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By O. Julio. Keiser University.

The statistical procedure for making such predictions is called linear regression purchase hydrea 500mg otc. In the following sections 500 mg hydrea mastercard, we’ll examine the logic behind regression and see how to use it to predict scores generic hydrea 500 mg. These involve the same formulas we used previ- ously buy 500 mg hydrea amex, except now we plug in Y scores hydrea 500 mg on line. This translates into predicting when someone has one score on a variable and when they have a different score. It’s important that you know about linear regression because it is the statistical procedure for using a relationship to predict scores. Linear regression is commonly used in basic and applied research, particularly in educational, industrial and clinical settings. This approach is also used when people take a test when applying for a job so that the employer can predict who will be better workers, or when clinical patients are tested to identify those at risk of developing emotional problems. While r is the statistic that summarizes the linear relationship, the regression line is the line on the scatterplot that summarizes the relationship. If the correlation coefficient is not 0 and passes the inferential test, then perform linear regression to further summa- rize the relationship. An easy way to understand a regression line is to compare it to a line graph of an experiment. In Chapter 4, we created a line graph by plotting the mean of the Y scores for each condition—each X—and then connecting adjacent data points with straight lines. Because the mean is the central score, we assume that those participants at X3 scored around a Y of 3, so (1) 3 is our best single description of their scores, and (2) 3 is our best prediction for anyone else at that X. It is difficult, however, to see the linear (straight-line) relationship in these data because the means do not fall on a straight line. Think of the regression line as a straightened-out version of the line graph: It is drawn so that it comes as close as possible to connecting the mean of Y at each X while still producing a straight line. Although not all means are on the line, the distance that some means are above the line averages out with the distance that other means are below the line. Thus, the regression line is called the best-fitting line because “on average” it passes through the center of the various Y means. Because each Y mean is located in the center of the cor- responding Y scores, the regression line also passes through the center of the Y scores. Thus, the linear regression line is the straight line that summarizes the linear relation- ship in a scatterplot by, on average, passing through the center of the Y scores at each X. Think of the regression line as reflecting the linear relationship hidden in the data. Because the actual Y scores fall above and below the line, the data only more or less fit this line. Therefore, the regression line is how we envision what a perfect version of the linear relationship in the data would look like. You should read the regression line in the same way that you read any graph: Travel vertically from an X until you intercept the regression line. A Y¿ is a summary of the Y scores for that X, based on the entire linear relationship. Likewise, any Y¿ is our best prediction of the Y scores at a corresponding X, based on the linear relationship that is summarized by the regression line. The regression line therefore consists of the data points formed by pairing each possible value of X with its corresponding value of Y¿. If you think of the line as reflecting a per- fect version of the linear relationship hidden in the data, then each Y¿ is the Y score everyone would have at a particular X if a perfect relationship were present. Therefore, we can measure the X scores of individuals who were not in our sample, and the corresponding Y¿ is our best prediction of their Y scores. The emphasis on prediction in correlation and regression leads to two important terms. The linear regression equation is the equation for a straight line that produces the value of Y¿ at each X and thus defines the straight line that summarizes a relationship. When we plot the data points formed by the X–Y¿ pairs and draw a line connecting them, we have the regression line. The regression equation describes two characteristics of the regression line: its slope and its Y intercept. The slope is a number that indicates how slanted the regression line is and the direc- tion in which it slants. When no relationship is present, the regression line is horizontal, such as line A, and the slope is zero.  Finally generic hydrea 500 mg on-line, think of s2 and s as the inferential variance and the inferential standard de- X X viation discount hydrea 500 mg on line, because the only time you use them is to infer the variance or standard devia- tion of the population based on a sample purchase hydrea 500mg overnight delivery. Think of S2 and S as the descriptive variance X X and standard deviation because they are used to describe the sample order hydrea 500mg on-line. The degrees of freedom is the number of scores in a sample that are free to reflect the variability in the population buy hydrea 500mg lowest price. Because N – 1 is a smaller number than N, dividing by N – 1 produces a slightly larger answer. Over the long run, this larger answer will prove to be a more accurate estimate of the population variability. Computing the Estimated Population Variance and Standard Deviation The only difference between the computational formula for the estimated population variance and the previous computational formula for the sample variance is that here the final division is by N 2 1. In previous examples, our age scores of 3, 5, 2, 4, 6, 7, and 8 produced ΣX2 5 203, and ΣX 5 35. Putting these quantities into the above formula gives 1ΣX22 13522 ΣX2 – 203 – 2 N 7 sX 5 5 N – 1 6 Work through this formula the same way you did for the sample variance: 352 is 1225, and 1225 divided by 7 equals 175, so 2 203 – 175 sX 5 6 Now 203 minus 175 equals 28, so 2 28 sX 5 6 and the final answer is S2 5 4. Although 4 accurately describes the sample, we estimate the variance of X the population is 4. In other words, if we could compute the true population vari- ance, we would expect σ2 to be 4. There- fore, the formula for the estimated population standard deviation involves merely adding the square root sign to the previous formula for the variance. The Population Variance and the Population Standard Deviation 99 The computational formula for the estimated population standard deviation is 1©X22 ©X2 – N sX 5 R N – 1 For our age scores, the estimated population variance was s2 5 4. Thus, if we could compute the standard deviation using the en- tire population of scores, we would expect σX to be 2. Interpreting the Estimated Population Variance and Standard Deviation Interpret the estimated population variance and standard deviation in the same way as S2 and S , except that now they describe how much we expect the scores to be spread X X out in the population, how consistent or inconsistent we expect the scores to be, and how accurately we expect the population to be summarized by. Notice that, assuming a sample is representative, we have pretty much reached our ultimate goal of describing the population of scores. If we can assume that the distribu- tion is normal, we have described its overall shape. So, for example, based on a statistics class with a mean of 80, we’d infer that the population would score at a µ of 80. The size of s (or s2) estimates how spread out the population is, so if s turned out to be 6, we’d X X X expect that the “average amount” the individual scores deviate from the of 80 is about 6. Further, we’d expect about 34% of the scores to fall between 74 and 80 (be- tween and the score at 21sX) and about 34% of the scores to fall between 80 and 86 (between and the score at 11sX) for a total of 68% of the scores between 74 and 86. With this picture in mind, and because scores reflect behaviors, we have a good idea of how most individuals in the population behave in this situation (which is why we con- duct research the first place). Compute the estimated population variance and 13522 standard deviation for the scores 1, 2, 2, 3, 4, 4, 255 – and 5. In every case, we are finding the difference between each score and the mean and then cal- culating an answer that is somewhat like the “average deviation. We compute the de- scriptive versions when the scores are available: When describing the sample, we cal- culate S2 and S. When the X X X X population of scores is unavailable, we infer the variability of the population based on a sample by computing the unbiased estimators, s2 and. With these basics in hand, you are now ready to apply the variance and standard de- viation to research. Thus, the mean from a study might describe the number of times that partic- ipants exhibited a particular behavior, but a small standard deviation indicates that they consistently did so. Or, in a survey, the mean might describe the typical opinion held by participants, but a large standard deviation indicates substantial disagreement among them. We also compute the mean and standard deviation in each condition of an experi- ment. For example, in Chapter 4 we tested the influence of recalling a 5- 10- or 15- item list.  Likewise proven hydrea 500 mg, any Y¿ is our best prediction of the Y scores at a corresponding X 500 mg hydrea, based on the linear relationship that is summarized by the regression line hydrea 500mg on-line. The regression line therefore consists of the data points formed by pairing each possible value of X with its corresponding value of Y¿ buy generic hydrea 500mg line. If you think of the line as reflecting a per- fect version of the linear relationship hidden in the data discount hydrea 500 mg mastercard, then each Y¿ is the Y score everyone would have at a particular X if a perfect relationship were present. Therefore, we can measure the X scores of individuals who were not in our sample, and the corresponding Y¿ is our best prediction of their Y scores. The emphasis on prediction in correlation and regression leads to two important terms. The linear regression equation is the equation for a straight line that produces the value of Y¿ at each X and thus defines the straight line that summarizes a relationship. When we plot the data points formed by the X–Y¿ pairs and draw a line connecting them, we have the regression line. The regression equation describes two characteristics of the regression line: its slope and its Y intercept. The slope is a number that indicates how slanted the regression line is and the direc- tion in which it slants. When no relationship is present, the regression line is horizontal, such as line A, and the slope is zero. A positive linear relationship produces regression lines such as B and C; each of these has a slope that is a positive number. A negative linear relationship, such as line D, yields a slope that is a negative number. The Y intercept is the value of Y at the point where the regression line intercepts, or crosses, the Y axis. If we extended line C, it would intercept the Y axis at a point below the X axis, so its Y intercept is a negative Y score. Because line D reflects a negative relationship, its Y intercept is the relatively high Y score of 9. When there is no relationship, the regression line is flat and every Y¿ equals the Y intercept. The regression equation works like this: The slope indicates the direction in which the Ys change as X increases and the rate at which they change. The Y intercept indicates the starting point from which the Y scores begin to change. Thus, together, the slope and intercept describe how, starting at a particular Y score, the Y scores tend to change by a specific amount as the X scores increase. As an example, say that we have developed a test to identify (predict) those indi- viduals who will be good or bad workers at a factory that makes “widgets. The predictor (X) variable is participants’ scores on the widget test, and the criterion (Y) variable is the number of widgets they produced. This is a very strong, positive linear relationship, and so the test will be what researchers call “a good predictor” of widget-making. The numerator of the formula for b is the same as the numerator in the formula for r, and the denominator of the formula for b is the left-hand quantity in the denominator of the formula for r. This positive slope indicates a positive relationship, which fits with the positive r of 1. Had the rela- tionship been negative, the formula would have produced a negative number here. Computing the Y Intercept The formula for the Y intercept of the linear regression line is a 5 Y 2 1b21X2 First, multiply the mean of all X scores times the slope of the regression line. Describing the Linear Regression Equation Once you have computed the Y intercept and the slope, rewrite the regression equation, substituting the computed values for a and b. Plotting the Regression Line We use the finished regression equation to plot our linear regression line. To draw a line, we need at least two data points, so choose a low and high X score, insert each into the regression equation, and compute the Y¿ for that X. Therefore, we also use the finished regression equation to predict anyone’s Y score if we know their X score. In fact, computing any Y¿ using the equation is the equivalent of going to the graph and traveling vertically from the X score up to the regression line and then left to the value of Y¿ on the Y axis. We can compute Y¿ for any value of X that falls within the range of Xs in our data, even if it’s a score not found in the original sample: No one scored an X of 1. Our regression equa- tion is based only on widget test scores between 1 and 4, so we shouldn’t predict a Y for an X of, for example, 6.

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