# Examine the degree of relationship between variables

discuss the meaning and interpretation of correlation coefficients, and relate this measure to .. degree of association between two sets of measures. The most. Types of research questions a Pearson correlation can examine: Is there a relationship between temperature, measured in degrees Fahrenheit, and ice cream. In this section we will first discuss correlation analysis, which is used to 0) between the extent of media exposure in adolescence and age at.

People use regression on an intuitive level every day. In business, a well-dressed man is thought to be financially successful. A mother knows that more sugar in her children's diet results in higher energy levels.

### Introduction to Correlation and Regression Analysis

The ease of waking up in the morning often depends on how late you went to bed the night before. Quantitative regression adds precision by developing a mathematical formula that can be used for predictive purposes. For example, a medical researcher might want to use body weight independent variable to predict the most appropriate dose for a new drug dependent variable.

**R tutorial for 2-2 Examining Relationships Between Two Categorical Variables**

The purpose of running the regression is to find a formula that fits the relationship between the two variables. Then you can use that formula to predict values for the dependent variable when only the independent variable is known. A doctor could prescribe the proper dose based on a person's body weight.

## Relationship Between Variables

The regression line known as the least squares line is a plot of the expected value of the dependent variable for all values of the independent variable. Technically, it is the line that "minimizes the squared residuals". 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. 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.

### The relationship between variables - Draw the correct conclusions

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.

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.

## Introduction to Correlation and Regression Analysis

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.

The covariance of gestational age and birth weight is: We now compute the sample correlation coefficient: Not surprisingly, the sample correlation coefficient indicates a strong positive correlation.

In practice, meaningful correlations i.