Chapter 4: Probability and Sampling Distributions

Section 4.1: Randomness

Definitions:

• parameter: a number that describes the population. Typically, the value of the parameter is not known because we cannot examine the whole population.
• statistic: a number that can be computed from the sample data. We often use the statistic to estimate the parameter.
  
  
• population mean, μ: is a fixed parameter that is unknown when we use a sample for inference
• sample mean, x-bar: is the average of the observations in the sample

The idea of probability

Sampling variabilty: the exact value of a statistic varies in repeated random sampling. This is not a problem if we have a random sample because chance behavior is unpredictable in the short run but has a regular pattern in the long run.

The idea of randomness is empirical. In other words, it is based on observation. Probability describes what happens in a large number of trials.

• random: a phenomenon is random if individual outcomes are uncertain but there is a regular distribution of outcomes in a large number of repetitions.
• probability: of any outcome of a random phenomenon is the proportion of times the outcome occurs in a very long series of repetition.

Thinking about randomness

We can never observe a probability exactly. Mathematical probability is an idealization based on imagining what would happen in an indefinitely long series of trials.

• independent trials: the outcome of one trial must not influence the outcome of any other.
• probability is empirical: we can estimate a real-world probability only by observing many trials. Short run trials only give an estimate of a probability.
• computer simulations are useful!