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Inferential Statistics for the Mean

Inferential Statistics for the Mean. Remember how a sampling distribution of means is created? Take a sample of size 500 from the US. Record the mean self-esteem. If the mean should be 25, you might get this. Self-esteem 15 20 25 30 35 40.

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Inferential Statistics for the Mean

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  1. Inferential Statistics for the Mean Remember how a sampling distribution of means is created? Take a sample of size 500 from the US. Record the mean self-esteem. If the mean should be 25, you might get this. Self-esteem 15 20 25 30 35 40

  2. Inferential Statistics for the Mean Remember how a sampling distribution of means is created? Take a sample of size 500 from the US. Record the mean self-esteem. If the mean should be 25, you might get this. Self-esteem 15 20 25 30 35 40

  3. Inferential Statistics for the Mean Remember how a sampling distribution of means is created? Take a sample of size 500 from the US. Record the mean self-esteem. If the mean should be 25, you might get this. Self-esteem 15 20 25 30 35 40

  4. Inferential Statistics for the Mean Remember how a sampling distribution of means is created? Take a sample of size 500 from the US. Record the mean self-esteem. If the mean should be 25, you might get this. Self-esteem 15 20 25 30 35 40

  5. Inferential Statistics for the Mean Remember how a sampling distribution of means is created? Take a sample of size 500 from the US. Record the mean self-esteem. If the mean should be 25, you might get this. Self-esteem 15 20 25 30 35 40

  6. Inferential Statistics for the Mean Remember how a sampling distribution of means is created? Take a sample of size 500 from the US. Record the mean self-esteem. If the mean should be 25, you might get this. Self-esteem 15 20 25 30 35 40

  7. Inferential Statistics for the Mean Remember how a sampling distribution of means is created? Take a sample of size 500 from the US. Record the mean self-esteem. If the mean should be 25, you might get this. Self-esteem 15 20 25 30 35 40

  8. Inferential Statistics for the Mean Remember how a sampling distribution of means is created? Take a sample of size 500 from the US. Record the mean self-esteem. If the mean should be 25, you might get this. Self-esteem 15 20 25 30 35 40

  9. Inferential Statistics for the Mean Remember how a sampling distribution of means is created? Take a sample of size 500 from the US. Record the mean self-esteem. If the mean should be 25, you might get this. Self-esteem 15 20 25 30 35 40

  10. Inferential Statistics for the Mean Remember how a sampling distribution of means is created? Take a sample of size 500 from the US. Record the mean self-esteem. If the mean should be 25, you might get this. The sample means would stack up in a normal curve. A normal sampling distribution. z -3 -2 -1 0 1 2 3 Self-esteem 15 20 25 30 35 40

  11. Inferential Statistics for the Mean Remember how a sampling distribution of means is created? Take a sample of size 500 from the US. Record the mean self-esteem. If the mean should be 25, you might get this. The sample means would stack up in a normal curve. A normal sampling distribution. 2.5% 2.5% z -3 -2 -1 0 1 2 3 Self-esteem 15 20 25 30 35 40

  12. Inferential Statistics for the Mean The sample size affects the sampling distribution: Standard error = population standard deviation / square root of sample size Y-bar= /n But in fact we use our sample’s standard deviation as an estimate of the population’s.

  13. Inferential Statistics for the Mean And if we increase our sample size (n)… Our repeated sample means will be closer to the true mean: 2.5% 2.5% Z-3 -2 -1 0 1 2 3 z -3 -2 -1 0 1 2 3

  14. Inferential Statistics for the Mean Means will be closer to the true mean, and our standard error of the sampling distribution is smaller: 2.5% 2.5% Z-3 -2 -1 0 1 2 3 z -3 -2 -1 0 1 2 3

  15. Inferential Statistics for the Mean The range of particular middle percentages gets smaller: Self-esteem 15 20 25 30 35 40 Z-3 -2 -1 0 1 2 3 95% Range z -3 -2 -1 0 1 2 3

  16. Inferential Statistics for the Mean …But we can say that 95% of the sample means in repeated sampling will always be in the range marked by -2 over to +2 standard errors. Self-esteem 15 20 25 30 35 40 Z-3 -2 -1 0 1 2 3 95% Range z -3 -2 -1 0 1 2 3

  17. Inferential Statistics for the Mean Ooops! Technically speaking, on a normal curve, 95% of cases always fall between +/- 1.96 standard deviations rather than 2. See comparison on next slide…

  18. Inferential Statistics for the Mean Empirical Rule vs. Actuality 68% 1z 68% 0.99z 95% 2z 95% 1.96z 99% 2.58z Almost all 3z 99.9973% 3z For evidence of this, check out Table A in A&F.

  19. Inferential Statistics for the Mean …But we can say that 95% of the sample means in repeated sampling will always be in the range marked by -1.96 over to +1.96 standard errors. Self-esteem 15 20 25 30 35 40 1.96 Z-3 -2 -1 0 1 2 3 -1.96 95% Range z -3 -2 -1 0 1 2 3

  20. Inferential Statistics for the Mean And remember: If we don’t know the true population mean, 95% of the time that range (confidence interval) would contain the true population mean! Self-esteem 15 20 25 30 35 40 95% Ranges for different samples.

  21. Inferential Statistics for the Mean If we want that range to contain the true population mean 99% of the time (99% confidence interval) we just construct a wider interval. Self-esteem 15 20 25 30 35 40 99% Ranges for different samples.

  22. Inferential Statistics for the Mean 1.96z The sampling distribution’s standard error is a measuring stick that we can use to indicate the range of a specified middle percentage of sample means in repeated sampling. 95% 1z 68% 3z 99.99% 25 -3 -1.96 -1 0 1 1.96 3 68% 95% 99.99%

  23. Inferential Statistics for the Mean • We use that measuring stick to say two things: • If my sample is in the middle specified percent, the population’s mean is within this range. (Confidence Interval) • If the population mean is the same as a guess of mine, then my sample’s mean would have to fall within this range to have been drawn from the middle specified percent. (Significance Test) 1.96z 95% 1z 68% 3z 99.99%  -3 -1.96 -1 0 1 1.96 3 68% 95% 99.99%

  24. Inferential Statistics for the Mean Confidence Interval Example: I collected a sample of 2,500 with an average self-esteem score of 28 with a standard deviation of 8. What if we want a 99% confidence interval? CI = Mean +/- z * s.e. • Find the standard error of the sampling distribution: s.d. / n = 8/50 = 0.16 • Build the width of the Interval. 99% corresponds with a z of 2.58. 2.58 * 0.16 = 0.41 • Insert the mean to build the interval: 99% C.I. = 28 +/- 0.41 The interval: 27.59 to 28.41 We are 99% confident that the population mean falls between these values.

  25. Inferential Statistics for the Mean And if we wanted a 95% Confidence Interval instead? I collected a sample of 2,500 with an average self-esteem score of 28 with a standard deviation of 8. What if we want a 99% confidence interval? CI = Mean +/- z * s.e. • Find the standard error of the sampling distribution: s.d. / n = 8/50 = 0.16 • Build the width of the Interval. 99% corresponds with a z of 2.58. 2.58 * 0.16 = 0.41 • Insert the mean to build the interval: 99% C.I. = 28 +/- 0.41 The interval: 27.59 to 28.41 We are 99% confident that the population mean falls between these values. 95% X 1.96 95% X X X X 0.31 1.96 X 95% X 0.31 X X 27.69 to 28.31 X 95%

  26. Inferential Statistics for the Mean By centering my sampling distribution’s +/- 1.96z range around my sample’s mean... • I can identify a range that, if my sample is one of the middle 95%, would contain the population’s mean. Or • I have a 95% chance that the population’s mean is somewhere in that range.

  27. Inferential Statistics for the Mean By centering my sampling distribution’s +/- 1.96z range around my sample’s mean... • I can identify a range that, if my sample is one of the middle 95%, would contain the population’s mean. Or • I have a 95% chance that the population’s mean is somewhere in that range. X 2.58z X 99% 99% X

  28. Inferential Statistics for the Mean Besides construct a confidence interval, we can also do a significance test.

  29. Inferential Statistics for the Mean • We use that measuring stick to say two things: • If my sample is in the middle specified percent, the population’s mean is within this range. (Confidence Interval) • If the population mean is the same as a guess of mine, then my sample’s mean would have to fall within this range to have been drawn from the middle specified percent. (Significance Test) 1.96z 95% 1z 68% 3z 99.99%  -3 -1.96 -1 0 1 1.96 3 68% 95% 99.99%

  30. Inferential Statistics for the Mean • We know that if you have your sampling distribution centered on the population mean: • 16% of samples’ means would be larger than  + 1z and 16% would be smaller than  - 1z, for a total of 32% outside that range. 1z 68%  -3 -1.96 -1 0 1 1.96 3 68%

  31. Inferential Statistics for the Mean • We know that if you have your sampling distribution centered on the population mean: • 2.5% of samples’ means would be larger than  + 1.96z and 2.5% would be smaller than  - 1.96z, for a total of 5% outside that range. 1.96z 95%  -3 -1.96 -1 0 1 1.96 3 95%

  32. Inferential Statistics for the Mean • We know that if you have your sampling distribution centered on the population mean: • 0.005% of samples’ means would be larger than  + 3z and 0.005% would be smaller than  - 3z, for a total of 0.01% outside that range. 3z 99.99%  -3 -1.96 -1 0 1 1.96 3 99.99%

  33. Inferential Statistics for the Mean But you remember that we don’t normally know the actual mean for the population. But what if we guessed? What if we specified a value that might be the population mean?

  34. Inferential Statistics for the Mean If we guessed a mean… If our guess is correct, our sample’s mean should be among the common samples that would have been drawn from a population with that mean. If it is not, it is likely that the sample did not come from such a population. What if my sample’s mean were here?  -3 -1.96 -1 0 1 1.96 3 guess

  35. Inferential Statistics for the Mean One way to tell whether our sample’s mean was generated by such a population is to place our sampling distribution over the guessed mean to see if the sample mean is among the middle 99% or 95% of samples that would be generated. 1.96z 95% What if my sample’s mean were here? It is among the rare 5% of possible means.  -3 -1.96 -1 0 1 1.96 3 guess 95%

  36. Inferential Statistics for the Mean Essentially, a significance test for a mean tells you what the odds are that your sample mean could have come from your guessed population mean. 1.96z 95% What if my sample’s mean were here? It is among the rare 5% of possible means.  -3 -1.96 -1 0 1 1.96 3 guess 95%

  37. Inferential Statistics for the Mean What you do is figure out what your sample’s z-score is relative to your guessed mean. If z is larger than 1.96 or smaller than -1.96, you have less than a 5% chance than your sample came from such a “guess population” —reject the guess! Essentially, a significance test for a mean tells you what the odds are that your sample mean could have come from your guessed population mean.  -3 -1.96 -1 0 1 1.96 3 guess 95% Sample mean

  38. Inferential Statistics for the Mean For example: If our guess was that self-doubt scores in the population averaged 20 on a scale from 1 – 50, we’d place a guess as below. Self-doubt 16 18 20 22 24 26 28

  39. Inferential Statistics for the Mean We guess 20, but our sample of size 100 has a mean of 25 and a standard deviation of 10. Guess,  Sample, Y-bar Self-doubt 16 18 20 22 24 26 28

  40. Inferential Statistics for the Mean Let’s build a sampling distribution around our guess, 20: sample of size 100; s.d. = 10. Sample, Y-bar s.e. = 10/100 = 10/10 = 1 Self-doubt 16 18 20 22 24 26 28 Z: -3 -2 -1 0 1 2 3 4 5

  41. Inferential Statistics for the Mean Our sample appears to be larger than a critical value of 1.96 (outer 5% of samples) or even 2.58 (outer 1% of samples). Sample, Y-bar s.e. = 10/100 = 10/10 = 1 Self-doubt 16 18 20 22 24 26 28 Z: -3 -2 -1 0 1 2 3 4 5

  42. Inferential Statistics for the Mean How many z’s is our sample mean away from our guess? Z = Y-bar –  / s.e. Z = 25 – 20 / 1 z = 5 s.e. = 10/100 = 10/10 = 1 Sample, Y-bar Self-doubt 16 18 20 22 24 26 28 Z: -3 -2 -1 0 1 2 3 4 5

  43. Inferential Statistics for the Mean Indeed, our sample z-score is 5, well above 1.96 or 2.58. Reject the guess! Looking in Table A… Our sample has a .0000287 % chance of having come from a population whose mean is 20! s.e. = 10/100 = 10/10 = 1 Sample, Y-bar Self-doubt 16 18 20 22 24 26 28 Z: -3 -2 -1 0 1 2 3 4 5

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