Foundations of Inferential Statistics

Faculty of Arts Part II: Foundations of Inferential Statistics PSYC 2101 Statistics in the Social Sciences

  • The course materials in PSYC 2101 have been provided for private study and educational use only.

  • TRU grants a limited, revocable license for personal use (including permission to print one copy) of Part II: Foundations of Inferential Statistics.

  • Materials are not to be further distributed.

  • Essential to access the online learning management system for supplementary materials like videos, web links, and interactive media.

Copyright & Credits

  • Copyright © 2021 (Revised), 2016 (Revised), 2013, 2007 Thompson Rivers University. All rights reserved.

  • The content is protected by copyright law worldwide.

  • Usage permitted for students enrolled at TRU for personal study purposes only.

  • No part of this work may be reproduced in any form without permission from TRU's Intellectual Property Office.

  • Course Revision Team includes John Marasigan, PhD, Cory Stumpf, BJ, Mark Wallin, PhD, Elizabeth Reimer, PhD, Michael Looney, MSc.

Table of Contents

  • Part II: Foundations of Inferential Statistics Weeks 3 & 4

    • Chapter 5: z-Scores: Location of Scores and Standardized Distributions

    • Learning Objectives

    • Supplement: The Standard Normal Curve

    • Study Plan

    • Practice Exercises for Chapter 5

    • SPSS (Optional Activity)

    • Review of Learning Objectives

    • Chapter 6: Probability

    • Learning Objectives

    • Study Plan

    • Practice Exercises for Chapter 6

    • SPSS (Optional Activity)

    • Review of Learning Objectives

    • Chapter 7: Probability and Samples: The Distribution of Sample Means

    • Learning Objectives

    • Study Plan

    • Practice Exercises for Chapter 7

    • SPSS

    • Review of Learning Objectives

    • Chapter 8: Introduction to Hypothesis Testing

    • Learning Objectives

    • Study Plan

    • Supplement and Extra Examples

    • Study Plan Continued

    • Practice Exercises for Chapter 8

    • SPSS

    • Review of Learning Objectives

  • Summary of Part II: Foundations of Inferential Statistics

  • Assignment 2: Foundations of Inferential Statistics

Part II: Foundations of Inferential Statistics (Weeks 3 & 4)

  • Many events (e.g., lightning strikes, severe wind changes) are rare, with small probabilities.

  • Understanding likelihood helps individuals and businesses make informed decisions.

    • Example: Decisions to not smoke are influenced by perceived risks of illness.

    • Individuals wear seatbelts to reduce injury probabilities in accidents.

Chapters Covered

  1. Chapter 5: z-Scores: Location of Scores and Standardized Distributions

  2. Chapter 6: Probability

  3. Chapter 7: Probability and Samples: The Distribution of Sample Means

  4. Chapter 8: Introduction to Hypothesis Testing

Chapter 5: z-Scores: Location of Scores and Standardized Distributions

Learning Objectives

  • After completing Chapter 5, students should be able to:

    • Explain z-scores: Indicate the exact location of scores within a normal distribution.

    • Transformations: Convert X values into z-scores and vice versa.

    • Distribution impacts: Describe how various distributions change when X values are transformed into z-scores.

    • Role in inferential stats: Articulate the significance of z-scores in inferential statistics.

Supplement: The Standard Normal Curve

  • Definition: The standard normal curve is a theoretical, symmetrical, bell-shaped distribution, often applied in inferential statistics.

  • Properties:

    • Symmetrical: Identical halves about the mean.

    • Unimodal: Has a single peak at the mean.

    • Central tendency: The mean, median, and mode coincide at the center.

    • Mean ($ ext{μ}$) = 0; Variance and Standard Deviation ($ ext{σ}$) = 1.

  • Abscissa & Ordinate:

    • z-scores extend continuously; the lower half is negative, the upper half is positive.

    • Uses relative frequency for ordinate so that the area under the curve equals 1.

  • Asymptotic nature: The curve approaches the x-axis but never touches it, implying an infinite number of terms.

  • Distribution applications: The curve demonstrates that measurements (e.g., height, intelligence) tend to cluster around the mean.

Study Plan

  • Read Chapter 5 (pages 150–171).

  • Complete practice exercises and refer back to learning objectives for self-assessment.

Practice Exercises for Chapter 5

  • Exercise 1: Find z-scores for average dress sales.

  • Exercise 2: Determine hours taken to sew a dress based on a given z-score.

  • SPSS: Steps for transforming raw scores into z-scores are outlined in the textbook.

Review of Learning Objectives

  • Evaluate progress on learning objectives, and seek further assistance from faculty if necessary.

Chapter 6: Probability

Learning Objectives

  • At the conclusion of this chapter:

    • Define probability and its relationship with random sampling.

    • Calculate probabilities for scores using the unit normal table.

    • Utilize the unit normal table to determine percentiles and percentile ranks.

    • Apply normal approximation to binomial distributions for calculating binomial probabilities.

Study Plan

  • Read Chapter 6 (pages 178–206).

  • Complete practice exercises.

Practice Exercises for Chapter 6

  • Exercise 1: Calculate retirement probabilities for factory workers.

  • Exercise 2: Determine sewing probabilities using the normal distribution.

  • Exercise 3: Calculate percentile rank for a sewer's production.

Review of Learning Objectives

  • Self-evaluate progress on learning objectives and seek assistance when needed.

Chapter 7: Probability and Samples: The Distribution of Sample Means

Learning Objectives

  • Upon completion of this chapter, students should be able to:

    • Define distribution of sample means, expected value, and the standard error of the mean.

    • Locate sample mean in distribution of sample means via z-scores.

    • Find probabilities for specific sample means using z-scores and unit normal table.

Study Plan

  • Read Chapter 7 (pages 214–239).

  • Complete practice exercises.

Practice Exercises for Chapter 7

  • Exercise 1: Evaluate expected error between sample mean and population mean for a factory's worker output.

  • Exercise 2: Calculate the likelihood of sample mean exceeding a certain number after performance evaluation adjustments.

Review of Learning Objectives

  • Reflect on achievement of learning objectives and seek help if unsure.

Chapter 8: Introduction to Hypothesis Testing

Learning Objectives

  • At the end of this chapter, students should be able to:

    • Understand the logic of hypothesis testing.

    • State hypotheses and identify the critical region.

    • Use the z-score statistic to test hypotheses with appropriate decisions.

    • Define Type I and Type II errors.

    • Explain power concepts and compute Cohen’s d.

    • Differentiate between one-tailed and two-tailed tests and apply them accordingly.

Study Plan

  • Read Chapter 8 (pages 244–287).

  • Follow specified steps in hypothesis testing.

Steps in Hypothesis Testing

  1. State the hypotheses: Null ($H0$) and alternative ($H1$) hypotheses.

  2. Locate the critical region: Determine corresponding test statistic and critical value.

  3. Compute the test statistic: Based on sample data.

  4. Make a statistical decision: Reject or fail to reject $H_0$ based on test statistic's location.

  5. State the conclusion: Communicate outcome in clear terms, beyond mere statistical jargon.

Example Hypothesis Test

  • Scenario: Compare current class average with historical data to assess if it has increased.

  • Follow the five steps with given parameters (e.g., sample mean, population mean, standard deviation).

Practice Exercises for Chapter 8

  • Exercise 1: Test effectiveness of a weight loss pill with mean weight comparison.

  • Exercise 2: Evaluate impact of a technique on student test scores and compute effect size using Cohen's d.

Review of Learning Objectives

  • Reflect on understanding and progress towards objectives.

Summary of Part II: Foundations of Inferential Statistics

  • Introduced fundamental concepts of inferential statistics, focusing on the importance of sample statistics concerning populations.

  • Discussed normal distributions, z-scores, and transformations of data.

  • Explored probability concepts and their relationship to inferential techniques.

  • Covered the distribution of sample means and the central limit theorem.

  • Presented hypothesis testing framework and error types, alongside steps vital for substantiating claims in statistical analysis.

Assignment 2: Foundations of Inferential Statistics

  • Assignment contributes 10% towards the final course grade.

  • Save completed document as a Word file and name appropriately before submission for marking.