Relation, Analyzing Linear Relations, and Modelling With Graphs
Linear Systems
A linear system is two or more linear equations together. In given equation, an ordered pair satisfies the equations when the left side of the equation equals the right side, when x and y are subsided into equations.
The solution of a linear system o 2 equations is the point where the graphs intercept.
Sampling Principles
There are 5 different types of sampling
Simple random sample
Systemic Random Sampling
Stratified Random Sampling
Non-Random Sampling
Bias
Simple random sampling is the basic sampling technique where we select a group of subjects (a sample) for study from a larger group (a population). Each individual is chosen entirely by chance and each member of the population has an equal chance of being included in the sample.
Systematic sampling a of probability sampling method where sample members from a larger population are selected according to a random starting point but with a fixed, periodic interval. This interval, called the sampling interval, is calculated by dividing the population size by the desired sample size.
A stratified sample is one that ensures that subgroups (strata) of a given population are each adequately shown in the whole sample population of a research study. For example, one might divide a sample of adults into subgroups by age, like 18–28, 10–29, 30-39–49, 40–58, and 67 and above.
Non-random sampling is any sampling method in which the process that determines whether a member of the population is selected for inclusion in the sample is guided by a non-chance or non-random process.
In statistics, sampling bias is a bias in which a sample is collected in such a way that some members of the intended population have a lower or higher sampling probability than others.
Analyzing Data
A scatter plot is a graph that shows the relationship between two sets of numeric data or variables. When graphing two sets of data, the independent variable is graphed along the x-axis and the dependent variable is graphed along the y-axis.
Points in a scatter plot often show a pattern or a trend. An outlier is a point that does not follow the trend.
When making a scatter plot graph you would use a line of best fit to show the data. The line of best fit is made with around the same number of points on the top and on the bottom of the line to make it equal (A line of best fit is used for making predictions).
When the prediction involves a point within the range of the values of the variable represented on the x-axis, this is called interpolating. When the prediction involves a point outside the range of the values of the variable represented on the x-axis, this is called extrapolating.
First Differences
In a linear relationship the points form a straight line have a formula of y=mx or y=mx+b. In a nonlinear relationship the points make a curved line and have a formula with an exponent in it Eg. y=mx^2+b.A difference table, is a table of values with 1 extra column. To find first differences, subtract consecutive values of y (dependent variable).
Rate of Change
Rate of change is change in one quantity relative to the change in another.
Hypothesis' and Sources of Data
A hypothesis is an educated guess. There are two sources of data. A primary source and a secondary source
Standard Form and Intercept form
The standard form is Ax + By + C = 0 where A, B and C are integers, A and B are not both zero, A must be › 0 (If the coefficient of the x-term is negative, you can multiply both sides by (-1) to express the equation in standard form)
Standard form is y=mx+b. You can change an equation in standard form to slope y-intercept form by rearranging the equation.
Distance Time Graphs
A distance-time- shows how distance varies with time. A rising line – increases as time increases. A falling line – decrease in distance with time. A horizontal line – distance remains constant as time passes
The rule of Four
The Rule of Four can be used to show a relation in four ways: words, a diagram or graph, numbers, and equation.
Graphing a line using Intercepts
The x-intercept is the x-coordinate of the point when the line crosses the x-axis. The y-intercept is the y-coordinate of the point when the line crosses the y-axis.
y=mx+b
Y and X represent the points that make the line. M represents the slope of the line and b represents the y intercept of the line Horizontal lines have a formula of y=b Vertical lines have a formula of x=a
Direct and Partial variation
Direct variation occurs when the dependent variable changes by the same factor as the independent variable. Direct variation lines formula is y=mx.
Partial variation lines have an equation of y=mx+b because they go trough b which is also known as the y intercept.
Slope
The slope of a line is the steepness of the line. The slope is calculated using the formula rise/run or rise over run. The other formula to calculate the slope is y2-y1/x2-x1
