Lecture_Stats4_handout

Page 2

Learning Objectives for This Lecture

• Deepen understanding of the normal distribution, including insights into the standard normal distribution.

• Acquire knowledge on different types of distributions.

• Explore aspects of the scientific research process, focusing on experimental design and the role of statistics.

• Understand statistical hypotheses.

• Gain insights into inferential statistics and statistical analyses.

• Learn general principles for writing up results.

Page 3

Descriptive Statistics Overview

• Purpose: Describing and summarizing data.

• Measures of Central Tendency:

  • Mean

  • Median (Interquartile range)

  • Less common: Mode

• Variability and Uncertainty:

  • Variance

  • Standard deviation

  • Standard error of the mean

  • 95% confidence interval

• Graphical Representation: Bar plots, box-and-whisker plots.

Page 4

Normal (Gaussian) Distribution

• A common continuous probability distribution.

• Characteristics:

  • Bell-shaped and asymptotic at the extremes.

  • Symmetrical around the mean; mean = median = mode.

  • Area under the curve relates to the relative frequency of observations and their probabilities.

Page 5

Normal Distribution Properties

• The area under the curve corresponds to the relative frequency.

• 68.3% of scores fall within ±1 standard deviation from the mean.

Page 6

Normal Distribution Insights

• Normal distribution frequently models continuous variables, such as weights and sizes.

• Very important for statistical analyses.

• Distribution Properties:

  • 68.3% of scores lie within ±1 standard deviation.

  • 95.5% lie within ±2 standard deviations.

Page 7

Normal Distribution Mathematical Definition

• Formula: f(x) = (1 / (2πσ²)) * e^(-((x−μ)²/(2σ²)))

• Normal distribution is described by its mean (µ) and standard deviation (σ); variance is σ².

Page 8

Standard Normal Distribution

• Independent mean and standard deviation allows for infinite combinations and distributions.

• Standard Normal Distribution (z-distribution): A normal distribution with mean 0 and standard deviation 1.

• Total area under the curve is 1.

Page 9

Example: Standard Normal Distribution

• Scenario: Measuring height of 100 people.

• Mean (𝑦𝑦�) = 170 cm, Standard deviation (s) = 8 cm.

• Calculate probabilities (a) taller, (b) shorter, (c) within given heights by converting to z-scores.

Page 10

Calculating z-value

• Scenario Continued: Finding probability for height less than 160 cm.

• Steps: 1. Convert height to z-score.

2. Find probability using R function: pnorm(-1.25) ≈ 0.106; therefore, ~11% chance for a person shorter than 160 cm.

Page 11

Probability Between Heights

• Height Range: 165 cm to 180 cm.

• Convert heights to z-scores:

  • z1 = (165 – 170) / 8 = -0.625

  • z2 = (180 – 170) / 8 = 1.25

• Find probability by subtracting pnorm(z2) from pnorm(z1).

Page 12

Probability Calculation Example Continued

• Calculating Probability: pnorm(1.25) – pnorm(-0.625) = 0.63.

• Result: Probability that a randomly selected person is between 165 cm and 180 cm is 63%.

Page 13

Summary of Probability Calculations

• Converting heights into z-scores is essential.

• Utilizing pnorm function allows for accurate probability assessments within ranges.

Page 14

Example: Career Restrictions Based on Height

• British Men's Mean Height: ~176 cm, standard deviation ~6 cm.

• Task: Determine the proportion of British men disqualified from spy careers based on height criteria.

Page 15

Blood Alcohol Concentration in Rats Study Example

• Study findings: Rats died at a mean concentration of 7.5 mg/ml (±0.2 SD).

• Task: Calculate the probability that a rat died with blood alcohol between 7.1 and 7.6 mg/ml.

Page 16

Normal Distributions Visualization

• Example histograms showcasing frequency distributions.

• Continuous data often illustrated through frequency graphs.

Page 17

Poisson Distribution

• Typically used for discrete count data (e.g., number of events in a fixed interval).

• Example histograms demonstrate Poisson randomness.

Page 18

Binomial Distribution

• Focuses on proportions and binary variables (e.g., success/failure).

• Example applications include animal survival rates post-treatment.

Page 19

Importance of Statistics Discussion Prompt

• Reflect on the necessity of statistics in research and encourage group discussion.

Page 20

Why Statistics Matter

• Variability exists in all measurements.

• Statistics enable us to extract meaningful signals from random noise.

Page 21

Scientific Research Process Framework

• Key steps:

  • Collect information.

  • Interpretation of data.

  • Develop hypotheses.

  • Design experiments to test hypotheses.

Page 22

Understanding Statistical Hypotheses

• Key definitions:

  • Statement about the world that can be tested/falsified.

  • Good vs. bad examples of hypotheses.

Page 23

Null Hypothesis Concept

• Null Hypothesis (H₀): No difference or effect.

• Example of hypothesis testing where no evidence found signifies absence of voles.

Page 24

Falsification Theory in Hypotheses

• Importance of a testable hypothesis.

• Good examples of falsifiable hypotheses vs. ambiguous statements.

Page 25

Types of Hypotheses

• Null Hypothesis (H₀): Indicates no effect.

• Alternative Hypothesis (H₁): Indicates the presence of effect or signal.

Page 26

Recap of Research Process

• Summarizing the scientific research process and its components, emphasizing hypothesis formulation and experimental design.

Page 27

Experiment Design

• Clearly define Y-variable (response) and X-variable (explanatory).

• Example hypotheses relating to foraging duration in ecological studies.

Page 28

Data Types in Hypothesis Testing

• Important to categorize variables as binary, proportion, continuous, discrete, or categorical.

Page 29

Correlative vs. Experimental Design

• Distinction between correlation (no manipulation) and experiments (manipulate variables to test causation).

Page 30

Selection of Statistical Analysis

• Considerations for sampling, signal vs. noise ratio, and the importance of sample size in reducing error.

Page 31

The Concept of Statistical Power

• Explain potential statistics errors and the importance of power in hypothesis testing. Type I & II errors' implications.

Page 32

Types of Statistical Errors

• Type I Error: Rejecting a true null hypothesis.

• Type II Error: Not rejecting a false null hypothesis.

Page 33

Power and Outcomes in Hypothesis Testing

• Explore outcomes of hypothesis testing and their probabilities.

• Successful rejection of H0 is desired.

Page 34

Summary of Probabilities in Hypothesis Testing

• Combination of H₀ acceptance/rejections and the associated probabilities for accurate hypothesis validation.

Page 35

Insight on Power and p-Values

• Balancing statistical power with the significance level of p-values in research analyses.

Page 36

Repetition of Statistical Concepts

• Emphasizing the key probabilities related to hypothesis testing outcomes to enhance understanding.

Page 37

Enhancing Statistical Power

• Strategies to Improve power in studies: increasing sample size, randomization, etc.

Page 38

Recap of Scientific Research Process

• Reinforcement of methodological principles guiding scientific inquiry in various research fields.

Page 39

Analysis and Interpretation of Statistical Tests

• Description of key statistical terms including test statistic and p-values.

• Introduction to p-value significance thresholds.

Page 40

Degrees of Freedom (DF)

• Explains the concept of degrees of freedom in relation to sample observations and parameters.

Page 41

Example of DF in Experimental Design

• Assess the number of degrees of freedom in a drug study involving multiple treatment levels.

Page 42

Extended Example of DF

• Analyze another experimental design with increased treatment levels impacting degrees of freedom calculations.

Page 43

Writing Up Scientific Results

• Guidance on presenting research findings effectively.

Page 44

Examples of Result Presentation

• Illustrates how to write results with statistical data, including figures and tables.

Page 45

Best Practices for Results Writing

• Details on results presentation methodologies to ensure clarity and conciseness.

Page 46

Key Elements in Result Documentation

• Essential elements to include in scientific reports concerning statistical results and visual evidence.

Page 47

Comparative Writing Analysis

• Comparative evaluation of strong versus weak result presentations.

Page 48

Summary of Normal Distributions

• Highlights of standard normal distribution methodology and probability conversion process.

Page 49

Summary of Research Methodology

• Comprehensive overview of the scientific research process, hypothesis testing, statistical significance, and their applications in studies.