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# Introduction to Statistical Analysis STAT 245

OSU

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This 16 page Class Notes was uploaded by Alison Vandervort on Monday September 21, 2015. The Class Notes belongs to STAT 245 at Ohio State University taught by Staff in Fall. Since its upload, it has received 61 views. For similar materials see /class/210005/stat-245-ohio-state-university in Statistics at Ohio State University.

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Date Created: 09/21/15

Review Chapters 3 8 Chapter 4 1 Random Phenomenon have longrun regularity a probability as longterm proportion b subjective interpretation 2 Probability Models a Sample Space i Event b probability rules i any probability is a number between 0 and 1 inclusive ii all possible outcomes together must have probability 1 iii ifA is an event then PAC 1 PA Complement Rule iv ifA and B are events with no outcomes in common disjointthen PA or B PA PB c Finite Sample Space PA is sum of probabilities of its component outcomes d Independent Events PA and B occur PAPB 3 Random Variables a Discrete and Continuous b Probability distributions give values of RV and corresponding probabilities i Probabilities as areas under the density curve use geometry or calculus to find them ii Probabilities from normal distributions 4 Means and Variances of RV a How to calculate for discrete RV and for continuous RV using calculus i Rules for linear functions aX b ii Rules for sum MW RV39s Law of Large Numbers i proportion of occurrences gets closer to true probability in the LONGRUN i7 gt p ii average ofa set of outcomes gets closerto the population mean in the LONG RUN 97 u iii beware the myth of small numbers 039 Chapter 3 A Experiments 1 Recognize whether a study is an observational study or an experiment an experiment must impose a treatment 2 Recognize bias due to confounding of explanatory variables with lurking variables in either an observational study or an experiment 3 Identify the factors explanatory variables levels treatments response variables and experimental units or subjects in an experiment 1 245 Final Review 4 5 Know and explain the basic principles of experimental design a Understand the importance of comparative experiments b Explain why a randomized comparative experiment can give good evidence for cause and effect relationships c Explain the need for replication Outline the design of a completely randomized experiment using a diagram The diagram should show the sizes ofthe groups the speci c treatments and the response variable Use Table B to carry out the random assignment of subjects to groups in a completely randomized experiment Recognize the placebo effect Recognize when the doubleblind technique should be used Know what a block design is and when it is appropriate to use this design Recognize a matched pairs experiment Understand to which groups results of an experiment can be generalized and that lack of realism can prevent us from generalizing results B Sampling FnPFDN 9 Identify the population in a sampling situation Identify the sampling frame in a sampling situation Know terms such a sample probability sampling and bias Distinguish sample from census Recognize bias due to voluntary response samples and other inferior sampling methods Understand the definition ofa simple random sample and be able to recognize if a sampling scheme produces an SRS Use Table B to select a simple random sample from a population Recognize the presence of undercoverage and nonresponse as sources of error in a sample survey Recognize the effect of the wording ofquestions on the responses Use random digits to select a stratified random sample from a population when the strata are de ned C Sampling Distributions 1 Identify parameters and statistics in a sample or experiment 2 Recognize the fact of sampling variability a statistic will take different values when you repeat a sample or experiment 3 Interpret a sampling distribution as describing the values taken by a statistic in all possible repetitions ofa sample or experiment under the same conditions 4 Interpret the sampling distribution ofa statistic as describing the probabilities of its possible values 5 Distinguish bias and variability in a sampling distribution 2 245 Final Review Chapter 5 Sampling Distributions A The Sampling Distribution for Counts and Proportions 1 The Binomial Setting Two possible outcomes a xed number n of independent observations with a xed probability p of success for each observation 2 Approximate Binomial setting SRS ofless than 5 of population 3 Exact Binomial Distribution of the count X of successes Use Table C Know the formulas for mean and standard deviation ofX 4Sample proportion Xn Know its mean and standard deviation 5 Normal approximation for distribution of counts and proportions Uses the exact means and variances 13 is approximately normal with mean p and standard deviation quotM n B The Sampling Distribution ofa Sample Mean 1 For a SRS f has mean yand standard deviation n 2 lfthe population is normal then the sampling distribution of f is exactly normal a Alinear combination of independent normal RV s is also normal 3lfthe population is not normal then the Central Limit Theorem says that the sampling distribution of f is approximately normal for SRS s of large size greater than 30 with mean 1 and standard deviation 1 J Chapter 6 A Con dence Intervals 1 State in nontechnical language what is mean by 95 con dence Be clear about the difference between con dence and probability A 95 or 90 or 80 etc confidence interval is an interval obtained by a method that in 95 or 90 or 80 etc ofall samples will produce an interval containing the true population parameter Calculate a con dence interval forthe mean u ofa normal population with known standard deviation 039 Know the formula forthe margin oferror and forthe 2 interval endpoints level C con dence interval f 21 where 2 is J obtained from Table D Recognize when you can safely use this method for computing con dence intervals and when the data collection design or a small sample from a skewed population makes it inaccurate Understand that the margin oferror does not include the effects of undercoverage nonresponse or other practical dif culties It just accounts for sampling variability Understand how the margin of error ofa con dence interval changes with the sample size standard deviation and the level of confidence C 3 4 245 Final Review 5 Understand how the critical value 2 is obtained 6 Find the sample size required to obtain a con dence interval of speci ed margin oferror m when the con dence level and other information are given B Signi cance Tests 1 Hypotheses a null hypothesis Ho i statement we assume is true amp assess strength of evidence against ii generally a statement of quotno effectquot or quotno differencequot iii general form population parameter speci c value eg y O or p 05 b alternate hypothesis Ha i statement which before we look at data we hope or suspect is true instead of Ho ii general forms a twosided population parameter at speci c value eg ya 0 or p 05 a onesided population parameter gt speci c value eg y gt O or p gt 05 population parameter lt speci c value eg u lt O or p lt 05 2 Test statistic Z a number calculated from the data that mesures how far the data differ from what we expect when the null hypothesis is true Usually a standardized value of an estimate ofthe population parameter 3 A Critical value approach Reject Ho ifZ is too far away from its expectation when H0 is true ie when Z is in the quotcritical regionquot The critical region is chosen so that the probability of rejecting Ho when it is true is a speci ed value a 3 B Pvalues a interpretation A probability computed assuming H0 is true that a sample would yield results as extreme or more extreme than the results we got Small Pvalues provide strong evidence against the null hypothesis It is computed as the value of a that would result if the observed value on were on the border ofthe critical region b often computed as some area under a density curve of a normal sampling distribution i for a onesided alternative hypothesis Pvalue involves area in only one tail ii for a twosided alternative hypothesis Pvalue involvestotal area in both tails c a test is statistically signi cant at level aie reject Ho if the Pvalue ofthe test is less than or equal to the signi cance level a This makes the two forms of signi cance testing completely equivalent 3 Tests for Population Means a assumptions needed for validity i data are from a SRS ii n is large iii population standard deviation 039 is known b state hypotheses i H0 1 go where go is some known value ii Ha y yo where is one oflt or gt depending on the particular problem x 0 64 c calculate test statistic the standard score of 7c 2 4 245 Final Review d Compare 2 to the critical value 2 and reject Ho when 2 lt z z gt z or z gt 2 depending on the alternative The value of z is chosen to make the probability of rejecting Ho or when H0 is true ALTERNATIVELY calculate Pvalue using Table A i Ha determines the appropriate area under the normal curve to calculate e Know the equivalence between two sided hypothesis tests and con dence intervals 4 Some Cautions on Tests of Significance a Pvalues are more informative than signi cance levels which have often been used inappropriately b lack of statistical significance can have practical signi cance and shouldn39t be ignored c beware of searching for signi cance a large set oftests will often produce one or more signi cant results by chance even though Ho really was true in every case d Recognize that signi cance testing does not measure the size or importance of an effect statistical signi cance vs practical significance Especially with very large sample sizes a result may be statistically signi cant without having much practical signi cance V th small sample sizes effects that are of practical significance may not be statistically signi cant e Recognize when you can use the 2 test and when the data collection design or a small sample from a skewed population makes it inappropriate Statistical signi cance means basically nothing if data was collected through a study with poor design Chapter 8 Con dence lntervals and Tests for Proportions 1 Know the assumptions for validity data are from a SRS n is large but not more than 5 ofthe population p is not too close to O or 1 Con dence intervals for Population Proportions p 1 Use the quotlarge samplequot method based on the sample proportion iv a level C con dence interval is i1lz M where 2 is obtained from Table D n This would be the default method provided that the conditions for its use are satis ed Know what those conditions are X2 2 Use the Plus Four estimate Z7 71 4 a level C con dence interval is iJiz pa if where zis obtained from Table D n Know the conditions for using this approach 3 nd the sample size needed to produce a speci ed margin of error and con dence level using the quotlarge samplequot method Know the conservative approach and the guessed p approach 5 245 Final Review Tests for Population Proportions 1 state hypotheses a H0 p p0 where p0 is some known value b Ha p p0 where is one oflt or gt depending on the particular problem 2 calculate test statistic the standard score of a z 3 calculate Pvalue using Table A a Ha determines the appropriate area under the normal curve to calculate b Know when this approach is justi ed Chapter 7 A OneSamplet Procedures 1 Recognize when the t procedures are appropriate in practice in particularthat they are moderately robust against lack of normality but are strongly in uenced by outliers 2 Also recognize when the design of the study outliers or a small sample from a skewed distribution make the t procedures risky 3 Use t to obtain a con dence interval at a stated level of con dence for the mean u ofa population when the population standard deviation 0 is unknown and the sample standard deviation s is used as an estimate of 039 Carry out a t test for the hypothesis that a population mean u has a specified value against either a onesided or a twosided alternative Use Table D oft distribution critical values to approximately give a range for the Pvalue or carry out a xed level a test 5 Know the equivalence between two sided hypothesis tests and con dence intervals 5 6 245 Final Review Review Chapters 3 8 Chapter 4 1 Random Phenomenon have longrun regularity a probability as longterm proportion b subjective interpretation 2 Probability Models a Sample Space i Event b probability rules i any probability is a number between 0 and 1 inclusive ii all possible outcomes together must have probability 1 iii ifA is an event then PAC 1 PA Complement Rule iv ifA and B are events with no outcomes in common disjointthen PA or B PA PB c Finite Sample Space PA is sum of probabilities of its component outcomes d Independent Events PA and B occur PAPB 3 Random Variables a Discrete and Continuous b Probability distributions give values of RV and corresponding probabilities i Probabilities as areas under the density curve use geometry or calculus to find them ii Probabilities from normal distributions 4 Means and Variances of RV a How to calculate for discrete RV and for continuous RV using calculus i Rules for linear functions aX b ii Rules for sum MW RV39s Law of Large Numbers i proportion of occurrences gets closer to true probability in the LONGRUN i7 gt p ii average ofa set of outcomes gets closerto the population mean in the LONG RUN 97 u iii beware the myth of small numbers 039 Chapter 3 A Experiments 1 Recognize whether a study is an observational study or an experiment an experiment must impose a treatment 2 Recognize bias due to confounding of explanatory variables with lurking variables in either an observational study or an experiment 3 Identify the factors explanatory variables levels treatments response variables and experimental units or subjects in an experiment 1 245 Final Review 4 5 Know and explain the basic principles of experimental design a Understand the importance of comparative experiments b Explain why a randomized comparative experiment can give good evidence for cause and effect relationships c Explain the need for replication Outline the design of a completely randomized experiment using a diagram The diagram should show the sizes ofthe groups the speci c treatments and the response variable Use Table B to carry out the random assignment of subjects to groups in a completely randomized experiment Recognize the placebo effect Recognize when the doubleblind technique should be used Know what a block design is and when it is appropriate to use this design Recognize a matched pairs experiment Understand to which groups results of an experiment can be generalized and that lack of realism can prevent us from generalizing results B Sampling FnPFDN 9 Identify the population in a sampling situation Identify the sampling frame in a sampling situation Know terms such a sample probability sampling and bias Distinguish sample from census Recognize bias due to voluntary response samples and other inferior sampling methods Understand the definition ofa simple random sample and be able to recognize if a sampling scheme produces an SRS Use Table B to select a simple random sample from a population Recognize the presence of undercoverage and nonresponse as sources of error in a sample survey Recognize the effect of the wording ofquestions on the responses Use random digits to select a stratified random sample from a population when the strata are de ned C Sampling Distributions 1 Identify parameters and statistics in a sample or experiment 2 Recognize the fact of sampling variability a statistic will take different values when you repeat a sample or experiment 3 Interpret a sampling distribution as describing the values taken by a statistic in all possible repetitions ofa sample or experiment under the same conditions 4 Interpret the sampling distribution ofa statistic as describing the probabilities of its possible values 5 Distinguish bias and variability in a sampling distribution 2 245 Final Review Chapter 5 Sampling Distributions A The Sampling Distribution for Counts and Proportions 1 The Binomial Setting Two possible outcomes a xed number n of independent observations with a xed probability p of success for each observation 2 Approximate Binomial setting SRS ofless than 5 of population 3 Exact Binomial Distribution of the count X of successes Use Table C Know the formulas for mean and standard deviation ofX 4Sample proportion Xn Know its mean and standard deviation 5 Normal approximation for distribution of counts and proportions Uses the exact means and variances 13 is approximately normal with mean p and standard deviation quotM n B The Sampling Distribution ofa Sample Mean 1 For a SRS f has mean yand standard deviation n 2 lfthe population is normal then the sampling distribution of f is exactly normal a Alinear combination of independent normal RV s is also normal 3lfthe population is not normal then the Central Limit Theorem says that the sampling distribution of f is approximately normal for SRS s of large size greater than 30 with mean 1 and standard deviation 1 J Chapter 6 A Con dence Intervals 1 State in nontechnical language what is mean by 95 con dence Be clear about the difference between con dence and probability A 95 or 90 or 80 etc confidence interval is an interval obtained by a method that in 95 or 90 or 80 etc ofall samples will produce an interval containing the true population parameter Calculate a con dence interval forthe mean u ofa normal population with known standard deviation 039 Know the formula forthe margin oferror and forthe 2 interval endpoints level C con dence interval f 21 where 2 is J obtained from Table D Recognize when you can safely use this method for computing con dence intervals and when the data collection design or a small sample from a skewed population makes it inaccurate Understand that the margin oferror does not include the effects of undercoverage nonresponse or other practical dif culties It just accounts for sampling variability Understand how the margin of error ofa con dence interval changes with the sample size standard deviation and the level of confidence C 3 4 245 Final Review 5 Understand how the critical value 2 is obtained 6 Find the sample size required to obtain a con dence interval of speci ed margin oferror m when the con dence level and other information are given B Signi cance Tests 1 Hypotheses a null hypothesis Ho i statement we assume is true amp assess strength of evidence against ii generally a statement of quotno effectquot or quotno differencequot iii general form population parameter speci c value eg y O or p 05 b alternate hypothesis Ha i statement which before we look at data we hope or suspect is true instead of Ho ii general forms a twosided population parameter at speci c value eg ya 0 or p 05 a onesided population parameter gt speci c value eg y gt O or p gt 05 population parameter lt speci c value eg u lt O or p lt 05 2 Test statistic Z a number calculated from the data that mesures how far the data differ from what we expect when the null hypothesis is true Usually a standardized value of an estimate ofthe population parameter 3 A Critical value approach Reject Ho ifZ is too far away from its expectation when H0 is true ie when Z is in the quotcritical regionquot The critical region is chosen so that the probability of rejecting Ho when it is true is a speci ed value a 3 B Pvalues a interpretation A probability computed assuming H0 is true that a sample would yield results as extreme or more extreme than the results we got Small Pvalues provide strong evidence against the null hypothesis It is computed as the value of a that would result if the observed value on were on the border ofthe critical region b often computed as some area under a density curve of a normal sampling distribution i for a onesided alternative hypothesis Pvalue involves area in only one tail ii for a twosided alternative hypothesis Pvalue involvestotal area in both tails c a test is statistically signi cant at level aie reject Ho if the Pvalue ofthe test is less than or equal to the signi cance level a This makes the two forms of signi cance testing completely equivalent 3 Tests for Population Means a assumptions needed for validity i data are from a SRS ii n is large iii population standard deviation 039 is known b state hypotheses i H0 1 go where go is some known value ii Ha y yo where is one oflt or gt depending on the particular problem x 0 64 c calculate test statistic the standard score of 7c 2 4 245 Final Review d Compare 2 to the critical value 2 and reject Ho when 2 lt z z gt z or z gt 2 depending on the alternative The value of z is chosen to make the probability of rejecting Ho or when H0 is true ALTERNATIVELY calculate Pvalue using Table A i Ha determines the appropriate area under the normal curve to calculate e Know the equivalence between two sided hypothesis tests and con dence intervals 4 Some Cautions on Tests of Significance a Pvalues are more informative than signi cance levels which have often been used inappropriately b lack of statistical significance can have practical signi cance and shouldn39t be ignored c beware of searching for signi cance a large set oftests will often produce one or more signi cant results by chance even though Ho really was true in every case d Recognize that signi cance testing does not measure the size or importance of an effect statistical signi cance vs practical significance Especially with very large sample sizes a result may be statistically signi cant without having much practical signi cance V th small sample sizes effects that are of practical significance may not be statistically signi cant e Recognize when you can use the 2 test and when the data collection design or a small sample from a skewed population makes it inappropriate Statistical signi cance means basically nothing if data was collected through a study with poor design Chapter 8 Con dence lntervals and Tests for Proportions 1 Know the assumptions for validity data are from a SRS n is large but not more than 5 ofthe population p is not too close to O or 1 Con dence intervals for Population Proportions p 1 Use the quotlarge samplequot method based on the sample proportion iv a level C con dence interval is i1lz M where 2 is obtained from Table D n This would be the default method provided that the conditions for its use are satis ed Know what those conditions are X2 2 Use the Plus Four estimate Z7 71 4 a level C con dence interval is iJiz pa if where zis obtained from Table D n Know the conditions for using this approach 3 nd the sample size needed to produce a speci ed margin of error and con dence level using the quotlarge samplequot method Know the conservative approach and the guessed p approach 5 245 Final Review Tests for Population Proportions 1 state hypotheses a H0 p p0 where p0 is some known value b Ha p p0 where is one oflt or gt depending on the particular problem 2 calculate test statistic the standard score of a z 3 calculate Pvalue using Table A a Ha determines the appropriate area under the normal curve to calculate b Know when this approach is justi ed Chapter 7 A OneSamplet Procedures 1 Recognize when the t procedures are appropriate in practice in particularthat they are moderately robust against lack of normality but are strongly in uenced by outliers 2 Also recognize when the design of the study outliers or a small sample from a skewed distribution make the t procedures risky 3 Use t to obtain a con dence interval at a stated level of con dence for the mean u ofa population when the population standard deviation 0 is unknown and the sample standard deviation s is used as an estimate of 039 Carry out a t test for the hypothesis that a population mean u has a specified value against either a onesided or a twosided alternative Use Table D oft distribution critical values to approximately give a range for the Pvalue or carry out a xed level a test 5 Know the equivalence between two sided hypothesis tests and con dence intervals 5 6 245 Final Review MIDTERM REVIEW CHAPTERS 1 and 2 and SEC 91 While there is no guarantee that everything on the midterm is covered here it should be a helpful guide in your review Chapter 1 A Data 1 Identify the individuals and variables in a set of data 2 Identify each variable as categorical or quantitative Identify the units in which each quantitative variable is measured B Displaying Distributions 1 Make a bar graph of the distribution of a categorical variable lnterpret bar graphs and pie charts 2 Make a histogram ofthe distribution ofa quantitative variable Identify whether a histogram is a frequency histogram or a relative frequency histogram Compute the percent ofobservations that fall in a given range 3 Make a stemplot ofthe distribution ofa small set ofobservations Round leaves or split stems as needed to make an effective stemplot 4 Know the characteristics of the different plots and in which situations each plot is preferred C Inspecting Distributions Quantitative Variable 1 Look for overall pattern and for major deviations from the pattern 2 Assess from a histogram or stemplot whether the shape ofa distribution is roughly symmetric distinctly skewed or neither Assess whether the distribution has one or more major peaks 3 Decide which measures of center and spread are more appropriate The mean and standard deviation especially for symmetric distributions or the ve number summary especially for skewed distributions 4 Recognize outliers D Measuring Center 1 Find the mean ofa set ofobservations 2 Find the median ofa set ofobservations 3 Understand that the median is more resistant less affected by extreme observations than the mean Recognize that skewness in a distribution moves the mean away from the median and more toward the long tail 4 Understand what happens to the mean when a data set undergoes transformations such as addition multiplication subtraction and division E Measuring Spread 1 Find the quartiles for a set of observations 2 Give the venumber summary and draw a boxplot assess center spread skewness and symmetry from a boxplot 1 245 Midterm Review 3 4 5 Given several lists of numbers assess which has the smallest or largest standard deviation and which lists have similar standard deviations Know the basic properties of s s is always greater than or equal to zero s is equal to zero if all the observations are identical s increases as the spread increases s has the same units as the original measurements s is pulled strongly up by outliers or skewness Understand what happens to the standard deviation when a data set undergoes transformations F Density Curves 1 2 3 Know that the areas under a density curve represent proportions of all observations and that the total area under a density curve is 1 A density curve always lies above the X axis Approximately locate the median equal areas point and the mean balance point on a density curve Know that the mean and median both lie at the center of a symmetric density curve and that the mean moves farthertoward the long tail of a skewed curve G Normal Distributions 1 Recognize the shape of normal curves and be able to estimate by A 01 0 eye both the mean and standard deviation from such a curve Use the 6895997 rule and symmetry to state what percent of the observations from a normal distribution fall between two points when both points lie at the mean or one two or three standard deviations on either side of the mean Shifting and rescaling a addsubtract a value tofrom each individual measurement i also addsubtract value tofrom center ii spread unchanged b multiplydivide each individual measurement by a constant i also multiplydivide center by the constant ii also multiplydivide spread by the absolute value of the constant Find the standardized value zscore of an observation lnterpret z scores and understand that any normal distribution becomes standard normal N01 when standardized Given that a variable has the normal distribution with a stated mean u and standard deviation 039 calculate the proportion of values above a stated number below a stated number or between two stated numbers Given that a variable has the normal distribution with a stated mean u and standard deviation 0 calculate the point having a stated proportion of all values above it or a stated proportion of all values below it Backwards normal calculations 2 245 Midterm Review w 0 U ITI Chapter 2 Data 1 Recognize whether each variable is quantitative or categorical 2 Identify the explanatory and response variables in situations where one variable explains or influences another Scatterplots 1 Make a scatterplot to display the relationship between two quantitative variables Place the explanatory variable ifany on the horizontal axis ofthe plot 2 Describe the form direction and strength ofthe overall pattern ofa scatterplot In particular recognize positive or negative association and linear straightline patterns Recognize outliers in a scatterplot Correlation 1 Understand the correlation measures the strength and direction of the linear relationship between two quantitative variables 2 Know the basic facts about correlation correlation is between 1 and 1 r 1 only for perfect straightline relations r moves away from O as the linear relation gets stronger r is unitless and changing the units ofyour observations will not change r the correlation between x and y is the same as the correlation between y and x and r is not resistant to outliers Straight Lines 1 Explain what the slope b and the intercept a mean in the equation ya bx ofa straight line 2 Draw a graph of the straight line when you are given its equation Regression 1 Find the slope and intercept of the leastsquares regression line from the means and standard deviations ofx and y and their correlation 2 Find the equation ofthe leastsquares regression line from looking at the computer output 3 Know that which variable is the response variable and which is the explanatory variable matters when we compute the regression line Changing units also changes the equation ofthe line 4 Use the regression line to predict y for a given x Recognize extrapolation and be aware of its dangers 5 Use r2 to describe how much of the variation in one variable can be accounted for by a straightline relationship with another variable 6 Recognize outliers and potentially in uential observations from a scatterplot with the regression line drawn on it 7 Calculate the residuals Recognize unusual patterns in a plot ofthe residuals against the x variable 3 245 Midterm Review F Limits of Correlation and Regression 1 Understand that both r2 and the leastsquares regression line can be strongly in uenced by a few extreme observations 2 Recognize that correlations based on averages of several observations are usually stronger than the correlation for individual observations 3 Recognize possible lurking variables that may explain the observed association between two variables X and y 4 Understand that even a strong correlation does not mean that there is a causeandeffect relationship between X and y G Causation 1 best established via carefully designed randomized comparative experiments see Ch 3 2 in absence of experimentation look for a strong association b consistent association c higher doses associated with stronger responses d alleged cause precedes effect in time e alleged cause is plausible 3 Common Response 4 Confounding Section 91 Categorical Data 1 From a twoway table of counts nd the marginal distributions of both variables by obtaining the row sums and the column sums if they are not given and then dividing by the overall total 2 Express any distribution in percents by dividing the category counts by their total 3 Describe the relationship between two categorical variables by computing and comparing percents Often this involves comparing the conditional distributions ofone variable for the different categories of the other variable 4 Recognize Simpson s paradox and be able to explain it 4 245 Midterm Review

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