Coefficient Of Determination Values Of Variables example essay topic

Relationships between variables are often examined in research. When two variables are related such that a change in one is associated with a congruent change in the other, it is called correlation. The direction and degree of correlation between two interval or ratio scale variables (X and Y) are graphically represented on a scatter diagram (1.1) by plotting the corresponding values (1.0). The coefficient provides important information about the relationship itself. For example, the sign of the coefficient indicates the direction of the relationship and the value of the coefficient indicates the degree of the relationship. When the sign is positive (1.1), it indicates that both variables vary in the same direction.

On the other hand, when the sign is negative, it indicates that the variables vary in opposite directions. If, however, the coefficient is zero, it indicates no relationship. When the correlation coefficient is squared, it is called the coefficient of determination (1.3) and referred to as the effect size because it indicates the amount of variance of one variable that can be explained by the variance of a related variable. Determining whether or not the correlation between two variables is statistically significant requires a comparison of the absolute value of the correlation coefficient to a critical value of r based on the degrees of freedom.

When r is greater than the critical value, it is said to be statistically significant. It is important to know that statistical significance is based on a certain probability, usually. 05, and that the outcome is merely due to chance and should not be confused with causality. Because correlation studies are not experimental designs, the causal explanation for this relationship cannot be determined.

Thus, correlation does not mean causation. (1.3) coefficient of determination = Values of variables can be predicted if the two variables are correlated. One of the simplest ways to make these predictions is by using linear regression (1.4). The term regression is used by statisticians to indicate a backward shift toward the mean when predicting an unknown value from a known value when the variables are correlated. The prediction involves a series of computations using the means and the standard deviations of the variables and the correlation coefficient.

Because a linear relationship is required, we use the equation for a straight line, called the regression equation. There are two equations: one to predict Y from X and one to predict X from Y. In both of these equations, the values of a and b are called regression coefficients. Since it is almost impossible for two variables to be perfectly correlated, there is always error in predictions. The amount of the error is reflected in the value of the standard error of the estimate (1.5). This is the standard deviation of the actual values of a variable from the predicted values.

In addition to the predicted value of a variable, we can use the logic of the normal curve to find the proportion of scores (n) that fall within a certain distance of the regression line. This can be very useful information. However, this type of analysis is only valid if the assumption of equal standard deviations, known as homoscedasticity, is met. Thus, linear prediction can be used not only to predict an unknown value from a known value, but also to find the proportion of scores that will fall within certain distances (standard deviations) from the predicted value.

(1.4) (1.5) Probability is a measure of how likely it is that a given event or behavior will happen and is measured in terms of numbers between 0 and 1. Probability is computed several different ways. The addition rule of probability is used to compute the probability of one event or another event occurring. Compute this by adding the probability of A and the probability of B. To compute the probability of two events occurring together, the multiplication rule of probability is used and the probability of A is multiplied by the probability of B. However, in order to use either of these rules, the events must be mutually exclusive, meaning they cannot occur simultaneously.

Another new area of discussion in this chapter is sampling. The manner in which participants are selected for a study is very important. Since a sample is used to represent a population, researchers should make every attempt to ensure all the significant subgroups of the population are represented. One way to increase the chances of obtaining a representative sample is by using a sampling technique called random sampling.

This technique assumes that by using a completely random process, such as a random numbers table, everyone in the population is equally likely to be selected to participate in the study. Unfortunately, there are many times and circumstances when this is not possible. However, researchers should do their best to see that the recruited sample is as representative as possible. Inferring population values from sample data is called statistical inference and it is critically important to statistics.

The basis for statistical inference comes from the central theorem, which states that as the sample size increases, the distributions steadily grow and approach a normal distribution. Thus, when the sample size is large, it is assumed that the distribution is normal. The central limit theorem is also an important element in a theoretical distribution, called the distribution of sample means. This distribution is made up of an unlimited number of sample means (all same size n).

The mean of the distribution of sample means is equal to the population mean. Like other distributions, the distribution of sample means also has a standard deviation. The standard deviation of the sampling distribution of the means is called the standard error of the mean. When this value is small, it indicates there is less error estimating the true population mean. Therefore, statistical inference is made possible by evidence from the central limit theorem and the distribution of sample means. Thus, the mean of a large sample is considered the same or inferred to be the same value as the population mean.

The test used to compare a particular sample mean to the population mean is called the z test. This test is used to determine whether a sample belongs or differs from a population. It is computed by subtracting the mean of the distribution of means from the sample mean and dividing by the standard error of the mean. When the means do not statistically differ from each other, it is because the sample was drawn from the population with that mean value.

On the other hand, when the means statistically differ, it indicates that the sample came from a different population with a mean different from the one to which is compared. Experimental studies begin with a question about behavior stated in the form of a specific prediction about the behavior being studied. That question is a hypothesis. There are two mutually exclusive hypotheses in a research study. One is the null hypothesis that predicts there is no difference between sample means. The second hypothesis, referred to as the research hypothesis, predicts that there is a difference between sample means.

These hypotheses are stated in terms of the effect of the independent variable on the dependent variable. If the direction of the experimental is specified, it is a one-tailed test. The researcher then selects a research design with which to conduct the study, as well as the manner in which participants are selected for the study and assigned to experimental condition. When participants are randomly selected from the population and randomly assigned to condition, and there is a single independent variable, the design is a completely randomized experimental design and a completely randomized factorial experimental design if there are two or more independent variables. Sometimes, researchers carry out a one-group experimental design and collect data from only one sample and compare this mean with the population mean. However, this design does not involve the manipulation of an independent variable nor random assignment and causality cannot be determined.

Thus, one of the other designs must be used to determine the cause of change in the dependent variable. Once the design is selected, the experimenter must determine how participants will be assigned to the experimental conditions. If participants are in only one level of the independent variable, the design is a between-subjects design. If the researcher wants the same participant in all the levels of the independent variables, the design is a within-subjects design. Sometimes, the researcher may decide it is best to assign participants to one level of one independent variable and all levels of another. This design is a mixed design.

Based on this decision, the researcher will know how many participants to recruit for the experiment. Along with selection and assignment decisions, the experimenter must consider issues related to experimental control and ways to keep extraneous variables from varying along with the independent variable. If extraneous variables are not controlled, the experimenter will not know if the extraneous variable or the independent variable caused the observed effect on the dependent variable. Thus, removing alternative explanations is critical since the purpose of a true experiment is to isolate and determine the cause of changes or differences in behavior. The experimenter, knowing the hypothesis, must limit his / her exposure and direct involvement in the study as well. Experimenter bias occurs when the experimenter knows the hypothesis and may, unknowingly, differ in facial expression, tone of voice, etc. when speaking with participants in the experimental conditions, thus influencing the results in favor of the hypothesis.

Even the behavior of participants can change if they think they know the hypothesis. This is called demand characteristics because participants may alter their behavior to support or not support what they believe is the hypothesis. When this occurs, the results of the study are meaningless. Besides design, assigning participants to condition, and issues of control, the experimenter considers the number of participants he / she needs in the study to maximize statistical power.

If power is low, true differences may not be detected. Thus, the researcher may commit a Type II error and fail to reject the null hypothesis when true differences really exist. If, however, the statistical decision is to reject the null when, in reality, there are no differences, a Type I error is made. As you can see, researchers must consider many factors when they design and conduct experimental research.