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# Study Guide 672

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This 31 page Study Guide was uploaded by Al Backey on Sunday September 27, 2015. The Study Guide belongs to 672 at Central Michigan University taught by Dr. Ray Allen in Fall 2015. Since its upload, it has received 44 views. For similar materials see Statistics in Professional Education Services at Central Michigan University.

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Date Created: 09/27/15

Module One Introduction to Statistics PES 672 What Are Statistics gt Techniques dealing with organizing summarizing and interpreting information NUMERICALLY gt Ways of making unlimited amounts of information empirical logical and useful gt Methods and rules that allow us to predict relationships and effects Why Study Statistics gt Gather an manage important information gt Make informed judgments gt Evaluate decisions that affect your personalprofessional life Characteristics of Use Statistics are 0 A way of learning from data Tool that uses logic and known phenomenon to predict and interpret unknown phenomenon Concerned with all elements of study design and analysis not just computations Statistics require judgment not merely calculations Role of Statistics 1 Describe data organize summarize simplify 2 Explain Phenomena Discover causes and relationships Data detectives 3 Predict Use current situations from sample observations to predict how populations will respond 0 Data judges Statistics and Professionalism Is what we are doing working Can we justify our contribution Are we using our resources wisely Would other methods work better Will what we are considering work in this situation Critical Statistical Issues Types of data groups Populations parameter Samples statistic Selection of data 0 Random sampling Sampling error 0 Significant findings Probability statement Observed differences are most likely to be real rather than chance or error Statistical vs Practical significanc Populations vs Samples Key to Success in All Inquiries Come up with a trustworthy finding about a population The Need for Good Sampling BIAS Sample may not accurately represent the intended population 0 Result A Test of Bias Question Suppose that instead of choosing random samples of 25 students from a population of 100 students you selected the first 25 students for the first sample the next 25 students for the second sample and so on How might this sampling procedure bias the statistical results What are Biostatistics Methods of Sampling Random Convenient Stratified Cluster Overview of the Research Process 1 Development of a hypothesis and then 2 Gather information that will allow rejection or acceptance of that hypothesis Research Types Nonexperimental QuasiexperimentalDescriptive Correlation Experimental Cause amp Effect Descriptive Research Question Observing things as they are Prevalence absence quantity or quality of a particular factor or occurrence No or minimal interaction Correlation Research Question How related are factors or occurrences If one factor has a certain value will another factor have a similar value or similar level of occurrence Cause amp Effect Research Question Determining effect of an intervention What happens when we intervene Assumes the intervention is the explanation for any change Classic experimental research Distinguishing Between Variables Values Observations Observation Individual discrete units upon which measurements are made Can be an individual e g a person Can be an aggregate of individuals e g a region Variable the generic factor we measure AGE of a person 0 HIV status of a person Value a realized measurement 27 positive male drunk SPSS BASICS Basic Functions Opening the program Entering data Defining variables Loading data les sav alternate sources Saving les Perusing data outputs Descriptive Statistics PES 672 Module 2 Samples vs Populations What is a sample 0 Types of samples Convenient Systematic Random Stratified Cluster OOOOO When is Incomplete Data a problem Three Major Uses for Statistics 0 Describe 0 Used to organize and describe characteristics of a collection of data Infer via Contrasts 0 Uses observations to make conclusions judgments or predictions Infer via Relationships Looking for commonalities 0 Important Definitions Variable Value Observation Observation the unit upon Which measurements are made Can be an individual eg a person Can be an aggregate of individuals eg a region Variable the generic thing we measure eg AGE of a person eg HIV status of a person Value a realized measurement eg 27 eg positive Data Analysis Measurement Scales The kinds of tools used to analyze data depends on the characteristics of the measure used to represent it Measurement Scales Nominalcategorical Ordinal 0 Interval Ratio Consequences of Measurement Scales Example Ask five children their favorite color Coded using the following Measurement Scale Example One Pain level Experimental subjects may be asked to rate the level of pain experiences when Tazerred vs Cow prodded May use a 5 point scale Measurement Scale Example Two Free Throws 0 Log the number of free throws in 10 attempts on three occasions per practice over 15 practices 0 Log the percent made each time 0 Measurement Scale Example Three 0 Perceived Level of Exertion 0 Measurement Scale Example Four 0 Brain Types amongst All Conference MAC basketball players Check and See Measurement Scales worksheet 1 Colleges and universities are requiring an increasing amount of information about applicants before making acceptance and financial aid decisions Classify each of the following types of data required on a college application as quantitative or qualitative a High school GPA b High school class rank c Applicant s score on the SAT or ACT d Gender of applicant e Parents income f Age of applicant Descriptive statistics Tool 1 Frequency Distributions 0 Simple form of descriptive statistic 0 Used to organize data 0 Organizes by the frequency of individual scores within a data set 0 Is usually organized from highest to lowest individual score or ranges of scores 0 Allows the ability to determine basic patterns of scores 0 Presents the following elements 0 Set of categories or scores 0 Frequency of each individual score Frequency distribution in practice 0 The following set of N 20 scores can be organized by frequency 0 8 9 8 7 10 9 6 4 9 8 7 8 10 9 8 6 9 7 8 8 0 The distribution table is organized by score and how often it occurred X f Proportions amp Percentages 0 Distribution of scores can described as proportions of the whole 0 Proportions measure the fraction of the total group that is associated with a certain score p fN 0 Measure describes the relative frequency of a score to the distribution 0 E g two subjects out of 20 had a score of 6 so the proportion would be 2 0 Percentage can also be used to describe frequency of scores 0 Percentage is p fN 100 Percentages are commonly displayed as part of a frequency distribution chart Grouped or Class Frequency Distribution Tables Large numbers of scores or a wide range of scores are often best grouped according to a range of values or intervals Four basic rules in constructing this type of table 1 Should have about 10 intervals or classes determine range 2 Width of interval should be a simple number 3 Bottom score of each interval should be a multiple of the width 4 All intervals should be the same width Constructing a Grouped Frequency Table 0 Scores N25 gathered are 82 75 88 93 53 84 87 58 72 94 69 84 61 91 64 87 84 70 76 89 75 80 73 78 6O 0 Determine range 0 Determine width of intervals so that we have approximately 10 intervals 0 Identify the intervals Frequency Distribution Graphs 0 A picture or graphical representation of a frequency distribution 0 Often uses a bar graph or histogram 0 The height of the bar corresponds to the frequency 0 The width of the bar corresponds to the limits of the score or interval of scores 0 A histogram is used with data on an interval or ratio scale 0 A bar graph is used with data on a nominal or ordinal scale 0 Pareto graph tracks frequency of events for comparison 0 Circle or pie graph show relative proportion of events 0 Time series graphs represent data over time Shapes of Frequency Distributions Distributions can fit two basic shapes 0 Symmetrical Skewed 0 Positive 0 Negative Used to describe scores relative to one another Percentile Ranks 0 Can be used to describe the position of an individual score relative to others in a set 0 Percentage of individuals with scores at or below a certain level 0 Must determine cumulative frequency or number of scores in each category relative to the others Cumulative Percentages Uses cumulative frequency to associate a score as being above or below a percentage of the remaining score ie a score of 4 is higher than 95 of all the other scores Is determined by cfN 100 Task Computer Exercise One frequency distributions Measures of Central Tendency PES 672 Module 3 Tool 2 Measures of Central Tendency 0 A single score to describe an entire data set most typical score 0 Most common method of describing and summarizing a set of scores 0 Most central single value core bullseye etc 0 Three main types Mean Median Mode 0 Advantages 0 Provides a single number that can describe an entire set 0 Allows comparisons with other distributions 0 Relatively simple to calculate Dah Mean 0 Commonly known as the average of a set of scores 0 Gives an indication of the score at which the majority of scores are clustered 0 Identified as u for a population and M for a sample 0 Mean EXn 0 For a sample of n4 scores 3 7 4 6 0 Mean 2 Characteristics of the Mean 0 The mean will change in a predictable way when a single variable is changed 0 Changing a score or adding a new score 0 Adding or subtracting a constant from each score 0 Multiplying or dividing each score by a constant Disadvantages o Affected by extreme values The Median 0 Defined as the score that divides the distribution exactly in half 0 Is the middle score and also represents the 50th percentile 0 Determines the precise midpoint of a distribution Determining the Median 0 When N is an odd number 0 Scores 11 10 5 4 3 o 0 When N is an even number Scores 11 10 6 5 4 3 Find the middle two scores add them together and then divide by two a mini mean O O The Mode 0 Defined as the most common score in a distribution or most frequent score 0 Most useful in determining most typical score when using nominal data 0 Is represented most easily in a frequency distribution graph Disadvantages May not be unique For no repeated values all of them are modes Multimodes are difficult to interpret 0 A distribution can have more than one mode Two modes is called bimodal 0 two modes is multimodal J Which measure is best 0 Mean 0 Most common 0 Uses all the scores 0 Most related to variance 0 Is best used With interval and ratio scale 0 Mode 0 Easy to compute 0 Can be used With any scale of measurement most common With nominal scale 0 Median 0 Useful when extreme ranges of scores are present 0 Useful if undetermined values or open ended distributions are present 0 Useful for scores using ordinal scale Central Tendency amp Distribution Shape 0 Normal distribution unimodal 0 Mean median amp mode are the same 0 Normal distribution bimodal 0 Mean amp median are the same 0 Positively Skewed 0 Median amp mean Will be to the right of the mode 0 Negatively Skewed 0 Median amp mean Will be to the left of the mode Comparison of the Mean Median Mode Tool 3 Measures of Variability 0 Provides a measure that describes how much scores are spread out dispersion Describes the distribution amp individual scores in terms of distance from the mean 0 Measures how well an individual score represents the distribution Kurtosis tail Width curve steepness 0 Most basic measure of variability 0 Distance between largest score XmaX and lowest score Xmin 0 Uses the concept of real limits of a number 0 Range URL XmaX LRL Xmin 0 Eg scores 3 7 l2 8 5 10 0 Range 0 Doesn t account for all scores in a distribution Interquartile Range 0 Divides a distribution into four parts by using quartiles 0 Each quartile represents 25 of the scores in a distribution 0 Is equivalent to percentile ranks 0 1st quartile Ql is the score that separates the lowest 25 of the scores from the rest 0 2nd quartile Q2 is the score that has 50 of the distribution below it same as the median 0 Can be determined using a percentile rank table or distribution graph 0 The interquartile range is the distance between the first quartile and the third quartile Interquartile range Q3 Q1 0 Describes the scores that bound the middle 50 of the scores in a distribution 0 Eg Ql 25 Q3 8 Interquartile range Standard Deviation 0 Most common measure of variability represented by SD or O 0 Uses the mean of the distribution as a reference point 0 Measures variance of individual scores from the mean 0 Is also the average distance of scores from the mean Variance amp Standard Deviation 0 The most common measures of spread 0 Based on deviations around the mean 0 Each data value has a deviation defined as X M Task Computer Exercise 2 Measures of Central Tendency SO WHAT IS A Z SCORE A way to standardize scores Analogous turning raw test scores to s Allows us to predict proportions of scores and compare different scores using different units of measure Distribution of scores must be normal to begin with Practice Use SPSS to Calculate a z score for Raw Data Access data in SPSS Analyze Descriptive Statistics Descriptives Click left hand corner standardized values Demonstration Calculating Z Scores Using Achievement Data SAT Scores Generate descriptive statistics for analytic and writing scores Standardize the scores z scores Generate descriptives Raw data Means and standard deviations Standard scores Standard mean and deviation Relative performance by subjects 1 and 2 Practice School Data Analysis What were the raw mean scores What were the standardized mean scores What were the raw scores for student 1 Did student 1 do better relative to the rest of the class on exibility or abd strength Convert to Z scores Flexibility Abdominal Strength Save and send 672 Lastname z score TASK EXERCISE 3 Calculating and Interpreting 2 scores PES 672 NOTES Module 4 Probability Hypothesis Testing and Comparing Two Groups Signi cance and Comparative Studies What Allows us to Predict from Samples lunch bag 0 Professionals are interested in drawing inferences about POPULATIONS 0 Observed mean and standard deviation is not the same as the true mean and SD 0 Each time you sample there will be variation in sample mean and standard deviation Probability 0 The basis of decisionmaking in inferential statistics 0 Inferential statistics uses a sample to make judgments about the population 0 Identify samples that would represent the population 0 Use the laws of probability to draw conclusions about the population How Does the Normal Curve Help us Find Answers with Con dence 0 If observations follow a normal pattern we can predict the odds of obtaining our value randomly IF an observation is drawn randole What percent of observations will fall within ONE standard deviation of the mean 0 TWO standard deviations THREE standard deviations Why the normal curve is so valuable Randomly drawn data allowed to happen naturally almost always follows a normal pattern Example Comparing Coaching Salaries Texas 4A and 5A Head Football Coaches Mean 82179 St Dev 10457 What are the odds of randomly selecting a sample of coaches with a mean salary of 55000 Central Limit Theorem For randomly selected samples Ngt25 The distribution of sample means is approximately normal Mean of the distribution of samples equals the population mean The sd of samples equal the Population SD Allows us to Predict accuracy of the sample mean using the Normal Curve Standard Error of the Mean 0 Measure of the variation of the distribution of SAMPLE MEANS 0 Measures the amount of sampling error 0 Sampling error can be reduced by increasing sampling size 0 Working with sample means meets the assumption of normality provided the sample size is sufficient ngt25 0 We normally work within 2 Sample Errors which accounts for 95 probability Hypothesis Testing Four Steps Hypothesis Testing 1 State the null and alternative hypotheses 2 Calculate the appropriate test statistic 3 Convert the test statistic to a PValue 4 Make a statement about the signi cance level of the results 1 Null and Alternative Hypotheses 0 Convert the research question to null and alternative hypotheses 0 The null hypothesis H0 is a claim of no difference in the population 0 The alternative hypothesis Ha is a claim of difference Ho Example Salary 0 Statement of the problem 0 Suppose the annual salary in the United States is 25000 0 A survey of employees for a small company is taken SalaryFactors 0 Is the sample significantly different from the national average 0 The null hypothesis 0 The alternative hypothesis 2 Test Statistic 0 Calculates where the sample mean falls relative to the comparative distribution 0 One sample compared to a population 0 Two samples compared to each other 0 Uses the statistical test that meets the necessary criteria 0 Parametric Statistics results in a z or t score 3 Convert the Statistic to a pvalue 0 Probability the observed test statistic is inaccurate or having an alternate explanation assuming H0 is true 0 Equals area in the tail of the distribution outside the zstatistic 0 Smaller pvalues provider stronger evidence against H0 4 State Level of Signi cance 0 Assign pvalue for a level of confidence acceptable for the circumstances 0 plt 010 marginally significant I common value for pilot studies 0 plt 005 significant and reasonably trustworthy 39 Solid levels for educational decisions 0 plt 001 Highly significant evidence I Appropriate for high stake decisions 0 Interpret statistical results relative to p IMPORTANT Using pvalues to Support Decisions 0 What does the pvalue tell you 0 What does setting a critical value do for you 0 When you run a statistical test what do you determine IF O The pvalue is SMALLER than the critical value 0 The pvalue is LARGER than the critical value Illustration Body Weight 0 Statement of the problem 0 1970s 20 29 yo men in the US had a mean body weight 11 of 170 pounds with O 40 Ogden 2004 0 Does the mean body weight in the population differ now 0 The null hypothesis 0 The alternative hypothesis can be stated in one of two ways Reasoning Behind the zstat 0 TwoSided PValue 0 For onesided Ha gt use the Area Under the Curve in one of the tails 0 For twosided Ha gt double the onesided Pvalue consider deviations up and down from expected 0 Illustrative example If the oneside Pvalue 00010 then the twosided Pvalue 2 x 00010 00020 9 USING SPSS TO CALCULATE SINGLESAMPLE SIGNIFICANCE Mean of a single SAMPLE is known Known population Assume normality robust Example One Sample 0 Statement of the problem I Suppose the annual salary in the United States is 25000 0 A survey of employees for a small company is taken SalaryFactors I Is the sample significantly different from the national average 0 The null hypothesis 0 The alternative hypothesis 2 Calculate a Test Statistic PRACTICE I School Achievement Data Worksheet Part I 0 Abdominal fitness 0 Fitness gram Hypothesis Testing Errors 0 Testing the Null Hypothesis or Rejecting it 0 Reject the null at pgt005 I What is the chance we are right I What is the chance we are wrong 0 Accept the null at pgt005 I What is the chance we are right I What is the chance we are wrong Errors in Hypothesis Testing Type I error I Reject a null hypothesis that is true really was no difference 0 Most common and most serious 0 If alpha level is set stringently enough Type I errors are limited 0 Probability of Type I error is the alpha level or p Type II Error Accept a null hypothesis that is false really was a difference Failure to detect a difference Occurs with small samples andor small treatment effects Less serious consequences Probability of Type II error is more difficult to calculate and is represented by 5 Selecting an alpha level Minimize the risk of a Type I error Ability to detect differences Practical significance vs statistical significance Assumptions of Hypothesis Testing 1 Random sampling 2 Independent observations 3 Value of SE is constant following treatment 4 Normal sampling distribution PES 672 NOTES Module 5 Probability Hypothesis Testing and Comparing Two Groups Signi cance and Comparative Studies What Allows us to Predict from Samples lunch bag 0 Professionals are interested in drawing inferences about POPULATIONS 0 Observed mean and standard deviation is not the same as the true mean and SD 0 Each time you sample there will be variation in sample mean and standard deviation Probability 0 The basis of decisionmaking in inferential statistics 0 Inferential statistics uses a sample to make judgments about the population 0 Identify samples that would represent the population 0 Use the laws of probability to draw conclusions about the population How Does the Normal Curve Help us Find Answers with Con dence 0 If observations follow a normal pattern we can predict the odds of obtaining our value randomly IF an observation is drawn randomlv What percent of observations will fall within ONE standard deviation of the mean 0 TWO standard deviations THREE standard deviations Why the normal curve is so valuable Randomly drawn data allowed to happen naturally almost always follows a normal pattern Example Comparing Coaching Salaries Texas 4A and 5A Head Football Coaches Mean 82179 St Dev 10457 What are the odds of randomly selecting a sample of coaches with a mean salary of 55000 Central Limit Theorem For randomly selected samples Ngt25 The distribution of sample means is approximately normal Mean of the distribution of samples equals the population mean The sd of samples equal the Population SD Allows us to Predict accuracy of the sample mean using the Normal Curve Standard Error of the Mean 0 Measure of the variation of the distribution of SAMPLE MEANS 0 Measures the amount of sampling error 0 Sampling error can be reduced by increasing sampling size 0 Working with sample means meets the assumption of normality provided the sample size is sufficient ngt25 0 We normally work within 2 Sample Errors which accounts for 95 probability Hypothesis Testing Four Steps Hypothesis Testing State the null and alternative hypotheses Calculate the appropriate test statistic Convert the test statistic to a PValue Make a statement about the signi cance level of the results 908091 1 Null and Alternative Hypotheses 0 Convert the research question to null and alternative hypotheses 0 The null hypothesis H0 is a claim of no difference in the population 0 The alternative hypothesis Ha is a claim of difference Ho Example Salary 0 Statement of the problem 0 Suppose the annual salary in the United States is 25000 0 A survey of employees for a small company is taken SalaryFactors 0 Is the sample significantly different from the national average 0 The null hypothesis 0 The alternative hypothesis 2 Test Statistic 0 Calculates Where the sample mean falls relative to the comparative distribution 0 One sample compared to a population 0 Two samples compared to each other 0 Uses the statistical test that meets the necessary criteria 0 Parametric Statistics results in a z or t score 3 Convert the Statistic to a pvalue 0 Probability the observed test statistic is inaccurate or having an alternate explanation assuming H0 is true 0 Equals area in the tail of the distribution outside the zstatistic 0 Smaller pValues provider stronger evidence against H0 4 State Level of Signi cance 0 Assign pValue for a level of confidence acceptable for the circumstances 0 plt 010 marginally significant I common value for pilot studies 0 plt 005 significant and reasonably trustworthy 39 Solid levels for educational decisions 0 plt 001 Highly significant evidence I Appropriate for high stake decisions 0 Interpret statistical results relative to p IMPORTANT Using pvalues to Support Decisions 0 What does the pValue tell you 0 What does setting a critical value do for you 0 When you run a statistical test what do you determine IF O The pvalue is SMALLER than the critical value 0 The pvalue is LARGER than the critical value Illustration Body Weight 0 Statement of the problem 0 1970s 20 29 yo men in the US had a mean body weight 11 of 170 pounds with O 40 Ogden 2004 0 Does the mean body weight in the population differ now 0 The null hypothesis 0 The alternative hypothesis can be stated in one of two ways Reasoning Behind the zstat 0 TwoSided PValue 0 For onesided Ha gt use the Area Under the Curve in one of the tails 0 For twosided Ha gt double the onesided Pvalue consider deviations up and down from expected 0 Illustrative example If the oneside Pvalue 00010 then the twosided Pvalue 2 x 00010 00020 USING SPSS TO CALCULATE SINGLESAMPLE SIGNIFICANCE Mean of a single SAMPLE is known Known population Assume normality robust Example One Sample 0 Statement of the problem 0 Suppose the annual salary in the United States is 25000 0 A survey of employees for a small company is taken SalaryFactors 0 Is the sample significantly different from the national average 0 The null hypothesis 0 The alternative hypothesis 2 Calculate a Test Statistic PRACTICE 0 School Achievement Data Worksheet Part I 0 Abdominal fitness 0 Fitnessgram Hypothesis Testing Errors 0 Testing the Null Hypothesis or Rejecting it 0 Reject the null at pgt005 I What is the chance we are right I What is the chance we are wrong 0 Accept the null at pgt005 I What is the chance we are right I What is the chance we are wrong Errors in Hypothesis Testing Type I error I Reject a null hypothesis that is true really was no difference 0 Most common and most serious 0 If alpha level is set stringently enough Type I errors are limited 0 Probability of Type I error is the alpha level or p Type II Error Accept a null hypothesis that is false really was a difference Failure to detect a difference Occurs with small samples andor small treatment effects Less serious consequences Probability of Type II error is more difficult to calculate and is represented by 5 Selecting an alpha level Minimize the risk of a Type I error Ability to detect differences Practical significance vs statistical significance Assumptions of Hypothesis Testing 5 Random sampling 6 Independent observations 7 Value of SE is constant following treatment 8 Normal sampling distribution How do We Look For Signi cance Between TWO Groups Comparing GroupszKey Terms to Know and Consider 0 Independent vs Related Samples 0 With Independent Samples I The value obtained in one observation is not in uenced or impacted by the other variable 0 Dependent Independent Variables 0 Independent Variable I The variable you use and control to group variables 0 Dependent Variable I The resultant values obtained I Score depend on the stuff you manipulate SPSS Two Sample tTest Related Samples 0 Analyze compare means paired Options confidence level Independent Samples 0 Independent variable is the grouping variable I IV must be only two groups Demonstration Two Related Samples 0 State the hypotheses 0 Identify the dependent variable 0 Identify the independent variable 0 Identify if samples are related or independent I Choose the test I Set the critical value Example 1 Related Samples From quotCollegeCourseStudentData Do Students have different numbers of younger than older siblings Null 0 Alternative 0 Dependent Variable 0 Independent Variable I Independentrelated Samples Example 2 Single Value Group divided by a Value From quotCollegeCourseStudentData Do those with a good GPA have the same number of younger SiBs as those with weak GPA s Null 0 Alternative 0 Dependent Variable 0 Independent Variable I Independentrelated Samples Example 3 Independent Samples From quotCollegeCourseStudentData Are the GPA s of those planning on pursuing a PhD different from those who are not Null 0 Alternative 0 Dependent Variable 0 Independent Variable Independentrelated Samples OneWay Analysis of Variance PES 672 Module 6 Review Hypothesis Testing 1 Identify your samples 2 Null Hypothesis vs Alternative 3 Select the correct statistic Number of comparison groups 0 Identify the scale for dependent variable 0 Data normally distributed 4 Calculate the test statistic and PValue What does the pvalue tell you 5 Make a statement about the significance of the results Review The Normal Curve If we know the scores are distributed normally What percent of the samples will be within One standard deviation of the mean 0 Two standard deviations of the mean 0 THREE standard deviations of the mean Review ttests Allows us to COMPARE two groups of data Does the data behave they way we think it should ie is it normal Three types One sample compared to a known population 0 Two related samples 0 Two independent samples How Confident Are we in Our Decisions Type I Error With alpha at 05 0 If we reject the null 0 Type I error 0 Assume a relationship exists when it really does not 0 Probability of Type I error determined by significance level Example How Confident Are we in Our Decisions Type 11 Error With alpha at 05 0 If we fail to reject the null 0 Type II error The chances that the two samples are drawn from the same population 0 Example 0 Pregnancy test 0 Woman B is not different from any other woman Is there a chance that the woman is pregnant even if the test says no Size of Beta depends on sample size 0 Power equals lBeta IMPORTANT Using pvalues to Support Decisions What does the pvalue tell you What does setting a critical value do for you When you run a statistical test what do you determine IF The pvalue is SMALLER than the critical value 0 The pvalue is LARGER than the critical value Hypothesis Testing Errors Testing the Null Hypothesis or Rejecting it Reject the null at plt005 What is the chance we are right What is the chance we are wrong Accept the null at pgt005 What is the chance we are right What is the chance we are wrong Errors in Hypothesis Testing Type I error Reject a null hypothesis that is true really was no difference Most common and most serious 0 If alpha level is set stringently enough Type I errors are limited 0 Probability of Type I error is the alpha level or p Type II Error Accept a null hypothesis that is false really was a difference 0 Failure to detect a difference Occurs with small samples andor small treatment effects Less serious consequences 0 Probability of Type II error is more difficult to calculate and is represented bYB Selecting an alpha level Minimize the risk of a Type I error Ability to detect differences Practical significance vs statistical significance WHAT IF WE HAVE MORE THAN TWO GROUPS WE WANT TO COMPARE Uses of the Ttest Does a sample differ from a population Are two independent groups different from each other Are two related samples different from each other Limitations with the Ttest Compare two groups gIV on m DV Assumptions 0 Random samples Samples are normally distributed 0 DV is on an interval scale Independent samples 0 Dependent variable is interval data 0 Independent variable in categories 0 Different populations Related samples Must be interval data Comparing More Than Two Groups Two groups plt005 Interpretation Type 1 error Three groups Interpretation Type 1 error Four groups Interpretation Type 1 error The Problem of Multiple Comparisons The more comparisons the greater the familvwise error rate Introduction to Analysis of Variance ANOVA Commonalities with ttests Used to evaluate mean differences between treatmentspopulations Uses sample data to make conclusions about populations Comparisons to ttests Compare two or more treatmentspopulations Greater exibility in design Can use either independent or dependent measures ANOVA Test Statistic F F de ned Variance between sample meansvariance expected by chance error F ratio variance between treatmentsvariance within treatments Denominator is called error term If numerator amp denominator are roughly equal there is little difference due to treatment and F will be close to 10 How ANOVA Works Compares Variability between groups 39 Variability within groups Computes F statistic F statistic is translated to a pvalue sign Variability Mean Square Between The variability between groups provides a signal of group difference This variance is quantified with a statistic called the Mean Square Between MSB The MSB is the variance of the group means around the grand mean Variability Mean Square Within The variability within group quantifies the noise of random group difference This variance is quantified with a statistic called the Mean Square Within MSW The MSW is a weighted average of the variance around group means F statistic and ANOVA Table The F statistic is converted to a Pvalue with an F table or computer program The F statistic and ANOVA Table The Pvalue corresponds to the area under the curve to the right of the F statistic FRatio Distributions Determines how much differences between groups is significant F Ratios will always be positive values When null is true the two variances will be about the same and the ratio will be near 10 and the scores will pile up around 10 Distribution will have a high point around 1 and taper off to the right Shape will depend on degrees of freedom smaller df results in atter curve Hypothesis Testing with ANOVA F Step 1 State null hypothesis and set alpha level Step 2 Compute F Rati0 for this data Step 3 Calculatedetermine a pvalue associated with the F distribution Step 4 Make hypothesis decision Illustration Pets moderating stress response Heart rates bpm monitored after being exposed to a psychological stressor Group 1 0 Presence of their pet dog Group 2 Presence of a human friend Group 3 0 Neither their dog or friend present SPSS Data Set Up Data should be set up in two columns One column is for the explanatory variable group One column is for the response variable heart rate USING SPSS TO SIGNIFICANCE BETWEEN gtTWO GROUPS One DV 0 Interval data One IV 0 Multiple groupings Independent groups Assume normality Descriptive Statistics Data should be described and explored before moving on to more complex inferential calculations F statistic and ANOVA Tam F 1193843 84793 1407 Interpretation of a Significant Fvalue Sign p gt alpha level 0 All of the means fall within the range of possible chance 0 Fail to conclude with any confidence that any of the means are significantly different at plt005 Sign p lt alpha level 0 The null hypothesis is false At least one of the means is probably not a result of random error Post Hoc Comparisons ONLY after concluding that at least one of the population means differs 0 After rejection of the ANOVA H0 Used to determine which means differ Many post hoc options SPSS Choices for Post Hoc Comparison Procedures Why Post Hoc Tests Basic ANOVA 0 Indicates if there are differences among the samples Does not indicate m samples are different from the others Pairwise comparisons Allows examination of individual samples two at a time Each of these comparisons includes risk of a Type I error 0 Risk increases with the number of pairwise comparisons What Post Hoc Tests Tukey s HSD Sample sizes must be equal Scheffe Test Used when sample sizes are unequal Bonferroni Correction Designed to correct for family wise error More conservative of the posthoe tests guards best against Type I error Used when multiple measurements are taken of the outcome e g pretest amp posttest TASK EXERCISE 6 ANOVA

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