Midterm Review Ch 7-8

Requests for Help by Chapter

Chapter 7 zDescribe the characteristics of the relationship between two variables. zDiscuss the null and research hypotheses for correlation. zDiscuss using the r table for determining significance.

Chapter 8 zDiscuss the steps in making raw data predictions from raw data values. zDiscuss the situations when you cannot use regression. zDiscuss the inappropriateness of predicting outside of the sample range. zDiscuss the null hypothesis in regression. zDiscuss alpha levels and critical values with respect to statistical significance. zDiscuss residual error when predicting from regression.

Describe the characteristics of the relationship between two variables.

Describe the characteristics of the relationship between two variables zThree dimensions characterize the relationship between two variables, ylinearity, ydirection, yand strength.

Linearity zThe relationship is either linear or some other curvilinear relationship. zIn a linear relationship, as scores on one variable increase, scores on the other variable either generally increase or generally decrease. zIn a curvilinear relationship, as scores on one variable increase, scores on the other variable move first in one direction, then in another direction.

Direction zThe direction of a relationship is either positive or negative. zIn a positive relationship, as scores on one variable increase, scores on the other variable increase. So a best fitting line rises from left to right on a graph and therefore has a positive slope. zIn a negative relationship, as scores on one variable increase, scores on the other variable decrease. So a best fitting line falls from left to right on a graph and therefore has a negative slope.

Strength zThe strength of a correlation indicates how predictable one variable is from another. zIn a strong relationship, t X and t Y scores are consistently similar or dissimilar. So, you are able to accurately predict one score from another. zIn a weak relationship, t X and t Y scores are in consistent in similarity or dissimilarity. So, you are only able to somewhat predict one score from another. zIn an independent relationship, there is no consistency in the relationship of the tX and tY scores So, it is impossible to predict one score from another.

Discuss alpha levels and critical values with respect to statistical significance. Discuss the null and research hypotheses for correlation. Discuss the null hypothesis in regression. Discuss using the r table for determining significance.

Alpha levels and significance zScientists are a careful bunch. They are very careful to not make a Type 1 error. zA Type 1 error is when you mistakenly say that you have found a relationship in a population, when the one you found in your random sample doesn’t exist in the population as a whole zTo be careful, scientists will only say there is a relationship, if the probability of a Type 1 error is very low (5 in 100 or lower.)

... Alpha levels and significance zThese probabilities are called alpha levels. zThe typical alpha levels are p .05 or p .01. zThese represent 5 out of 100 or 1 out of 100. zA sample r that is far enough from to occur 5 or fewer times in 100 when rho actually equals zero is called significant.

The Null Hypothesis zThe null hypothesis (H 0 ) states that a non-zero correlation in a sample between two variables is the result of random sampling fluctuation. zTherefore, there is no underlying relationship in the population as a whole. zIn mathematical terms, rho =

The Alternative Hypothesis zThe alternative hypothesis is the opposite of the null hypothesis. zIt states that there is an underlying relationship in the population as a whole and that it is reflected by a non-zero correlation in your random sample.

Rejecting the Null Hypothesis zThe purpose of research is to reject the null hypothesis. zWe reject the null hypothesis, when the correlation is significant. zThe correlation is significant, when the probability that the result is due to an error is less than the.05 or.01 alpha level.

Using the r Table to Determine Significance zFirst, calculate r. zThen, determine the degrees of freedom, (n p -2). zLook in the r table to see if r falls outside the CI.95 in Column 2 of the r table. If r does, it is significant.

to to to to to to to to to to to to to to to to to to to df nonsignificant Look in this column for the row that has your degrees of freedom. Does r fall here? Or here? Non-significant Significant

Significant Correlation zIf r is non-significant, we continue to accept the null hypothesis and say that rho = in the population. zIf r is significant, we reject the null hypothesis at the.05 or.01 alpha level. We assume that rho is best estimated by r, the correlation found in the random sample.

Discuss the situations when you cannot use regression. Discuss the inappropriateness of predicting outside of the sample range.

Use Regression Carefully zWhen we have the entire population and compute rho, yWe know all of the values of X and Y. yWe know the direction and strength of the relationship between X and Y variables. zTherefore, we can safely use the regression equation to predict Y from X. zEven when rho is exactly zero, the regression equation is still right. It tells us to predict that everyone will score at the mean of Y.

Samples and Regression zWhen we have a random sample from a population, we can only predict when yr is significant, otherwise we assume that rho is 0, ythe relationship between the variables is linear, otherwise it is inappropriate to use correlation at all, ythe X score is within the range of X scores in the sample, because for values outside of the range, you do not know if the linear relationship holds.

Discuss the steps in making raw data predictions from raw data values.

Describe the steps in making predictions from raw data. Scientists are interested in taking the score for one variable and then predicting the score for another variable. If you want to predict, you must first ensure that there is a linear relationship between the two variables. Then, you must calculate the correlation coefficient and check that it is significant. You also must check that the score you are predicting from is within the original range of scores. If these conditions are met, then you can use the regression equation to predict. You first convert the predicting score to a t score. Then you plug the t score and the correlation value into the regression equation. You solve the regression equation for the predicted t score. Finally, you convert the predicted t score into the predicted score.

Discuss residual error when predicting from regression.

zThe average squared error when we predict from the mean is the variance, also called the mean square error. zThe average squared error when we predict from the regression equation is called the residual mean square.

Residual Square Error zA significant correlation will always yield a better prediction than the mean. zTherefore, the residual mean square is always better, that is, smaller than the variance.

Steps in calculating Residual Square Error zTo calculate the variance take the deviations of Y from the mean of Y, square them, add them up, and divide by degrees of freedom. zTo calculate the residual mean square take the deviations of each Y from its predicted value, square them, add them up, and divide by degrees of freedom.

Short way to do that zr 2 equals the proportion of error that is gotten rid of when you use the regression equation rather than the mean as your prediction. zSo the amount of error you get rid of equals the original sum of squares for Y times r 2. zSo the remaining error, SS RESID, equals the amount of error you get by using the mean as your predictor (SS Y ) minus the amount you get rid of by using the regression equation, r 2 SS Y

SS RESID =SS Y –r 2 SS Y To get average squared error when using the regression equation, divide by df REG (MS RESID =SS RESID /(n p -2) The standard error of the estimate is simply the square root of MS RESID