1 / 60

Where We Left Off

Where We Left Off. What is the probability of randomly selecting a sample of three individuals, all of whom have an I.Q. of 135 or more? Find the z -score of 135, compute the tail region and raise it to the 3 rd power.

unity-bonner
Download Presentation

Where We Left Off

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. Where We Left Off • What is the probability of randomly selecting a sample of three individuals, all of whom have an I.Q. of 135 or more? • Find the z-score of 135, compute the tail region and raise it to the 3rd power. So while the odds chance selection of a single person this far above the mean is not all that unlikely, the odds of a sample this far above the mean are astronomical z = 2.19 P = 0.0143 0.01433 = 0.0000029 X X Anthony J Greene

  2. Sampling Distributions I What is a Sampling Distribution? A If all possible samples were drawn from a population B A distribution described with Central Tendency µM And dispersion σM ,the standard error II The Central Limit Theorem Anthony J Greene

  3. Sampling Distributions • What you’ve done so far is to determine the position of a given single score, x, compared to all other possible x scores x Anthony J Greene

  4. Sampling Distributions • The task now is to find the position of a group score, M, relative to all other possible sample means that could be drawn M Anthony J Greene

  5. Sampling Distributions • The reason for this is to find the probability of a random sample having the properties you observe. M Anthony J Greene

  6. Sampling Distributions • Any time you draw a sample from a population, the mean of the sample, M , it estimates the population mean μ, with an average error of: • We are interested in understanding the probability of drawing certain samples and we do this with our knowledge of the normal distribution applied to the distribution of samples, or Sampling Distribution • We will consider a normal distribution that consists of all possible samples of size n from a given population Anthony J Greene

  7. Sampling Error Sampling error is the error resulting from using a sample to estimate a population characteristic. Anthony J Greene

  8. Sampling Distribution of the Mean For a variable x and a given sample size, the distribution of the variable M (i.e., of all possible sample means) is called the sampling distribution of the mean. The sampling distribution is purely theoretical derived by the laws of probability. A given score x is part of a distribution for that variable which can be used to assess probability A given mean M is part of a sampling distribution for that variable which can be used to determine the probability of a given sample being drawn Anthony J Greene

  9. The Basic Concept • Extreme events are unlikely -- single events • For samples, the likelihood of randomly selecting an extreme sample is more unlikely • The larger the sample size, the more unlikely it is to draw an extreme sample Anthony J Greene

  10. The original distribution of x: 2, 4, 6, 8 Now consider all possible samples of size n = 2What is the distribution of sample means M Anthony J Greene

  11. The Sampling Distribution For n=2Notice that it’s a normal distribution with μ = 5 Anthony J Greene

  12. Heights of the five starting players Anthony J Greene

  13. Possible samples and sample means for samples of size two M Anthony J Greene

  14. Dotplot for the sampling distribution of the mean for samples of size two (n = 2) M Anthony J Greene

  15. Possible samples and sample means for samples of size four M Anthony J Greene

  16. Dotplot for the sampling distribution of the mean for samples of size four (n = 4) M Anthony J Greene

  17. Sample size and sampling error illustrations for the heights of the basketball players Anthony J Greene

  18. Dotplots for the sampling distributions of the mean for samples of sizes one, two, three, four, and five M M M M M Anthony J Greene

  19. Sample Size and Standard Error The possible sample means cluster closer around the population mean as the sample size increases. Thus the larger the sample size, the smaller the sampling error tends to be in estimating a population mean, m, by a sample mean, M. For sampling distributions, the dispersion is called Standard Error. It works much like standard deviation. Anthony J Greene

  20. Standard Error of M For samples of size n, the standard error of the variable x equals the standard deviation of x divided by the square root of the sample size: In other words, for each sample size, the standard error of all possible sample means equals the population standard deviation divided by the square root of the sample size. Anthony J Greene

  21. The Effect of Sample Size on Standard ErrorThe distribution of sample means for random samples of size (a) n = 1, (b) n = 4, and (c) n = 100 obtained from a normal population with µ = 80 and σ = 20. Notice that the size of the standard error decreases as the sample size increases. Anthony J Greene

  22. Mean of the Variable M For samples of size n, the mean of the variable M equals the mean of the variable under consideration: mM= m. In other words, for each sample size, the mean of all possible sample means equals the population mean. Anthony J Greene

  23. The standard error of M for sample sizes one, two, three, four, and five Standard error = dispersion of MσM Anthony J Greene

  24. The sample means for 1000 samples of four IQs. The normal curve for x is superimposed Anthony J Greene

  25. Sampling Distribution of the Mean for a Normally Distributed Variable Suppose a variable x of a population is normally distributed with mean m and standard deviation s. Then, for samples of size n, the sampling distribution of M is also normally distributed and has mean mM = m and standard error of Anthony J Greene

  26. (a) Normal distribution for IQs(b) Sampling distribution of the mean for n = 4(c) Sampling distribution of the mean for n = 16 Anthony J Greene

  27. Samples Versus Individual Scores

  28. Frequency distribution for U.S. household size Anthony J Greene

  29. Relative-frequency histogram for household size Anthony J Greene

  30. Sample means n = 3,for 1000 samples of household sizes. Anthony J Greene

  31. The Central Limit Theorem For a relatively large sample size, the variable M is approximately normally distributed, regardlessof the distribution of the underlying variable x. The approximation becomes better and better with increasing sample size. Anthony J Greene

  32. M M M M M M Sampling distributions fornormal, J-shaped, uniform variable M Anthony J Greene M M M

  33. APA Style: TablesThe mean self-consciousness scores for participants who were working in front of a video camera and those who were not (controls). Anthony J Greene

  34. APA Style: Bar GraphsThe mean (±SE) score for treatment groups A and B. Anthony J Greene

  35. APA Style: Line GraphsThe mean (±SE) number of mistakes made for groups A and B on each trial. Anthony J Greene

  36. Summary • We already knew how to determine the position of an individual score in a normal distribution • Now we know how to determine the position of a sample of scores within the sampling distribution • By the Central Limit Theorem, all sampling distributions are normal with Anthony J Greene

  37. Sample Problem 1 • Given a distribution with μ = 32 and σ = 12 what is the probability of drawing a sample of size 36 where M > 48 Does it seem likely that M is just a chance difference? Anthony J Greene

  38. Sample Problem 2 • In a distribution with µ = 45 and σ = 45 what is the probability of drawing a sample of 25 with M >50? Anthony J Greene

  39. z -1.96 +1.96 M 84.12 95.88 Sample problem 3 • In a distribution with µ = 90 and σ = 18, for a sample of n = 36, what sample mean M would constitute the boundary of the most extreme 5% of scores? zcrit = ± 1.96 Anthony J Greene

  40. Sample Problem 4 • In a distribution with µ = 90 and σ = 18, what is the probability of drawing a sample whose mean M > 93? What information are we missing? n = 9 Anthony J Greene

  41. Sample Problem 5 • In a distribution with µ = 90 and σ = 18, what is the probability of drawing a sample whose mean M > 93? n = 16 Anthony J Greene

  42. Sample Problem 6 • In a distribution with µ = 90 and σ = 18, what is the probability of drawing a sample whose mean M > 93? n = 25 Anthony J Greene

  43. Sample Problem 7 • In a distribution with µ = 90 and σ = 18, what is the probability of drawing a sample whose mean M > 93? n = 36 Anthony J Greene

  44. Sample Problem 8 • In a distribution with µ = 90 and σ = 18, what is the probability of drawing a sample whose mean M > 93? n = 81 Anthony J Greene

  45. Sample Problem 9 • In a distribution with µ = 90 and σ = 18, what is the probability of drawing a sample whose mean M > 93? n = 169 Anthony J Greene

  46. Sample Problem 10 • In a distribution with µ = 90 and σ = 18, what is the probability of drawing a sample whose mean M > 93? n = 625 Anthony J Greene

  47. Sample Problem 10 • In a distribution with µ = 90 and σ = 18, what is the probability of drawing a sample whose mean M > 93? n = 1 Anthony J Greene

  48. Anthony J Greene

  49. z = ±2.58 Sample Problem 11 • In a distribution with µ = 200 and σ = 20, what sample mean M corresponds to the most extreme 1% ? n = 1 Anthony J Greene

  50. z = ±2.58 Sample Problem 12 • In a distribution with µ = 200 and σ = 20, what sample mean M corresponds to the most extreme 1% ? n = 4 Anthony J Greene

More Related