1.4904 P(107.5<Xmean<109) = ? Step 1: visualize and calculate z scores = = P 1 =0.4535 P 2 =0.4963 P=0.4963-0.4535=0.0428 Chapter 6 Sampling - Review: Sampling and Standard Error - Calculating Standard Error-Infinite Population - Calculating Stanard Error-Finite Population Chapter 7 Introduction of Estimation - Types of Estimates - Interval Estimates SPSS Tips for t-test

20. Calculating the Standard Error SC 6-8 Chapter 6, SC No. 6-8 P.327 Known: n=36 μ =? σ =1.25 pounds What is the probability that the sample mean is within one-half pound of the population mean? = = Step 1: Visualize and calculate z scores Step 2: Calculate the standard error of sample means Chapter 6 Sampling - Review: Sampling and Standard Error - Calculating Standard Error-Infinite Population - Calculating Stanard Error-Finite Population Chapter 7 Introduction of Estimation - Types of Estimates - Interval Estimates SPSS Tips for t-test

21. Calculating the Standard Error SC 6-8 Chapter 6, SC No. 6-8 P.327 Known: n=36 μ =? σ =1.25 pounds What is the probability that the sample mean is within one-half pound of the population mean? = = Step 2: Calculate the standard error of sample means Step 3: Calculate the z scores P z1 =0.4918 P z2 =0.4918 + = 0.9836 Step 4: convert to P value Step 5: Finalize your answer Chapter 6 Sampling - Review: Sampling and Standard Error - Calculating Standard Error-Infinite Population - Calculating Stanard Error-Finite Population Chapter 7 Introduction of Estimation - Types of Estimates - Interval Estimates SPSS Tips for t-test

22. Chapter 7 Introduction of Estimation confidence level confidence interval Chapter 6 Sampling - Review: Sampling and Standard Error - Calculating Standard Error-Infinite Population - Calculating Stanard Error-Finite Population Chapter 7 Introduction of Estimation - Types of Estimates - Interval Estimates SPSS Tips for t-test

23. Types of Estimates Interval Estimates Point Estimates Chapter 6 Sampling - Review: Sampling and Standard Error - Calculating Standard Error-Infinite Population - Calculating Stanard Error-Finite Population Chapter 7 Introduction of Estimation - Types of Estimates - Interval Estimates SPSS Tips for t-test

24. Interval Estimates: Basic Concepts P.354 Interval Estimates standard deviation is 10 interviewed 200 person according to them, the mean is 36 months Stardard error of the mean from an infinite population standard deviation of the population sample size 36+0.707=36.707 36-0.707=35.293 Chapter 6 Sampling - Review: Sampling and Standard Error - Calculating Standard Error-Infinite Population - Calculating Stanard Error-Finite Population Chapter 7 Introduction of Estimation - Types of Estimates - Interval Estimates SPSS Tips for t-test

25. Interval Estimates: Basic Concepts P.354 Interval Estimates standard deviation is 10 interviewed 200 person according to them, the mean is 36 months z =1.0 P=0.3413 68.3% of the actual mean lie between 35.293 and 36.707 Chapter 6 Sampling - Review: Sampling and Standard Error - Calculating Standard Error-Infinite Population - Calculating Stanard Error-Finite Population Chapter 7 Introduction of Estimation - Types of Estimates - Interval Estimates SPSS Tips for t-test

26. Interval Estimates: Basic Concepts P.354 Interval Estimates standard deviation is 10 interviewed 200 person according to them, the mean is 36 months z =2.0 P=0.4775 95.5% of the actual mean lie between _______ and_________ 95.5% of the actual mean lie between 34.586 and 37.414 Chapter 6 Sampling - Review: Sampling and Standard Error - Calculating Standard Error-Infinite Population - Calculating Stanard Error-Finite Population Chapter 7 Introduction of Estimation - Types of Estimates - Interval Estimates SPSS Tips for t-test

27. Interval Estimates: Basic Concepts P.354 Interval Estimates standard deviation is 10 interviewed 200 person according to them, the mean is 36 months z =3.0 P=0.4987 99.7% of the actual mean lie between 33.879 and 38.121 Chapter 6 Sampling - Review: Sampling and Standard Error - Calculating Standard Error-Infinite Population - Calculating Stanard Error-Finite Population Chapter 7 Introduction of Estimation - Types of Estimates - Interval Estimates SPSS Tips for t-test

28. Interval Estimates: Basic Concepts P.354 Interval Estimates Chapter 6 Sampling - Review: Sampling and Standard Error - Calculating Standard Error-Infinite Population - Calculating Stanard Error-Finite Population Chapter 7 Introduction of Estimation - Types of Estimates - Interval Estimates SPSS Tips for t-test

29. Interval Estimates: Basic Concepts Interval Estimates Chapter 6 Sampling - Review: Sampling and Standard Error - Calculating Standard Error-Infinite Population - Calculating Stanard Error-Finite Population Chapter 7 Introduction of Estimation - Types of Estimates - Interval Estimates SPSS Tips for t-test Interval =

30. Interval Estimates: Basic Concepts Interval Estimates EX 7-27a Chapter 7, No. 7-27 P. 365 Known: n=40 EX 7-27b P=90%  z=1.645 Upper limit =1416+7.8029 Lower limit=1416-7.8029 =1424 =1408 90% confident that our population mean lies between 1408 and 1424. Interval Estimates Chapter 6 Sampling - Review: Sampling and Standard Error - Calculating Standard Error-Infinite Population - Calculating Stanard Error-Finite Population Chapter 7 Introduction of Estimation - Types of Estimates - Interval Estimates SPSS Tips for t-test = Interval =

31. Interval Estimates: Basic Concepts Interval Estimates If the σ is unknown , ? Chapter 6 Sampling - Review: Sampling and Standard Error - Calculating Standard Error-Infinite Population - Calculating Stanard Error-Finite Population Chapter 7 Introduction of Estimation - Types of Estimates - Interval Estimates SPSS Tips for t-test Estimated standard error of mean Estimated standard error of proportion P.368 Interval = Interval = Interval =

32. Interval Estimates: Basic Concepts Interval Estimates EX 7-35a EX 7-35b z=2.33 Answer: 0.01 ~0.09 Chapter 6 Sampling - Review: Sampling and Standard Error - Calculating Standard Error-Infinite Population - Calculating Stanard Error-Finite Population Chapter 7 Introduction of Estimation - Types of Estimates - Interval Estimates SPSS Tips for t-test Chapter 7, No. 7-35 P. 369 known: n=200 p=0.05 q=0.95 known: n=200 p=0.05 q=0.95 P=98% Interval = 

33. Interval Estimates: Basic Concepts Interval Estimates If the sample size is =< 30, AND σ is unknown, ? t - distribution You can read the t value from Appendix Table 2 Chapter 6 Sampling - Review: Sampling and Standard Error - Calculating Standard Error-Infinite Population - Calculating Stanard Error-Finite Population Chapter 7 Introduction of Estimation - Types of Estimates - Interval Estimates SPSS Tips for t-test

34. Interval Estimates: Basic Concepts Interval Estimates How to read the t-table ? t - distribution e.g. n=10  df= 9 P=90% 0.05 0.05 Chapter 6 Sampling - Review: Sampling and Standard Error - Calculating Standard Error-Infinite Population - Calculating Stanard Error-Finite Population Chapter 7 Introduction of Estimation - Types of Estimates - Interval Estimates SPSS Tips for t-test

35. Interval Estimates: Basic Concepts Interval Estimates How to use t value ? t - distribution Chapter 6 Sampling - Review: Sampling and Standard Error - Calculating Standard Error-Infinite Population - Calculating Stanard Error-Finite Population Chapter 7 Introduction of Estimation - Types of Estimates - Interval Estimates SPSS Tips for t-test interval = interval =

36. Summary Chapter 6 Sampling - Review: Sampling and Standard Error - Calculating Standard Error-Infinite Population - Calculating Stanard Error-Finite Population Chapter 7 Introduction of Estimation - Types of Estimates - Interval Estimates

37. The Normal Distribution SPSS Tip: t-test The data can be downloaded from: Blackboard – Inductive Statsitics STA2—SPSS-- Week 3

38. The Normal Distribution SPSS Tip: t-test 3 types of t-test One Sample t-test Paired-Samples t-test Independent Samples t-test test whether the population mean is different from a constant test whether the population mean of differences between paired scores is equal to zero test the relationship between two categories and a quantitative variable

39. The Normal Distribution SPSS Tip: t-test One Sample t-test Example: A researcher wants to evaluate whether customers believe price change is more a function of natural fluctuations in inflation or due to effects caused by human interventions. Thirty customers are assessed on the Price Change Attitude Scale , which yields scores that range from 0 ( due solely to natural fluctuations in inflation ) to 100 ( due solely to human interventions ). A score of 50 is the test value and represents an equal contribution of the two effects. The data can be downloaded from: Blackboard – Inductive Statsitics STA2—SPSS-- Week 3 One-Sample t-test.sav Variable Description PCAS Price Change Attitude Scale

40. The Normal Distribution SPSS Tip: t-test One Sample t-test Example: A researcher wants to evaluate whether customers believe price change is more a function of natural fluctuations in inflation or due to effects caused by human interventions. Thirty customers are assessed on the Price Change Attitude Scale , which yields scores that range from 0 ( due solely to natural fluctuations in inflation ) to 100 ( due solely to human interventions ). A score of 50 is the test value and represents an equal contribution of the two effects. Null Hypothesis: The population mean is equal to 50. Variable Description PCAS Price Change Attitude Scale

41. The Normal Distribution SPSS Tip: t-test Step 1: Choose Analyze--> Compare Means --> One-Sample T Test One Sample t-test

42. The Normal Distribution SPSS Tip: t-test Step 2: Move the variable you want to test into the box ”Test Variable(s)”. Enter the value in the box “Test Value”. In this example, the PCAS middle value is 50. Click OK and you will see a popup window. One Sample t-test

43. The Normal Distribution SPSS Tip: t-test One Sample t-test Read the next slide to know how to interpret it !

45. The Normal Distribution SPSS Tip: t-test Example: A researcher is interested in determining whether customers’ satisfaction with DOVE body lotion improves when exposed to a new TV commercial. Thirty customers are assessed on the Satisfaction Scale for Customers (SSC) prior to and after the new TV commercial. The data can be downloaded from: Blackboard – Inductive Statsitics STA2—SPSS-- Week 3 Paired-Sample t-test.sav Paired-Samples t-test Variable Description Pre_SSC Percent correct on the Sales Scale for Customers prior to the new TV commercial Post_SSC Percent correct on the Sales Scale for Customers after the new TV commercial

46. The Normal Distribution SPSS Tip: t-test Example: A researcher is interested in determining whether customers’ satisfaction with DOVE body lotion improves when exposed to a new TV commercial. Thirty customers are assessed on the Satisfaction Scale for Customers (SSC) prior to and after the new TV commercial. Paired-Samples t-test Null Hypothesis: The population means’ difference is zero. Variable Description Pre_SSC Percent correct on the Sales Scale for Customers prior to the new TV commercial Post_SSC Percent correct on the Sales Scale for Customers after the new TV commercial

47. The Normal Distribution SPSS Tip: t-test Paired-Samples t-test Step 1: Choose Analyze--> Compare Means --> Paired-Samples T Test

48. The Normal Distribution SPSS Tip: t-test Paired-Samples t-test Step 2: Move the first variable the box “Paired Variables”, to the location Variable 1, and the second variable to Variable 2. Click OK and you will see a popup window.

49. The Normal Distribution SPSS Tip: t-test Paired-Samples t-test Read the next slide to know how to interpret it !