Arguments and Reasoning

Reasoning

Arguments

• An argument is a set of statements, called premises, intended to support a conclusion
• Arguments can be:
  • Deductive: The conclusion follows necessarily from the premises
  • Inductive: The conclusion is probable, but not certain
• Components of an argument:
  • Premises: Statements that provide evidence for the conclusion
  • Inference: The process of drawing a conclusion from the premises
  • Conclusion: The statement being supported by the premises
• Types of arguments:
  • Categorical: Involves statements about categories or classes
  • Hypothetical: Involves statements about hypothetical situations
  • Analogical: Involves comparisons between similar things

Hypothesis Testing

• A statistical method for testing a hypothesis about a population based on a sample of data
• Steps in hypothesis testing:
  1. Null hypothesis (H0): A statement of no effect or no difference
  2. Alternative hypothesis (H1): A statement of an effect or difference
  3. Test statistic: A numerical value that measures the difference between the sample data and the null hypothesis
  4. P-value: The probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true
  5. Alpha level (α): The maximum probability of type I error (rejecting the null hypothesis when it is true)
  6. Decision: Reject the null hypothesis if the p-value is less than α, otherwise fail to reject the null hypothesis
• Common types of errors:
  • Type I error: Rejecting the null hypothesis when it is true
  • Type II error: Failing to reject the null hypothesis when it is false

Arguments

• An argument consists of premises, inference, and a conclusion
• Types of arguments:
  • Deductive: Conclusion follows necessarily from premises
  • Inductive: Conclusion is probable, but not certain
• Components of an argument:
  • Premises: Statements providing evidence for the conclusion
  • Inference: Process of drawing a conclusion from premises
  • Conclusion: Statement being supported by premises

Types of Arguments

• Categorical: Involves statements about categories or classes
• Hypothetical: Involves statements about hypothetical situations
• Analogical: Involves comparisons between similar things

Hypothesis Testing

• A statistical method for testing a hypothesis about a population based on a sample of data
• Steps in hypothesis testing:
  • Null hypothesis (H0): Statement of no effect or no difference
  • Alternative hypothesis (H1): Statement of an effect or difference
  • Test statistic: Numerical value measuring the difference between sample data and null hypothesis
  • P-value: Probability of obtaining a test statistic at least as extreme as the one observed, assuming null hypothesis is true
  • Alpha level (α): Maximum probability of type I error
  • Decision: Reject null hypothesis if p-value is less than α, otherwise fail to reject null hypothesis

Common Errors in Hypothesis Testing

• Type I error: Rejecting the null hypothesis when it is true
• Type II error: Failing to reject the null hypothesis when it is false