CHAPTER 7CHAPTER 7PProbability and robability and Samples:Samples:

Distribution of Sample MeansDistribution of Sample Means

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Probability and Samples:Probability and Samples:Chap 7Chap 7

Sampling ErrorSampling Error

The amount of error between a The amount of error between a sample sample statistic (M) statistic (M) and and population parameter (µ).population parameter (µ).

Distribution of Sample Means: Distribution of Sample Means: is the is the collection of sample means for all the collection of sample means for all the possible random samples of a particular size possible random samples of a particular size (n) that can be obtained (n) that can be obtained from a population.from a population.

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Sampling DistributionSampling Distribution Sampling Distribution Sampling Distribution is a is a

distribution of statistics obtained distribution of statistics obtained by selecting all the possible by selecting all the possible samples of a specific size from a samples of a specific size from a population. Ex. Every distribution population. Ex. Every distribution has a mean and standard deviation. has a mean and standard deviation. The mean of all sample means is The mean of all sample means is called called Sampling Distribution. Sampling Distribution. The The mean of all standard deviations is mean of all standard deviations is called called Standard Error Standard Error of Mean of Mean ((σσMM))44

Expected Value of MExpected Value of MThe mean of the distribution of The mean of the distribution of (M) (M) sample sample means means (statistics) (statistics) is equal to the mean of the is equal to the mean of the Population of scores Population of scores (µ) (µ) and is called the and is called the Expected Value of M M= µ Expected Value of M M= µ

And, the average standard deviation (And, the average standard deviation (SS) for all ) for all of these means is called of these means is called Standard Error Standard Error of of Mean, Mean, σσMM. It provides a measure of how much . It provides a measure of how much distance is expected on average between a distance is expected on average between a sample mean (M) sample mean (M) and the and the population mean (µ)population mean (µ)

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The Law of Large NumbersThe Law of Large Numbers

The Law of Large Numbers The Law of Large Numbers states that the larger the states that the larger the sample size (n), the more sample size (n), the more probable it is that the sample probable it is that the sample mean mean (M)(M) will be close to the will be close to the population mean population mean (µ)(µ)

n≈ Nn≈ N66

Probability and SamplesProbability and Samples The The CCentral entral LLimit imit TTheorem:heorem:

Describes the distribution of sample means Describes the distribution of sample means by identifying 3 basic characteristics that by identifying 3 basic characteristics that describe any distribution:describe any distribution:1. 1. The The shape of the distribution shape of the distribution of sample of sample mean has 2 conditions mean has 2 conditions 1a1a. The population . The population from which the samples are selected is from which the samples are selected is normal normal distribution. distribution. 1b1b. The number of . The number of scores scores (n) (n) in each sample is relatively in each sample is relatively largelarge

(30 or more) (30 or more) The larger theThe larger the n n the shape of the the shape of the distribution tends to be more normal.distribution tends to be more normal.

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The The CCentral entral LLimit imit TTheorem:heorem: 2. 2. Central Tendency: Central Tendency: Stats that the mean of Stats that the mean of

the distribution of sample means the distribution of sample means M M is equal is equal to the population mean to the population mean µµ and is called the and is called the expected value of M. expected value of M. M= µ M= µ

3. 3. Variability: Variability: or the or the standard error standard error of mean of mean σσMM..

The standard deviation of the distribution of The standard deviation of the distribution of sample means is called the standard error sample means is called the standard error of mean of mean σσMM..

It measures the standard amount of It measures the standard amount of difference one should expect between difference one should expect between M M andand µ µ simply due to chance.simply due to chance. 88

Computations/ Calculations orComputations/ Calculations or Collect Collect Data and Compute Data and Compute Sample StatisticsSample Statistics

Z Score for ResearchZ Score for Research

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Computations/ Calculations or Collect Computations/ Calculations or Collect Data and Compute Sample StatisticsData and Compute Sample Statistics

Z Score for ResearchZ Score for Research

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Computations/ Calculations or Collect Computations/ Calculations or Collect Data and Compute Sample StatisticsData and Compute Sample Statistics

Z Score for ResearchZ Score for Research

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Computations/ Calculations or Collect Data Computations/ Calculations or Collect Data and Compute Sample Statisticsand Compute Sample Statistics

d=d=Effect Size/Cohn dEffect Size/Cohn dIs the difference between the Is the difference between the means in a treatment condition.means in a treatment condition.It means that the result from a It means that the result from a research study is not just by research study is not just by chance alonechance alone

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Computations/ Calculations or Collect Computations/ Calculations or Collect Data and Compute Sample StatisticsData and Compute Sample Statistics

d=d=Effect SizeEffect Size

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Problem 1Problem 1 The population of scores on the The population of scores on the

SAT forms a normal distribution SAT forms a normal distribution with with µ=500 µ=500 andand σσ=100.=100. If you take If you take a random sample of a random sample of n=25 n=25 students, students, what is the probability what is the probability (%) (%) that the sample mean will be that the sample mean will be greater than 540. greater than 540. M=540?M=540?

First calculate the Z Score then, look for proportion and convert into First calculate the Z Score then, look for proportion and convert into percentage.percentage.

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Problem 2Problem 2 Once again, the distribution of SAT forms a Once again, the distribution of SAT forms a

normal distribution with a mean of normal distribution with a mean of µ=500 µ=500 andand σσ=100.=100. For this example we are going For this example we are going to determine to determine what kind of sample mean (M) what kind of sample mean (M) is likely to be obtained as the average SAT is likely to be obtained as the average SAT score for a random sample of score for a random sample of n=25n=25 students. Specifically, we will determine the students. Specifically, we will determine the exact range of values that is expected for exact range of values that is expected for the sample mean the sample mean 80% 80% of the time.of the time.

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CHAPTER 8CHAPTER 8Hypothesis Hypothesis TestingTesting

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Chap 8Chap 8Hypothesis TestingHypothesis Testing Hypothesis : Hypothesis : Statement such as “The Statement such as “The

relationship between IQ and GPA.relationship between IQ and GPA. Topic of Topic of a research.a research.

Hypothesis Test: Hypothesis Test: Is a statistical method Is a statistical method that uses sample data to evaluate a that uses sample data to evaluate a hypothesis about a population. hypothesis about a population.

The statistics used to Test a hypothesis The statistics used to Test a hypothesis isis called “Test Statistic” called “Test Statistic” i.e., Z, t, r, F, etc.i.e., Z, t, r, F, etc. 1919

Hypothesis TestingHypothesis Testing The Logic of Hypothesis: The Logic of Hypothesis: If the If the

sample mean is sample mean is consistentconsistent with with the prediction we conclude that the prediction we conclude that the the hypothesishypothesis is is reasonablereasonable but, if there is a big but, if there is a big discrepancy discrepancy we decide that hypothesis iswe decide that hypothesis is not not reasonable. reasonable.

Ex. Registered Voters are Smarter than Average People. Ex. Registered Voters are Smarter than Average People.

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Role of Role of StatisticsStatistics in in Research Research

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Steps in Steps in Hypothesis-TestingHypothesis-TestingStep 1: State The Hypotheses Step 1: State The Hypotheses

HH00=Null Hypothesis=Null Hypothesis

HH11 :Alternative or :Alternative or Researcher HypothesisResearcher Hypothesis

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Steps in Steps in Hypothesis-TestingHypothesis-TestingStep 1: State The Hypotheses Step 1: State The Hypotheses

HH00 : µ ≤ 100 : µ ≤ 100 averageaverage

HH11 : µ > 100 : µ > 100 averageaverage

Statistics:Statistics: Because the Because the Population mean Population mean or or µ is µ is

known known the the statistic of choice is statistic of choice is zz-Score-Score

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Hypothesis TestingHypothesis TestingStep 2: Locate the Critical Region(s) or Step 2: Locate the Critical Region(s) or

Set the Criteria for a DecisionSet the Criteria for a Decision

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Directional Hypothesis TestDirectional Hypothesis Test

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None-directional None-directional Hypothesis TestHypothesis Test

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Hypothesis TestingHypothesis TestingStep 3: Computations/ Calculations or Step 3: Computations/ Calculations or

Collect Data and Compute Sample Collect Data and Compute Sample StatisticsStatistics

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Hypothesis TestingHypothesis TestingStep 4: Make a DecisionStep 4: Make a Decision

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Hypothesis TestingHypothesis TestingStep 4: Make a DecisionStep 4: Make a Decision

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Uncertainty and Errors in Uncertainty and Errors in Hypothesis TestingHypothesis Testing

Type I ErrorType I Error Type II Error Type II Error see next slide see next slide

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True H0 False H0

Reject Type I Error α Correct DecisionPower=1-β

Retain Correct Decision Type II error β

True State of the WorldTrue State of the World

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True H0 False H0

RejectReject Type I Type I

Error Error ααCorrect DecisionCorrect DecisionPower=1-Power=1-ββ

Retain Correct Correct DecisionDecision

Type II error Type II error β

PowerPower Power:Power: The The powerpower of a statistical test is the of a statistical test is the

probability that the test will correctly probability that the test will correctly reject a false null hypothesis. reject a false null hypothesis.

That is, That is, powerpower is the probability that is the probability that the test will identify a treatment effect the test will identify a treatment effect if one really exists.if one really exists.

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The The αα level or the level of level or the level of significance:significance:

The The αα level for a hypothesis test is level for a hypothesis test is the probability that the test will the probability that the test will lead to a lead to a Type I Type I error. error.

That is, the That is, the alpha level alpha level determines determines the the probability of obtaining sample probability of obtaining sample data in the critical region data in the critical region even even though the null hypothesis is true.though the null hypothesis is true.

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The The αα level or the level of level or the level of significance:significance:

It is a It is a probability value probability value which is used to define which is used to define the concept of the concept of ““highly highly unlikely”unlikely” in a hypothesis in a hypothesis test. test.

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The Critical RegionThe Critical Region Is composed of the extreme sample Is composed of the extreme sample

values that are highly unlikely (as values that are highly unlikely (as defined by thedefined by the αα level or the level of level or the level of significance) significance) to be obtained if the to be obtained if the null hypothesis is true. null hypothesis is true.

If sample data fall in the critical If sample data fall in the critical region, the null hypothesis is region, the null hypothesis is rejected.rejected.

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Computations/ Calculations or Collect Data Computations/ Calculations or Collect Data and Compute Sample Statisticsand Compute Sample Statistics

d=d=Effect Size/Cohn dEffect Size/Cohn dIt is the difference between the It is the difference between the means in a treatment condition.means in a treatment condition.It means that the result from a It means that the result from a research study is not just by research study is not just by chance alonechance alone

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Effect Size=Cohn’s dEffect Size=Cohn’s d

Effect Size=Cohn’s d= Effect Size=Cohn’s d= Result from the research Result from the research study is bigger than what study is bigger than what we expected to be just by we expected to be just by chance alone.chance alone.

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Cohn’s d=Cohn’s d=Effect SizeEffect Size

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Evaluation of Cohn’s d Effect Size Evaluation of Cohn’s d Effect Size with Cohn’s dwith Cohn’s d

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Magnitude of d Evaluation of Effect Size

d≈0.2 Small Effect Size

d≈0.5 Medium Effect Size

d≈0.8 Large Effect Size

ProblemsProblems Researchers have noted a decline in cognitive Researchers have noted a decline in cognitive

functioning as people age (Bartus, 1990) functioning as people age (Bartus, 1990) However, the results from other research However, the results from other research suggest that the antioxidants in foods such as suggest that the antioxidants in foods such as blueberriesblueberries can reduce and even reverse can reduce and even reverse these age-related declines, at least in these age-related declines, at least in laboratory rats (Joseph, Shukitt-Hale, laboratory rats (Joseph, Shukitt-Hale, Denisova, et al., 1999). Based on these results Denisova, et al., 1999). Based on these results one might theorize that the same antioxidants one might theorize that the same antioxidants might also benefit might also benefit elderly humans. elderly humans. Suppose a Suppose a researcher is interested in testing this theory. researcher is interested in testing this theory. Next slideNext slide

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ProblemsProblems Standardized neuropsychological tests such as Standardized neuropsychological tests such as

the the Wisconsin Card Sorting Test Wisconsin Card Sorting Test WCSTWCST can be use can be use to measure conceptual thinking ability and mental to measure conceptual thinking ability and mental flexibility (Heaton, Chelune, Talley, Kay, & Kurtiss, flexibility (Heaton, Chelune, Talley, Kay, & Kurtiss, 1993). Performance on this type of test declines 1993). Performance on this type of test declines gradually with age. Suppose our researcher selects a gradually with age. Suppose our researcher selects a test for which adults older than 65 have an average test for which adults older than 65 have an average score of score of μμ=80=80 with a standard deviation of with a standard deviation of σσ=20=20. . The The distribution of test score is approximately normal. The distribution of test score is approximately normal. The researcher plan is to obtain a sample of researcher plan is to obtain a sample of n=25n=25 adults adults who are older than 65, and give each participants a who are older than 65, and give each participants a daily dose of blueberry supplement that is very high in daily dose of blueberry supplement that is very high in antioxidants. After taking the supplement for 6 months antioxidants. After taking the supplement for 6 months

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ProblemsProblems The participants were given the The participants were given the

neuropsychological tests to measure their level neuropsychological tests to measure their level of cognitive function. of cognitive function. M=92M=92, , 2 tailed2 tailed, ,

αα = 0.05 = 0.05

The The hypothesishypothesis is that the is that the blubberyblubbery supplement supplement does appear does appear to have an effect to have an effect on cognitive functioning.on cognitive functioning.

Step 1Step 1

HH0 0 :: μμ with supplement with supplement = 80= 80

HH11 :: μμ with supplementwith supplement ≠ 80 ≠ 804343

None-directional None-directional Hypothesis TestHypothesis Test

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ProblemsProblems M=92M=92, , one tailed, one tailed,

αα = 0.05 = 0.05

If the If the hypothesishypothesis stated that the consumption stated that the consumption of of blubbery blubbery supplement will supplement will increaseincrease the the cognitive functioning (test scores) cognitive functioning (test scores) then,then,

Step 1Step 1

HH0 0 :: μμ ≤ ≤ 8080

HH11 :: μμ > > 8080

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Directional Hypothesis TestDirectional Hypothesis Test

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ProblemsProblems M=92, M=92, one tailed, one tailed,

αα = 0.05 = 0.05

If the If the hypothesishypothesis stated that the consumption stated that the consumption of of blubbery blubbery supplement will supplement will decreasedecrease the the cognitive functioning (test scores) cognitive functioning (test scores) then,then,

Step 1Step 1

HH0 0 :: μμ ≥ ≥ 8080

HH11 :: μμ < < 8080

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Directional Hypothesis TestDirectional Hypothesis Test

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ProblemsProblems Alcohol appears to be involve in a variety Alcohol appears to be involve in a variety

of birth defectsof birth defects, including low birth weight including low birth weight and retarded growth.and retarded growth. A researcher would A researcher would like to investigate like to investigate the effect of prenatal the effect of prenatal alcohol on birth weight alcohol on birth weight . A random . A random sample of sample of n=16n=16 pregnant rats is obtained. pregnant rats is obtained. The mother rats are given daily dose of The mother rats are given daily dose of alcohol. At birth, one pop is selected from alcohol. At birth, one pop is selected from each litter to produce a sample of each litter to produce a sample of n=16 n=16 newborn rats. The average weight for the newborn rats. The average weight for the sample is sample is M=16M=16 grams.grams. 4949

ProblemsProblems The researcher would like to compare the The researcher would like to compare the

sample with the general population of rats. sample with the general population of rats. It is known that regular new born rats have It is known that regular new born rats have an average weight of an average weight of μμ=18 =18 grams. The grams. The distribution of weight is normal distribution of weight is normal with with σσ=4=4, , set set αα=0.01=0.01, , and we use a and we use a 2 tailed test 2 tailed test consequently, 0.005 on each tail and the critical consequently, 0.005 on each tail and the critical value for Z=2.58 value for Z=2.58 Step 1Step 1

HH0 0 :: μμ alcohol exposure alcohol exposure = 18 = 18 gramsgrams

HH11 :: μμ alcohol exposurealcohol exposure ≠ 18 ≠ 18 gramsgrams

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Degrees of FreedomDegrees of Freedomdf=n-1df=n-1

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Standard Deviation of SampleStandard Deviation of Sample

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