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    Aron chpt 6 ed Aron chpt 6 ed Presentation Transcript

    • Hypothesis Tests with Means of Samples
      • Chapter 6
      Copyright © 2011 by Pearson Education, Inc. All rights reserved
    • 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
    • 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
    • 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
    • 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
    • Example 2 Copyright © 2011 by Pearson Education, Inc. All rights reserved
    •   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
    • 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
    • 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
    • 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
    • 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
      • 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.
    • 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
    • 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
    • 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
    • Variance of a Distribution of Means
      • SD of women’s height = 2.5 in.
          • Population SD 2 M = Population SD 2
              • N
    • Standard Deviation of a Distribution of Means Copyright © 2011 by Pearson Education, Inc. All rights reserved Pop. SD = 2.5 in.
    • 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
      • What is the distribution if you only throw one die at a time?
        • (hint-think about the probability of getting each number)
    •   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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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.
    • 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.
    • 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?
    • 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
    • Example of Steps for Hypothesis Testing: Step 4
      • Step 4: Determine your sample’s score on the comparison distribution.
        • Sample 1 mean = 65 in.
    • Example of Steps for Hypothesis Testing: Step 5
      • Step 5: Decide whether to reject the null hypothesis.
      Z =3.75
    • 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
    • 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
    • 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
      • 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.
      • 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.
    • 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.
    • 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?
    • 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
    • Example of Steps for Hypothesis Testing: Step 4
      • Step 4: Determine your sample’s score on the comparison distribution.
        • Sample 1 mean = 107
    • Example of Steps for Hypothesis Testing: Step 5
      • Step 5: Decide whether to reject the null hypothesis.
    • 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.
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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
    • 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