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Probability Standard Error of the Mean

Definition Review. Population: all possible casesParameters describe the populationSample: subset of cases drawn from the populationStatistics describe the sample. Statistics = Parameters. . Why Sample????. Can afford it. Can afford itCan do it in reasonable time. Why Sample????. Can afford it

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Probability Standard Error of the Mean

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    1. Probability & Standard Error of the Mean

    2. Definition Review Population: all possible cases Parameters describe the population Sample: subset of cases drawn from the population Statistics describe the sample

    3. Why Sample???? Can afford it

    4. Can afford it Can do it in reasonable time Why Sample????

    5. Can afford it Can do it in reasonable time Can estimate the amount of error (uncertainty) in statistics, allowing us to generalize (within limits) to our population Why Sample????

    6. Even with True Random Selection Some error (inaccuracy) associated with the statistics (will not precisely match the parameters) sampling error: everybody is different The whole measured only if ALL the parts are measured.

    7. With unbiased sampling Know that the amount of error is reduced as the n is increased statistics more closely approximate the parameters Amount of error associated with statistics can be evaluated estimate by how much our statistics may differ from the parameters

    8. Sample size Rules of thumb Larger n the better law of diminishing returns ie 100 to 200 vs 1500 to 1600 $$$ and time constraints Less variability in population => better estimate in statistics reduce factors affecting variability control and standardization

    10. True Random sampling: rare What population is the investigator interested in??? Getting a true random sample of any population is difficult if not impossible subject refusal to participate

    11. Catch 22 NEVER know our true population parameters, so we are ALWAYS at risk of making an error in generalization

    12. Probability

    13. Probability: the number of times some event is likely to occur out of the total possible events Backbone of inferential stats

    14. The classic: flip a coin heads vs tails: each at 1/2 (50%) flip 8x: what possible events (outcomes)?? flip it 8 million times: what probable distribution of heads/tails? Backbone of inferential stats

    15. Wayne Gretzky

    16. Wayne Gretzky & probability

    17. Wayne’s famous quote:

    18. Wayne Gretzky redux.

    19. Life with Probability life insurance rates obesity smoking car insurance rates age previous accidents driving demerits flood insurance

    20. The Ever-Changing Nature of %s

    21. How to Count Cards

    26. Probability & the Normal Curve Normal Curve mathematical abstraction unimodal symmetrical (Mean = Mode = Md) Asymptotic (any score possible) a family of curves Means the same, SDs are different Means are different, SDs the same both Means & SDs are different

    27. Dice Roll Outcomes

    28. Dice Roll Outcomes

    29. 99.7% of ALL cases within plus or minus 3 Standard Deviations Any score is possible but some more likely than others (which one?) Using the NC table Mean = 50 SD = 7 What is probability of getting a score > 64? one-tailed probability Probability & the Normal Curve

    30. Using the NC table What is probability of getting a score that is more than one SD above OR more than one SD below the mean? two-tailed probability Probability & the Normal Curve

    31. Defining probable or likely What risk are YOU willing to take? Fly to Europe for $1,000,000 BUT… 50% chance plane will crash 25% chance 1%chance .001% chance .000000001% chance

    32. Defining probable or likely In science, we accept as unlikely to have occurred at random (by chance) 5% (0.05) 1% (0.01) 10% (0.10)

    34. Six monkeys fail to write Shakespeare Pantagraph, May 2003

    35. Any score is possible, but some more likely than others Key to any problem in statistical inference is to discover what sample values will occur in repeated sampling and with what probability. Probability & the Normal Curve

    36. Statistics Humour

    37. Sampling Distributions: Standard error of the mean

    38. Recall With sampling, we EXPECT error in our statistics statistics not equal to parameters cause: random (chance) errors

    39. Recall With sampling, we EXPECT error in our statistics statistics not equal to parameters cause: random (chance) errors Unbiased sampling: no factor(s) systematically pushing estimate in a particular direction

    40. Recall With sampling, we EXPECT error in our statistics statistics not equal to parameters cause: random (chance) errors Unbiased sampling: no factors systematically pushing estimate in a particular direction Larger sample = less error

    41. Central Limit Theorem Consider (conceptualize) a distribution of sample means drawn from a distribution repeated sampling (calculating mean) from the same population produces a distribution of sample means

    42. Central Limit Theorem A distribution of sample means drawn from a distribution (the sampling distribution of means) will be a normal distribution class: from list of 51 state taxes, each student create 5 random samples of n = 6. Look at distribution in SPSS Mp = 32.7 cents, SD = 18.1 cents

    43. Central Limit Theorem Mean of distribution of sampling means equals population mean if the n of means is large

    44. Central Limit Theorem Mean of distribution of sampling means equals population mean if the n of means is large true even when population is skewed if sample is large (n > 60)

    45. Central Limit Theorem Mean of distribution of sampling means equals population mean if the n of means is large true if population when skewed if sample is large (n > 60) SD of the distribution of sampling means is the Standard Error of the Mean

    46. Take home lesson We have quantified the expected error (estimate of uncertainty) associated with our sample mean Standard Error of the Mean SD of the distribution of sampling means

    47. Typical procedure Sample calculate mean & SD

    48. Typical procedure Sample calculate mean & SD KNOW & RECOGNIZE that

    49. Typical procedure Sample calculate mean & SD KNOW & RECOGNIZE that statistics are not exact estimates of parameters

    50. Typical procedure Sample calculate mean & SD KNOW & RECOGNIZE that statistics are not exact estimates of parameters a larger n provides a less variable measure of the mean

    51. Central Limit Theorem

    52. Typical procedure Sample, calculate mean & SD KNOW & RECOGNIZE that statistics are not exact estimates of the parameters a larger n provides a less variable measure of the mean sampling from a population with low variability gives a more precise estimate of the mean

    53. Estimating Sample SEm

    54. Example Calculation

    55. Confidence Interval for the Mean

    56. Confidence Interval for the Mean

    57. Confidence Interval for the Mean

    58. Example Calculation

    59. Example Calculation

    60. Example Calculation

    61. Example Calculation

    63. 95 % Confidence Interval for the Mean

    64. 95 % Confidence Interval for the Mean

    65. 95 % Confidence Interval for the Mean

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