Lecture 22 Dustin Lueker

 The sample mean of the difference scores is an estimator for the difference between the population means  We can now use exactly the same methods as for one sample ◦ Replace X i by D i 2STA 291 Summer 2010 Lecture 22

 Small sample confidence interval Note: ◦ When n is large (greater than 30), we can use the z- scores instead of the t-scores 3STA 291 Summer 2010 Lecture 22

 Small sample test statistic for testing difference in the population means ◦ For small n, use the t-distribution with df=n-1 ◦ For large n, use the normal distribution instead (z value) 4STA 291 Summer 2010 Lecture 22

 Variability in the difference scores may be less than the variability in the original scores ◦ This happens when the scores in the two samples are strongly associated ◦ Subjects who score high before the intensive training also tend to score high after the intensive training  Thus these high scores aren’t raising the variability for each individual sample 5STA 291 Summer 2010 Lecture 22

 If we wanted to examine the improvement students made after taking a class we would hope to see what type of value for ? Assuming we take X 1 -X 2 with X 1 being the student’s first exam score. 1.Positive 2.Negative STA 291 Summer 2010 Lecture 226

 Assuming we match people of similar health into 2 groups and gave group 1 a cholesterol medication and measured each groups cholesterol level after 8 weeks, what would we hope would be if we are subtracting group 2 from group 1? 1.Positive 2.Negative 3.Zero STA 291 Summer 2010 Lecture 227

 Regression ◦ The process of using sample information about explanatory variables (independent variables) to predict the value of a response variable (dependent variable)  Many types of regression ◦ One response variable to many response variables ◦ Linear, quadratic, cubic, logistic, exponential, etc. 8STA 291 Summer 2010 Lecture 22

 Uses one explanatory variable to predict a response variable ◦ Only type of regression we will look at in here  Model  y = Dependent (response) variable  x = Independent (explanatory) variable  β 0 =y-intercept  β 1 =Slope of the line (defined as rise/run)  ε=Error variable 9STA 291 Summer 2010 Lecture 22

 Model we will use in problems  y-hat = Dependent variable  x = Independent variable  b 0 =y-intercept  b 1 =Slope of the line (defined as rise/run)  Example:  Estimating college GPA by ACT score  College GPA would be our dependent (response) variable  ACT score would be our independent (explanatory) variable 10STA 291 Summer 2010 Lecture 22

 Notice that the equation is for y-hat which is an estimator of y ◦ When using a regression model it is important to remember that it will not exactly predict y, but rather give an estimate of what we would expect y to be  This is the reason we don’t have to have the error (ε) in the model we use, because error is accepted since we are simply what we would expect the value of y to be given x, basically estimating y 11STA 291 Summer 2010 Lecture 22

 Correlation Coefficient ◦ R = (-1,1)  Sometimes referred to as a lower case “r”  How strong the linear relationship is between the response and explanatory variable as well as the direction  ± indicates a positive relationship or a negative relationship  positive means our estimate of y goes up as x goes up  negative means our estimate of y goes down as x goes up  The closer the |R| is to one, the stronger the relationship is between the response and explanatory variables  R=0 indicates no relationship 12STA 291 Summer 2010 Lecture 22

 Coefficient of Determination ◦ Denoted by R 2  Calculated by squaring the correlation coefficient  Interpretation  The percent of variation in the response variable that is explained by the model  Simple Linear Regression  The percent of variation in y that is explained by x  This is because our model only has one variable ◦ The higher the R 2 value the better because we can explain more of the variation in our response variable, which is the one we are wanting to examine 13STA 291 Summer 2010 Lecture 22

 If the correlation coefficient is -.7, what would be the coefficient of determination?  Would larger values for the explanatory variable (x) yield larger or smaller values for the response variable (y)? STA 291 Summer 2010 Lecture 2214

 If model A has a correlation coefficient of.7, what would the correlation coefficient of model B need to be for us to be able to say that B is the better model? STA 291 Summer 2010 Lecture 2215

 If the slope of our simple linear regression equation is 13 and the y-intercept is -2, what would y-hat be if x=3?  What would y be? STA 291 Summer 2010 Lecture 2216