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### Aron chpt 6 ed

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