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# Aron chpt 6 ed

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## Aron chpt 6 edPresentation Transcript

• Hypothesis Tests with Means of Samples
• Chapter 6
• 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)
• 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
• 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.
• 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.
•   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
• .
• 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
• Variance of a Distribution of Means
• SD of women’s height = 2.5 in.
• Population SD 2 M = Population SD 2
• N
• .
• Variance of a Distribution of Means
• SD of women’s height = 2.5 in.
• Population SD 2 M = Population SD 2
• N
• 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.
• 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
• 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.
• 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
• 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
• 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.
• 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.
• 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
• 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.
• 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).
• 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
• 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 .
• 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.