Aron chpt 6 ed revised

Hypothesis Tests with Means of Hypothesis Tests with Means of SamplesSamples

Chapter 6

Copyright © 2011 by Pearson Education, Inc. All rights reserved

The Distribution of MeansThe Distribution of MeansBegin 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 Building a Distribution of MeansMeans

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 Determining the Characteristics of a Characteristics of a Distribution of MeansDistribution 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 Mean of a Distribution of MeansMeans

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 MM = Population M

Population MM 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.


Example 2Example 2


die 1 die 2 Ave. die 1 die 2 Ave. die 1 die 2 Ave.

1 1 1 1 13 3 1 2 25 5 1 3

2 1 2 1.5 14 3 2 2.5 26 5 2 3.5

3 1 3 2 15 3 3 3 27 5 3 4

4 1 4 2.5 16 3 4 3.5 28 5 4 4.5

5 1 5 3 17 3 5 4 29 5 5 5

6 1 6 3.5 18 3 6 4.5 30 5 6 5.5

7 2 1 1.5 19 4 1 2.5 31 6 1 3.5

8 2 2 2 20 4 2 3 32 6 2 4

9 2 3 2.5 21 4 3 3.5 33 6 3 4.5

10 2 4 3 22 4 4 4 34 6 4 5

11 2 5 3.5 23 4 5 4.5 35 6 5 5.5

12 2 6 4 24 4 6 5 36 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 samplesmean 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 Variance of a Distribution of MeansMeansThe variance of a distribution of means is the

variance of the population of individuals divided by the number of individuals in each sample.

Population SD2M = Population SD2

N Population SD2

M = the variance of the distribution of means

Population SD2 = the variance of the population of individuals

N = number of individuals in each sample.


Standard Deviation of a Standard Deviation of a Distribution of MeansDistribution 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 SDM = √Population SD2

M

Population SDM = standard deviation of the distribution of means

Population SDM 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 Variance of a Distribution of MeansMeansSD of women’s height = 2.5 in.


N

.


Variance of a Distribution of Variance of a Distribution of MeansMeansSD of women’s height = 2.5 in.


N

Standard Deviation of a Standard Deviation of a Distribution of MeansDistribution of Means


Pop. SD = 2.5 in.

The Shape of a Distribution of The Shape of a Distribution of MeansMeans

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 12 2 13 3 14 4 15 5 16 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 samplesmean 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 Review of the Three Kinds of DistributionsDistributions• 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 SD2 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• SD2 = [∑(X – M)2] / N• SD = √SD2

• 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 normalCopyright © 2011 by Pearson Education, Inc. All rights reserved

Hypothesis Testing with a Hypothesis Testing with a Distribution of Means: The Z Distribution of Means: The Z TestTest

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 Figuring the Z Score of a Sample’s Mean on the Sample’s Mean on the Distribution of MeansDistribution 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 MM) Population SDM

Z = (25 – 15) = 2 5



◦Pop. MM = Pop. M mean height = 63.8◦Pop. SD = .32◦Sample 1 mean = 65◦Sample size = 60



◦Pop. MM = Pop. M mean height = 63.8◦Pop. SD = 1.44◦Sample 1 mean = 65

Steps for Hypothesis Steps for Hypothesis TestingTesting

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 Example of Steps for Hypothesis Testing: Step 1Hypothesis 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

◦ Ha = Women at BAC are not equal in height to women in general

◦ H0 = Women at BAC are equal in height to women in general

Example of Steps for Example of Steps for Hypothesis Testing: Step 2Hypothesis Testing: Step 2Step 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 SD2 =____, sample size = ___◦Population SD2

M = ________◦Population SDM =◦The shape of the distribution will be

approximately normal because the sample size is larger than 30.


◦ 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 SD2 = (2.52) 6.25, sample size = 60◦ Population SD2

M = 6.25 / 60 = .10◦ Population SDM = √.10 = .32◦ The shape of the distribution will be


Example of Steps for Example of Steps for Hypothesis Testing: Step 3Hypothesis Testing: Step 3Step 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?


◦Significance level p<.05◦Two-Tailed◦Cutoff Z = -1.96 & +1.96

Example of Steps for Example of Steps for Hypothesis Testing: Step 4Hypothesis Testing: Step 4Step 4: Determine your sample’s score on the comparison distribution.

◦ Sample 1 mean = 65 in.

Example of Steps for Example of Steps for Hypothesis Testing: Step 5Hypothesis Testing: Step 5Step 5: Decide whether to reject the null

hypothesis.

Z =3.75

Example of Steps for Example of Steps for Hypothesis Testing: Step 5Hypothesis Testing: Step 5Step 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 Steps for Hypothesis TestingTesting

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 exampleAfter-school example◦ Ha = Children in academic after-school

programs will have higher IQ scores than children in the general population.

◦ H0 = 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

◦ Ha = Children in academic after-school programs will have higher IQ scores than children in the general population.

◦ H0 = 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 SD2 =____, sample size = ___◦Population SD2

M = ________◦Population SDM =◦The shape of the distribution will be



◦ The mean is 100 (the same as the population mean).

◦ Population SD2 = (152) or 225 , sample size = 35◦ Population SD2

M = 225 / 35 = 6.43◦ Population SDM = √6.43 = 2.54◦ The shape of the distribution will be



◦Significance level p<.01◦One-tailed or Two-Tailed?◦What is the cutoff Z?


◦Significance level p<.01◦One-Tailed◦Cutoff Z = +2.32

Example of Steps for Example of Steps for Hypothesis Testing: Step 4Hypothesis Testing: Step 4Step 4: Determine your sample’s score on the comparison distribution.

◦ Sample 1 mean = 107

Example of Steps for Example of Steps for Hypothesis Testing: Step 5Hypothesis Testing: Step 5Step 5: Decide whether to reject the null

hypothesis.

Example of Steps for Example of Steps for Hypothesis Testing: Step 5Hypothesis Testing: Step 5Step 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 Hypothesis Tests about Means of Samples in Means of Samples in Research ArticlesResearch 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 Advanced Topic: Estimation and Confidence IntervalsConfidence IntervalsEstimating 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 Range of Possible Means Likely to Include the Likely to Include the Population MeanPopulation 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% Figuring the 95% and 99% Confidence IntervalsConfidence IntervalsEstimate 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 S2M = Population SD2 / N

Take the square root of the variance of the distribution of means to find the standard deviation of the distribution of means.

Population SDM = √Population SD2M

◦ 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% Example of Figuring the 99% Confidence IntervalConfidence 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 S2M = Population SD2 / N = 2.52 / 60= .10


Population SDM = √Population SD2M = √.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% Example of Figuring the 95% Confidence IntervalConfidence 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 S2M = Population SD2 / N = 152 / 35= 6.43


Population SDM = √Population SD2M = √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 Confidence Intervals In Research ArticlesResearch Articles

Confidence intervals are becoming more common in research articles in some fields.


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Aron chpt 6 ed revised

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