# Linear Regression and Correlation

## Linear Equations

Linear regression for two variables is based on a linear equation with one independent variable. The equation has the form
where a and b are constant numbers.
The variable x is the independent variable; y is the dependent variable. Typically, you choose a value to substitute for the independent variable and then solve for the dependent variable.
The graph of a linear equation of the form y = a + bx is a straight line. Any line that is not vertical can be described by this equation.

### Slope and y-interceptof a Linear Equation

For the linear equation y = a + bx, b = slope and a = y-inttercept. From algebra, recall that the slope is a number that describes the steepness of a line; the y-intercept is the y-coordinate of the point (0, a), where the line crosses the y-axis.
Please note that in previous courses you learned 
 
   was the slope-intercept form of the equation, where m represented the slope and b represented the y-intercept. In this text, the form 
 
   is used, where a is the y-intercept and b is the slope. The key is remembering the coefficient of x is the slope, and the constant number is the y-intercept.

### References

Centers for Disease Control and Prevention. (n.d.). Retrieved from https://www.cdc.gov/
National Center for HIV/AIDS, Viral Hepatitis, STD, and TB Prevention. (n.d.). Centers for Disease Control and Prevention. Retrieved from https://www.cdc.gov/nchhstp/default.htm

### Chapter Review

The most basic type of association is a linear association. This type of relationship can be defined algebraically by the equations used (numerically with actual or predicted data values) or graphically from a plotted curve. Lines are classified as straight curves. Algebraically, a linear equation typically takes the form y = mx + b, where m and b are constants, x is the independent variable, and y is the dependent variable. In a statistical context, a linear equation is written in the form y = a + bx, where a and b are the constants. This form is used to help you distinguish the statistical context from the algebraic context. In the equation y = a + bx, the constant b that multiplies the x variable (b is called a coefficient) is called the slope. The slope describes the rate of change between the independent and dependent variables; in other words, the rate of change describes the change that occurs in the dependent variable as the independent variable is changed. In the equation y = a + bx, the constant a is called the y-intercept. Graphically, the y-intercept is the y-coordinate of the point where the graph of the line crosses the y-axis.  At this point, x = 0.
The slope of a line is a value that describes the rate of change between the independent and dependent variables. The slope tells us how the dependent variable (y) changes for every one-unit increase in the independent (x) variable, on average. The y-intercept is used to describe the dependent variable when the independent variable equals zero. Graphically, the slope is represented by three line types in elementary statistics.

### Formula Review

y = a + bx, where a is the y-intercept and b is the slope. The variable x is the independent variable and y is the dependent variable.
Use the following information to answer the next three exercises. A vacation resort rents scuba equipment to certified divers. The resort charges an up-front fee of $25 and another fee of $12.50 an hour.
Use the following information to answer the next two exercises. A credit card company charges $10 when a payment is late and $5 a day each day the payment remains unpaid.
Use the following information to answer the next exercise.  contains real data for the first two decades of flu reporting.
Use the following information to answer the next two exercises. A specialty cleaning company charges an equipment fee and an hourly labor fee. A linear equation that expresses the total amount of the fee the company charges for each session is y = 50 + 100x.
Use the following information to answer the next three questions. As a result of erosion, a river shoreline is losing several thousand pounds of soil each year. A linear equation that expresses the total amount of soil lost per year is y = 12,000x.
Use the following information to answer the next two exercises. The price of a single issue of stock can fluctuate throughout the day. A linear equation that represents the price of stock for Shipment Express is y = 15 – 1.5x, where x is the number of hours passed in an eight-hour day of trading.

### Homework
