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ruf.rice/~lane/stat_sim/sampling_dist/index.html

http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/index.html. Problem. What is the probability of drawing a card that is higher than a jack or lower than a 3?

james-bennett
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ruf.rice/~lane/stat_sim/sampling_dist/index.html

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  1. http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/index.htmlhttp://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/index.html

  2. Problem • What is the probability of drawing a card that is higher than a jack or lower than a 3? • If I gave only gave A’s to students scoring two standard devastations above the mean on the next test – what proportion of students would get an A? • If the next test had a mean of 56 and a SD of 4.2 – how many points would you need to get in order to receive an A?

  3. Example: IQ • Mean IQ = 100 • Standard deviation = 15 • What is the probability that a person you randomly bump into on the street has an IQ of 110 or higher?

  4. Step 1: Sketch out question -3 -2 -1 1 2  3 

  5. Step 1: Sketch out question 110 -3 -2 -1 1 2  3 

  6. Step 2: Calculate Z score (110 - 100) / 15 = .66 110 -3 -2 -1 1 2  3 

  7. Step 3: Look up Z score in Table Z = .66; Column C = .2546 110 .2546 -3 -2 -1 1 2  3 

  8. Example: IQ • You have a .2546 probability (or a 25.56% chance) of randomly bumping into a person with an IQ over 110.

  9. Now. . . . • What is the probability that the next 5 people you bump into on the street will have a mean IQ score of 110? • Notice how this is different!

  10. Population • You are interested in the average self-esteem in a population of 40 people • Self-esteem test scores range from 1 to 10.

  11. 1,1,1,1 2,2,2,2 3,3,3,3 4,4,4,4 5,5,5,5 6,6,6,6 7,7,7,7 8,8,8,8 9,9,9,9 10,10,10,10 Population Scores

  12. Histogram

  13. What is the average self-esteem score of this population? • Population mean = 5.5 • What if you wanted to estimate this population mean from a sample?

  14. Group Activity • Randomly select 5 people and find the average score

  15. Group Activity • Why isn’t the average score the same as the population score? • When you use a sample there is always some degree of uncertainty! • We can measure this uncertainty with a sampling distribution of the mean

  16. EXCEL

  17. Characteristics of a Sampling Distribution of the means • Every sample is drawn randomly from a population • The sample size (n) is the same for all samples • The mean is calculated for each sample • The sample means are arranged into a frequency distribution (or histogram) • The number of samples is very large

  18. INTERNET EXAMPLE

  19. Sampling Distribution of the Mean • Notice: The sampling distribution is centered around the population mean! • Notice: The sampling distribution of the mean looks like a normal curve! • This is true even though the distribution of scores was NOT a normal distribution

  20. Central Limit Theorem For any population of scores, regardless of form, the sampling distribution of the means will approach a normal distribution as the number of samples get larger. Furthermore, the sampling distribution of the mean will have a mean equal to  and a standard deviation equal to / N

  21. Mean • The expected value of the mean for a sampling distribution • E (X) = 

  22. Standard Error • The standard error (i.e., standard deviation) of the sampling distribution x = / N

  23. Standard Error • The  of an IQ test is 15. If you sampled 10 people and found an X = 105 what is the standard error of that mean? x = / N

  24. Standard Error • The  of an IQ test is 15. If you sampled 10 people and found an X = 105 what is the standard error of that mean? x = 15/ 10

  25. Standard Error • The  of an IQ test is 15. If you sampled 10 people and found an X = 105 what is the standard error of that mean? 4.74= 15/ 3.16

  26. Standard Error • The  of an IQ test is 15. If you sampled 10 people and found an X = 105 what is the standard error of that mean? What happens to the standard error if the sample size increased to 50? 4.74= 15/ 3.16

  27. Standard Error • The  of an IQ test is 15. If you sampled 10 people and found an X = 105 what is the standard error of that mean? What happens to the standard error if the sample size increased to 50? 4.74= 15/ 3.16 2.12 = 15/7.07

  28. Standard Error • The bigger the sample size the smaller the standard error • Makes sense!

  29. Question • For an IQ test •  = 100 •  = 15 • What is the probability that in a class the average IQ of 54 students will be below 95? • Note: This is different then the other “z” questions!

  30. Z score for a sample mean Z = (X - ) / x

  31. Step 1: Sketch out question -3 -2 -1 0 1 2 3

  32. Step 2: Calculate the Standard Error 15 / 54 = 2.04 -3 -2 -1 0 1 2 3

  33. Step 3: Calculate the Z score (95 - 100) / 2.04 = -2.45 -3 -2 -1 0 1 2 3

  34. Step 4: Look up Z score in Table Z = -2.45; Column C =.0071 .0071 -3 -2 -1 0 1 2 3

  35. Question • From a sample of 54 students the probability that their average IQ score is 95 or lower is .0071

  36. Practice • Page 154 • 7.12 • 7.13

  37. As N (sample size) increases the standard error decreases!

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