Intro to Statistical Inference

• See Homework #1 assignment on Blackboard
• Reading assignment for Lab #1: OpenIntro Stats, Chapters 1 and 2
• Reading assignment for Wednesday: OpenIntro Stats, Chapter 3: Distributions of random variables (optional)
  • 4.1: Normal distribution

Multiple representations are important!!!

Definition: Modeling is 1) the process of moving from observations of the real world to a model, 2) moving from one model representation to another model representation, or 3) comparing different models.

Discuss: Why are models and modeling useful?

Answer: Models can be used for prediction, explanation and understanding.

Note: You are already constructing models and doing modeling!!!

In this course, we will work to become proficient in understanding, analyzing, and creating models - in particular mathematical models.

• models can make accurate predictions,
• models can generate causal explanations.
• simple, unrealistic models help scientists explore complex systems,
• models can be used to explore unknown possibilities,
• models can lead to the development of conceptual frameworks,
• models help expose assumptions,
• models help define expectations,
• models help us to devise new tests.

Definition: A parameter is a quantity describing a population, whereas an estimate or statistic is a related quantity calculated from a sample.

Parameter examples: Averages, proportions, measures of variation, and measures of relationship

Statistics is a technology that describes and measures aspects of nature from samples.

Statistics lets us quantify the uncertainty of these measures.

Statistics makes it possible to determine the likely magnitude of measurements departure from the “truth”.

Statistics is about estimation, the process of inferring an unknown quantity of a target population using sample data.

The two sides of the statistical coin:

• Parameter estimation
• Hypothesis testing

Definition: A statistical hypothesis is a specific claim regarding a population parameter.

Definition: Hypothesis testing uses data to evaluate evidence for or against statistical hypotheses.

The two sides of the statistical coin:

• Parameter estimation
• Hypothesis testing

Example: A trapping study measures the rate of fruit fall in forest clear-cuts.

The two sides of the statistical coin:

• Parameter estimation
• Hypothesis testing

Example: A clinical trial is carried out to determine whether taking large doses of vitamin C benefits health of advanced cancer patients.

…well, most of the time.

• Many statistical techniques require assumptions about where your data is coming from (i.e. properties of the population)
• In other words, an assumed probability model describes the population
• Statistical techniques that are based on probability models are called parametric techniques, while those that are not are called non-parametric techniques.

For your question, there is desired and undesired information in your data.

Goals:

• Get accurate information by reducing bias
• Get precise information by reducing sampling error due to random variation (increase signal-to-noise ratio)

Definition: Bias is a systematic discrepancy between the estimates we would obtain, if we could sample a population again and again, and the true population characteristic.