# The table below shows a linear relationship between x and y theory

### Graphing With Excel - Linear Regression

Let's pick two points off of the table: (1, 0) and (0, -6). slope of f(x) = The y- intercept is the value of the function when x = 0. On the table, there. In regression analysis, the dependent variable is denoted "y" and the A correlation close to zero suggests no linear association between two where Cov(x,y) is the covariance of x and y defined as The variance of birth weight is computed just as we did for gestational age as shown in the table below. y x y. 4) The data below are the number of absences and the final grades of 9 randomly selected students from a The data below show the relationship between the number of years that . 2 Describe the properties of the linear correlation coefficient. .. 52) The table below shows the scores on an end-of- year project of

The figure below shows four hypothetical scenarios in which one continuous variable is plotted along the X-axis and the other along the Y-axis. Scenario 3 might depict the lack of association r approximately 0 between the extent of media exposure in adolescence and age at which adolescents initiate sexual activity. Example - Correlation of Gestational Age and Birth Weight A small study is conducted involving 17 infants to investigate the association between gestational age at birth, measured in weeks, and birth weight, measured in grams.

We wish to estimate the association between gestational age and infant birth weight. In this example, birth weight is the dependent variable and gestational age is the independent variable.

## Graphing with Excel

The data are displayed in a scatter diagram in the figure below. Each point represents an x,y pair in this case the gestational age, measured in weeks, and the birth weight, measured in grams. Note that the independent variable is on the horizontal axis or X-axisand the dependent variable is on the vertical axis or Y-axis.

The scatter plot shows a positive or direct association between gestational age and birth weight. Infants with shorter gestational ages are more likely to be born with lower weights and infants with longer gestational ages are more likely to be born with higher weights. The formula for the sample correlation coefficient is where Cov x,y is the covariance of x and y defined as are the sample variances of x and y, defined as The variances of x and y measure the variability of the x scores and y scores around their respective sample meansconsidered separately.

The covariance measures the variability of the x,y pairs around the mean of x and mean of y, considered simultaneously.

### Introduction to Correlation and Regression Analysis

To compute the sample correlation coefficient, we need to compute the variance of gestational age, the variance of birth weight and also the covariance of gestational age and birth weight. We first summarize the gestational age data. The mean gestational age is: To compute the variance of gestational age, we need to sum the squared deviations or differences between each observed gestational age and the mean gestational age.

The computations are summarized below. The variance of gestational age is: Next, we summarize the birth weight data. The mean birth weight is: The variance of birth weight is computed just as we did for gestational age as shown in the table below.

The variance of birth weight is: Next we compute the covariance, To compute the covariance of gestational age and birth weight, we need to multiply the deviation from the mean gestational age by the deviation from the mean birth weight for each participant i.

Notice that we simply copy the deviations from the mean gestational age and birth weight from the two tables above into the table below and multiply.

**Production Possibilities Curve- Econ 1.1**

The covariance of gestational age and birth weight is: Once we draw the scatterplot, we can have a good idea of the relationship between the two variables. We often materialize this relationship with a trendline.

The easiest interpretation of the trendline would be to say that there is a positive, negative, or no relationship between the two variables.

- Introduction to Correlation and Regression Analysis
- Statistics review 7: Correlation and regression

That is what the correlation coefficient does. The correlation coefficient is usually written as r. Positive Negative When looking at a scatterplot, the overall pattern includes: The trend can be linear, curved, or no pattern at all.

The direction can be positive, negative or no direction at all. By strength, we refer to how closely the points fit the trend. The figure below shows a few scatterplots and the correlation values: It makes no difference which variable you call x and which you call y in calculating the correlation.

Correlation requires that both variables be quantitative, so that it makes sense to do the arithmetic indicated by the formula for r. We cannot calculate a correlation between the incomes of a group of people and what city they live in, because city is a categorical variable a boxplot of income by city would be a much better tool to use in that case.

## Lesson 12: Correlation & Simple Linear Regression

Because r uses the standardized values of the observations, r does not change when we change the units of measurement of x, y, or both.

Measuring height in inches rather than centimeters and weight in pounds rather than kilograms does not change the correlation between height and weight.

The correlation r itself has no unit of measurement; it is just a number. Positive r indicates positive association between the variables, and negative r indicates negative association. Values of r near 0 indicate a very weak linear relationship. Correlation measures the strength of only the linear relationship between two variables.