A probability simulation estimates a probability by imitating a random event many times with random numbers. You assign numbers to each outcome, run many trials with a random digit table, a calculator or a computer, and count how often the event happens. The fraction of trials where it happens is your estimate.
A randomized block design is an experiment where you first sort subjects into groups of similar subjects, called blocks, and then randomly assign treatments within each block. A matched pairs design is a block design with blocks of two. Blocking removes variation the blocks cause, so the treatment effect is easier to see.
Experimental design is the plan for how an experiment gives treatments and measures results. A good design compares two or more treatments, often including a control group, uses random assignment to form the groups, and has enough subjects in each group. Blinding and placebos keep expectations from changing the results.
An observational study watches individuals and measures variables without changing anything. An experiment deliberately gives treatments to individuals to see how they respond. Only a well-designed experiment with random assignment can show cause and effect. An observational study can show a link, but other variables might explain it.
Sampling bias happens when the way a sample is collected makes it differ from the population in the same direction again and again. The main types are undercoverage, nonresponse, response bias, voluntary response and convenience sampling. A biased sample gives misleading results, and a larger sample does not fix it.
Sampling methods are ways to choose a sample, a smaller group, from a population. The four main random methods are the simple random sample, the stratified random sample, the cluster sample and the systematic sample. Random sampling helps the sample represent the whole population, so its results can be trusted.
A transformation to achieve linearity changes curved data so that it follows a straight line. You take the logarithm, square root or another function of x, y or both. Then you fit a least-squares line to the new values and undo the transformation to make predictions in the original units.
An influential point is a data point that changes the least-squares regression line a lot when you remove it. Points with x-values far from the rest, called high-leverage points, are often influential. An outlier in regression is a point with a large residual. To check influence, find the line with and without the point.
The coefficient of determination, r², is the fraction of the variation in y that a least-squares regression line explains. To find it, square the correlation r, or use 1 − SSE/SST. For example, r² = 0.81 means the line explains 81% of the variation in y.
A residual is the actual y-value minus the predicted y-value: y − ŷ. It shows how far a data point is above or below the regression line. A residual plot graphs every residual against x. If the plot shows no pattern, a linear model fits the data well.