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Aron chpt 6 ed

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  • 1. Hypothesis Tests with Means of Samples
    • Chapter 6
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 2. The Distribution of Means
    • Begin with an example
      • Randomly sample three people from population of women at BAC
      • Compute mean height of sample
      • Population mean = 63.8 in
      • Sample 1 – 67, 66, 62 (ave = 65 in)
      • Sample II – 63, 62, 61 (ave = 62 in)
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 3. Building a Distribution of Means
    • Think of a distribution of means as if you kept randomly choosing samples of equal sizes from a population and took the means of those samples.
      • Those means are what make up a distribution of means.
    • The characteristics of a distribution of means can be calculated from:
      • characteristics of the population of individuals
      • number of scores in each sample
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 4. Determining the Characteristics of a Distribution of Means
    • Characteristics of the comparison distribution that you need are:
      • the mean
      • the variance and standard deviation
      • the shape
    • The mean of the distribution of means is about the same as the mean of the original population of individuals.
      • This is true for all distributions of means.
    • The spread of the distribution of means is less than the spread of the distribution of the population of individuals.
      • This is true for all distributions of means.
    • The shape of the distribution of means is approximately normal.
      • This is true for most distributions of means.
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 5. Mean of a Distribution of Means
    • The mean of a distribution of means of samples of a given size from a particular population
    • It is the same as the mean of the population of individuals.
      • Population M M = Population M
        • Population M M is the mean of the distribution of means.
    • Because the selection process is random and because we are taking a very large number of samples, eventually the high means and the low means perfectly balance each other out.
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 6. Example 2 Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 7.   die 1 die 2 Ave.   die 1 die 2 Ave.   die 1 die 2 Ave. 1 1 1 1 13 3 1 2 13 5 1 3 2 1 2 1.5 14 3 2 2.5 14 5 2 3.5 3 1 3 2 15 3 3 3 15 5 3 4 4 1 4 2.5 16 3 4 3.5 16 5 4 4.5 5 1 5 3 17 3 5 4 17 5 5 5 6 1 6 3.5 18 3 6 4.5 18 5 6 5.5 7 2 1 1.5 19 4 1 2.5 19 6 1 3.5 8 2 2 2 20 4 2 3 20 6 2 4 9 2 3 2.5 21 4 3 3.5 21 6 3 4.5 10 2 4 3 22 4 4 4 22 6 4 5 11 2 5 3.5 23 4 5 4.5 23 6 5 5.5 12 2 6 4 24 4 6 5 24 6 6 6
  • 8. 20   19   18   17   16   15   14   13   12   11   10   9   8   7   6   5     4         3             2                 1                                                 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 36 samples mean  f 1 1 1.5 2 2 3 2.5 4 3 5 3.5 6 4 5 4.5 4 5 3 5.5 2 6 1
  • 9. 20   19   18   17   16   15   14   13   12   11     10     9         8         7             6             5                 4                 3                     2                     1                                                     1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 72 samples   mean f 1 2 1.5 4 2 6 2.5 8 3 10 3.5 12 4 10 4.5 8 5 6 5.5 4 6 2
  • 10. 20   19   18   17     16     15     14         13         12         11             10             9             8                 7                 6                 5                     4                     3                     2                         1                                                     1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 108 samples   mean f 1 3 1.5 6 2 9 2.5 12 3 15 3.5 18 4 15 4.5 12 5 9 5.5 6 6 3
  • 11. 48     47     46     45     44     43     42     41     40     39         38         37         36         35         34         33         32         31             30             29             28             27             26             25             24             23                 22                 21                 20                 19                 18                 17                 16                 15                     14                     13                     12                     11                     10                     9                     8                     7                         6                         5                         4                         3                         2                         1                                                     1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 288 samples   mean f 1 8 1.5 16 2 24 2.5 32 3 40 3.5 48 4 40 4.5 32 5 24 5.5 16 6 8
  • 12.
    • The spread of the distribution of means is less than the spread of the distribution of the population of individuals.
      • This is true for all distributions of means.
    The distribution of a pop. of individuals The distribution of a sample taken from pop. The distribution of means of samples taken from pop.
  • 13. Variance of a Distribution of Means
    • The variance of a distribution of means is the variance of the population of individuals divided by the number of individuals in each sample.
        • Population SD 2 M = Population SD 2
            • N
          • Population SD 2 M = the variance of the distribution of means
          • Population SD 2 = the variance of the population of individuals
          • N = number of individuals in each sample
      • .
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 14. Standard Deviation of a Distribution of Means
    • The standard deviation of a distribution of means is the square root of the variance of the distribution of means comparison distribution.
      • Population SD M = √Population SD 2 M
        • Population SD M = standard deviation of the distribution of means
        • Population SD M is also known as the standard error of the mean.
          • tells you how much the means in the distribution of means deviate from the mean of the population
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 15. Variance of a Distribution of Means
    • SD of women’s height = 2.5 in.
        • Population SD 2 M = Population SD 2
            • N
      • .
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 16. Variance of a Distribution of Means
    • SD of women’s height = 2.5 in.
        • Population SD 2 M = Population SD 2
            • N
  • 17. Standard Deviation of a Distribution of Means Copyright © 2011 by Pearson Education, Inc. All rights reserved Pop. SD = 2.5 in.
  • 18. The Shape of a Distribution of Means
    • The shape of a distribution of means is approximately normal if either:
      • each sample is of 30 or more individuals or
      • the distribution of the population of individuals is normal
    • Regardless of the shape of the distribution of the population of individuals, the distribution of means tends to be unimodal and symmetrical.
      • Middle scores for means are more likely and extreme means are less likely.
      • A distribution of means tends to be symmetrical because lack of symmetry is caused by extremes.
        • Since there are fewer extremes in a distribution of means, there is less asymmetry.
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 19.
    • What is the distribution if you only throw one die at a time?
      • (hint-think about the probability of getting each number)
  • 20.   die 1 f 1 1 1 2 2 1 3 3 1 4 4 1 5 5 1 6 6 1 6               5               4               3               2               1                                 1 2 3 4 5 6
  • 21. 20   19   18   17   16   15   14   13   12   11   10   9   8   7   6   5     4         3             2                 1                                                 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 36 samples mean  f 1 1 1.5 2 2 3 2.5 4 3 5 3.5 6 4 5 4.5 4 5 3 5.5 2 6 1
  • 22. Review of the Three Kinds of Distributions
    • Population’s Distribution
      • made up of scores of all individuals in the population
      • could be any shape, but is often normal
      • Population M represents the mean.
      • Population SD 2 represents the variance.
      • Population SD represents the standard deviation.
    • Particular Sample’s Distribution
      • made up of scores of the individuals in a single sample
      • could be any shape
      • M = (∑X) / N calculated from scores of those in the sample
      • SD 2 = [∑(X – M) 2 ] / N
      • SD = √SD 2
    • Distribution of Means
      • means of samples randomly taken from the population
      • approximately normal if each sample has at least 30 individuals or if population is normal
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 23. Hypothesis Testing with a Distribution of Means: The Z Test
    • Z Test
      • Hypothesis-testing procedure in which there is a single sample and the population variance is known
      • The comparison distribution for the Z test is a distribution of means.
        • The distribution of means is the distribution to which you compare your sample’s mean to see how likely it is that you could have selected a sample with a mean that extreme if the null hypothesis were true.
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 24. Figuring the Z Score of a Sample’s Mean on the Distribution of Means
    • If you had a sample with a mean of 25, a distribution of means with a mean of 15, and a standard deviation of 5, the Z score of the sample’s mean would be 2.
      • Z = (M - Population M M )
        • Population SD M
        • Z = (25 – 15) = 2
        • 5
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 25. Figuring the Z Score of a Sample’s Mean on the Distribution of Means
      • Pop. M M = Pop. M mean height = 63.8
      • Pop. SD = 1.44
      • Sample 1 mean = 65
      • Sample II mean = 62
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 26. Figuring the Z Score of a Sample’s Mean on the Distribution of Means
      • Pop. M M = Pop. M mean height = 63.8
      • Pop. SD = 1.44
      • Sample 1 mean = 65
  • 27. Steps for Hypothesis Testing
    • The steps for hypothesis testing are the same for a sample of more than 1 as they are for a sample of 1.
      • Step 1: Restate the question as a research hypothesis and a null hypothesis about the population.
      • Step 2: Determine the characteristics of the comparison distribution.
      • Step 3: Determine the cutoff sample score on the comparison distribution at which the null hypothesis should be rejected.
      • Step 4: Determine your sample’s score on the comparison distribution.
      • Step 5: Decide whether to reject the null hypothesis.
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 28. Example of Steps for Hypothesis Testing: Step 1
    • Step 1: Restate the question as a research hypothesis and a null hypothesis about the population.
      • Population 1: Women at BAC
      • Population 2: Women in general
      • H a = Women at BAC are not equal in height to women in general
      • H 0 = Women at BAC are equal in height to women in general
  • 29. Example of Steps for Hypothesis Testing: Step 2
    • Step 2: Determine the characteristics of the comparison distribution.
      • The comparison distribution is a distribution of means of samples of 60 individuals each.
      • The mean is ______(the same as the population mean).
      • Population SD 2 =____, sample size = ___
      • Population SD 2 M = ________
      • Population SD M =
      • The shape of the distribution will be approximately normal because the sample size is larger than 30.
  • 30. Example of Steps for Hypothesis Testing: Step 2
    • Step 2: Determine the characteristics of the comparison distribution.
      • The comparison distribution is a distribution of means of samples of 60 individuals each.
      • The mean is 63.8 in. (the same as the population mean).
      • Population SD 2 = (2.5 2 ) 6.25 , sample size = 60
      • Population SD 2 M = 6.25 / 60 = .10
      • Population SD M = √ .10 = .32
      • The shape of the distribution will be approximately normal because the sample size is larger than 30.
  • 31. Example of Steps for Hypothesis Testing: Step 3
    • Step 3: Determine the cutoff sample score on the comparison distribution at which the null hypothesis should be rejected.
      • Significance level p<.05
      • One-tailed or Two-Tailed?
      • What is the cutoff Z?
  • 32. Example of Steps for Hypothesis Testing: Step 3
    • Step 3: Determine the cutoff sample score on the comparison distribution at which the null hypothesis should be rejected.
      • Significance level p<.05
      • Two-Tailed
      • Cutoff Z = -1.96 & +1.96
  • 33. Example of Steps for Hypothesis Testing: Step 4
    • Step 4: Determine your sample’s score on the comparison distribution.
      • Sample 1 mean = 65 in.
  • 34. Example of Steps for Hypothesis Testing: Step 5
    • Step 5: Decide whether to reject the null hypothesis.
    Z =3.75
  • 35. Example of Steps for Hypothesis Testing: Step 5
    • Step 5: Decide whether to reject the null hypothesis.
      • Reject the null hypothesis
      • Find support for the research hypothesis that BAC women’s height is not equal to women in general
  • 36. Steps for Hypothesis Testing
    • The steps for hypothesis testing are the same for a sample of more than 1 as they are for a sample of 1.
      • Step 1: Restate the question as a research hypothesis and a null hypothesis about the population.
      • Step 2: Determine the characteristics of the comparison distribution.
      • Step 3: Determine the cutoff sample score on the comparison distribution at which the null hypothesis should be rejected.
      • Step 4: Determine your sample’s score on the comparison distribution.
      • Step 5: Decide whether to reject the null hypothesis.
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 37. After-school example
      • H a = Children in academic after-school programs will have higher IQ scores than children in the general population.
      • H 0 = Children in academic after-school programs will not have higher IQ scores than children in the general population.
      • Population mean = 100
      • Population SD = 15
      • Sample mean = 107
      • Sample size = 35 children
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 38.
    • Step 1: Restate the question as a research hypothesis and a null hypothesis about the population.
      • Population 1: Children who participate in academic after-school program
      • Population 2: Children in general
      • H a = Children in academic after-school programs will have higher IQ scores than children in the general population.
      • H 0 = Children in academic after-school programs will not have higher IQ scores than children in the general population.
  • 39.
    • Step 2: Determine the characteristics of the comparison distribution.
      • The mean is ______(the same as the population mean).
      • Population SD 2 =____, sample size = ___
      • Population SD 2 M = ________
      • Population SD M =
      • The shape of the distribution will be approximately normal because the sample size is larger than 30.
  • 40. Example of Steps for Hypothesis Testing: Step 2
    • Step 2: Determine the characteristics of the comparison distribution.
      • The mean is 100 (the same as the population mean).
      • Population SD 2 = (15 2 ) or 225 , sample size = 35
      • Population SD 2 M = 225 / 35 = 6.43
      • Population SD M = √ 6.43 = 2.54
      • The shape of the distribution will be approximately normal because the sample size is larger than 30.
  • 41. Example of Steps for Hypothesis Testing: Step 3
    • Step 3: Determine the cutoff sample score on the comparison distribution at which the null hypothesis should be rejected.
      • Significance level p<.01
      • One-tailed or Two-Tailed?
      • What is the cutoff Z?
  • 42. Example of Steps for Hypothesis Testing: Step 3
    • Step 3: Determine the cutoff sample score on the comparison distribution at which the null hypothesis should be rejected.
      • Significance level p<.01
      • One-Tailed
      • Cutoff Z = +2.32
  • 43. Example of Steps for Hypothesis Testing: Step 4
    • Step 4: Determine your sample’s score on the comparison distribution.
      • Sample 1 mean = 107
  • 44. Example of Steps for Hypothesis Testing: Step 5
    • Step 5: Decide whether to reject the null hypothesis.
  • 45. Example of Steps for Hypothesis Testing: Step 5
    • Step 5: Decide whether to reject the null hypothesis.
      • A mean of 107 is 2.76 standard deviations above the mean of the distribution of means
      • Reject the null hypothesis (support the research hypothesis)
      • Children who attend academic after-school programs have higher IQ scores than children who do not attend the programs.
  • 46. Hypothesis Tests about Means of Samples in Research Articles
    • Z tests are not often seen in research articles because it is rare to know a population’s mean and standard deviation.
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 47. Advanced Topic: Estimation and Confidence Intervals
    • Estimating the population mean based on the scores in a sample is an important approach in experimental and survey research.
      • When the population mean is unknown, the best estimate of the population mean is the sample mean.
        • The accuracy of the population mean estimate is the standard deviation of the distribution of means (standard error).
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 48. Range of Possible Means Likely to Include the Population Mean
    • Confidence Interval
      • used to get a sense of the accuracy of an estimated population mean
      • It is the range of population means from which it is not highly unlikely that you could have obtained your sample mean.
      • 95% confidence interval
        • confidence interval for which there is approximately a 95% change that the population mean falls in this interval
          • Z scores from -1.96 to +1.96 on the distribution of means
      • 99% confidence interval
        • confidence interval for which there is approximately a 99% chance that the population mean falls in this interval
          • Z scores from -2.58 to +2.58
      • confidence limit
        • upper and lower value of a confidence interval
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 49. Figuring the 95% and 99% Confidence Intervals
    • Estimate the population mean and figure the standard deviation of the distribution of means.
      • The best estimate of the population mean is the sample mean.
      • Find the variance of the distribution of means.
        • Population S 2 M = Population SD 2 / N
        • Take the square root of the variance of the distribution of means to find the standard deviation of the distribution of means.
        • Population SD M = √Population SD 2 M
      • Find the Z scores that go with the confidence interval you want.
        • 95% CI Z scores are +1.96 and -1.96
        • 99% CI Z scores are +2.58 and -2.58
      • To find the confidence interval, change these Z scores to raw scores .
    Copyright © 2011 by Pearson Education, Inc. All rights reserved
  • 50. Example of Figuring the 99% Confidence Interval
    • If we used the earlier example of 60 BAC women
      • The population mean is 63.8 in and the standard deviation is 2.5 in.
      • The sample mean is 65.
    • Estimate the population mean and figure the standard deviation of the distribution of means.
      • The best estimate of the population mean is the sample mean of 65.
      • Find the variance of the distribution of means.
        • Population S 2 M = Population SD 2 / N = 2.5 2 / 60= .10
        • Take the square root of the variance of the distribution of means to find the standard deviation of the distribution of means.
        • Population SD M = √Population SD 2 M = √.10 = .32
      • Find the Z scores that go with the confidence interval you want .
        • 99% CI Z scores are +2.58 and -2.58
      • To find the confidence interval ,change these Z scores to raw scores.
        • lower limit = (-2.58)(.32) + 65 = -.83 + 65 = 64.17
        • upper limit = (+2.58)(.32) + 65 = .83 + 65 = 65.83
  • 51. Example of Figuring the 95% Confidence Interval
    • If we used the earlier example of 35 children who participated in academic after-school program
      • The population mean is 100 and the standard deviation is 15
      • The sample mean is 107.
    • Estimate the population mean and figure the standard deviation of the distribution of means.
      • The best estimate of the population mean is the sample mean of 107.
      • Find the variance of the distribution of means.
        • Population S 2 M = Population SD 2 / N = 15 2 / 35= 6.43
        • Take the square root of the variance of the distribution of means to find the standard deviation of the distribution of means.
        • Population SD M = √Population SD 2 M = √6.43=2.54
      • Find the Z scores that go with the confidence interval you want .
        • 99% CI Z scores are +1.96 and -1.96
      • To find the confidence interval ,change these Z scores to raw scores.
        • lower limit = (-1.96)(2.54.) + 107 = -.4.98 + 107 = 102.02
        • upper limit = (+1.96)(2.54) + 107 = .4.98 + 107 = 111.98
  • 52. Confidence Intervals In Research Articles
    • Confidence intervals are becoming more common in research articles in some fields.
    Copyright © 2011 by Pearson Education, Inc. All rights reserved