Statistical Methods in PSYC HW

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Ch1. Operationalizing Independent & Dependent variables HWCh2. Hypothesis Testing for a Related Samples Research Design HW I would appreciate it if you could solve the HW questions by referring to the lecture materials I attached below. I also attached the HW1,2 files below. Thank you again.

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PSYC 300B — Homework Assignment #2
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PSYCHOLOGY 300B
Homework Assignment 2:
Related Sample Design
1. The Paris 2024 Summer Olympics are just a year and half away and are constantly on the
minds of athletes & coaches. The coach wants to maximize the probability that her athletes
will do their best and represent Canada well. Everyone has been, however, like, so totally
stressed out that they are making lots of errors (e.g., dropping their beer jugs as they walk
down the street, handing in their own urine samples for drug testing). The coach decides to
introduce Transcendental Meditation (TM) training to her nine star athletes. The coach hires
Moonblossom Ponnietayle to teach the 9 athletes TM & enforce its application 3 times a day.
Scores on a well standardized stress test (the NOPROB scale ranges from 0 for “no stress” to
20 for “totally freaking out”) will be taken prior to the introduction of TM and then 4 weeks
after the training & practice sessions.
a. Name the independent variable & the dependent variable & identify the scale of
measurement for the DV.
b. Apply the 6 Steps of Hypothesis Testing to these data
Athlete
Joc
Jyll
Nic
Nak Paddy Tinki Winki Dipsi
Po
Before TM:
16
18
13
19
18
14
20
11
15
After TM:
15
12
10
12
19
12
13
9
15
c. Do you think the coach will decide to incorporate TM into her athletes’ training regime?
2. A study was carried out to address the endangered species status of Vancouver Island (VI)
Marmots in the spring where your colleague fed 9 VI marmots 10mg of Viagra® for one
month. The number of attempted matings by each marmot was recorded before and after the
administration of Viagra®. In that study, the # of matings significantly increased (D = 2.00),
t (8) = 3.00, p < .05. Because the Null Hypothesis was rejected, your colleague correctly determined that Viagra® appeared to increase marmot mating behaviour. When critiquing the study, you note that study was carried out in the late spring, and marmots typically mate in during this time, so you were not surprised that the number of matings increased. Consequently, you decide to gather more data to see if Viagra would be able to extend the mating season. In early summer, you carry out a second study. You have no trouble getting 9 of the little furballs to participate. You replicate the study using the same research design and same level of Viagra® as your colleague. Carry out the appropriate test of the null hypothesis by hand. Assume p(α) = .052-tailed Page 1 of 2 © 2024 David A. Medler - PSYC 300B — Homework Assignment #2 a. What is the IV and how is it operationalized? Is it measured or manipulated? What is its scale of measurement? b. What is the dependent variable? What is its scale of measurement? c. What type of research design is this? d. Define the 3 distributions associated with this research design. For each distribution, give the symbol & value (if known) for its measure of central tendency and the symbol & value (if known) for its measure of variability. e. Carry out an inferential test on these data by hand. Marmots A B C D E F G H I Before Viagra®: 9 3 4 3 2 3 1 7 4 After Viagra®: 8 9 4 4 9 5 10 2 3 f. What do you conclude about the effect of Viagra® on mating behaviour for VI marmots? Page 2 of 2 © 2024 David A. Medler Operationalizing Independent & Dependent Variables Chapter 01 Chapter 01 PSYC 300B — Statistical Methods in Psychology II — © 2024 David A. Medler 1 Lecture Outline ‣ Making a Science ‣ Defining the Independent Variable ‣ Defining the Dependent Variable Chapter 01 PSYC 300B — Statistical Methods in Psychology II — © 2024 David A. Medler 2 IV & DV in Research Hypotheses Three Levels of Definition ‣ Construct (Concept) ➠ Research Question / Hypothesis ➠ Operational Definition ‣ Every independent variable (IV) & dependent variable (DV) can be defined on all three levels. Chapter 01 PSYC 300B — Statistical Methods in Psychology II — © 2024 David A. Medler 3 Conceptual Variables (Constructs) ‣ Vague, not well-defined or in meaningful terms. ‣ Very high level terms (incredibly broad) ‣ Abstract or theoretical level definition. ‣ Measurement depends on how variable is defined. ‣ Intelligence, Memory, Personality, Creativity Chapter 01 PSYC 300B — Statistical Methods in Psychology II — © 2024 David A. Medler 4 Research Question/Hypothesis ‣ Abstract concept becomes more specific, more concrete. ➙ ‣ Memory ‣ Intelligence ➙ Picture Recognition Memory. Musical Intelligence. ‣ Posits a relationship between the IV and the DV ‣ Research Questions are the reason why we are doing the study ‣ Research Hypotheses (technically) are how we statistically test the relationship ‣ In practice, the Research Question and Research Hypothesis are often conflated Chapter 01 PSYC 300B — Statistical Methods in Psychology II — © 2024 David A. Medler 5 Operational Definition ‣ “Operational” ➙ Method used to measure the concept. ‣ IV ➙ How variable is defined in experiment. ‣ Either manipulated or measured ‣ DV ➙ The participant’s behaviour that is measured. ‣ The data that we are collecting. Chapter 01 PSYC 300B — Statistical Methods in Psychology II — © 2024 David A. Medler 6 The Science Cycle 1. Start with a Theory Theory 2. Develop a Research Question to test the Theory 3. State a Specific Hypothesis based on Research Question Research Question Revise 4. Carry out the Experiment 5. Collect and Analyze the Data 6. Revise the Theory According to the Observed Data 7. Repeat! Chapter 01 Data Hypothesis Experiment PSYC 300B — Statistical Methods in Psychology II — © 2024 David A. Medler 7 Defining a Research Question ‣ Start with a Theory based on Previous Research/Observations ‣ Intelligence and Memory are related ‣ Define your Hypothesis by making the Theory more concrete ‣ “Academic” Intelligence is associated with Recall Memory ‣ Define a testable hypothesis by operationalizing your variables ‣ Intelligence as measured by IQ is associated with the number of nouns recalled. “Participants with high IQ SCORES will RECALL MORE NOUNS than those with low IQ SCORES.” Chapter 01 PSYC 300B — Statistical Methods in Psychology II — © 2024 David A. Medler 8 From Concept to Measurement RELATIONSHIP BETWEEN CONCEPTS & MEASURES OF CONCEPTS LEVEL OF ANALYSIS Chapter 01 TYPE OF VARIABLE Independent Variable Dependent Variable Conceptual Hypothesis “Intelligence” Memory Research Hypothesis “Academic” (school- based) Recall Memory Operationalized as… IQ Scores on WAIS-R # Nouns recalled (out of 50) PSYC 300B — Statistical Methods in Psychology II — © 2024 David A. Medler 9 Lecture Outline ‣ Making a Science ‣ Defining the Independent Variable ‣ Defining the Dependent Variable Chapter 01 PSYC 300B — Statistical Methods in Psychology II — © 2024 David A. Medler 10 Independent Variables (IV): Predictor ‣ Feature of study used to predict or explain behaviour. ‣ Researcher has some control ‣ Degree of control may vary. ‣ Manipulated IV (precise control) → Measured IV (no control) ‣ The experimenter determines the IV prior to the experiment being conducted or data collected. Chapter 01 PSYC 300B — Statistical Methods in Psychology II — © 2024 David A. Medler 11 Manipulated vs. Measured IV Manipulated IV ‣ Researcher controls what Participants are exposed to. ‣ Amount of… ; Type of… ‣ Examples? Measured IV ‣ Characteristic or trait inherent to Participants used ‣ to classify Participants into groups, or ‣ to represent a continuous measure (score) of their behaviour. ‣ Examples? Chapter 01 PSYC 300B — Statistical Methods in Psychology II — © 2024 David A. Medler 12 Interpretation of Outcome Based on Degree of Control ‣ Measured IV ‣ Association or Relationship. ‣ The experimenter has no control over the IV, so causal inferences cannot be made. ‣ Manipulated IV ‣ Cause & Effect if and only if ‣ Data are scores, & ‣ Participants randomly sampled/assigned into the manipulation conditions (levels) of the IV ‣ If both conditions are not met, then only an Association or Relationship can be inferred. Chapter 01 PSYC 300B — Statistical Methods in Psychology II — © 2024 David A. Medler 13 Examples of IVs on Four Scales of Measurement SCALE OF MEASUREMENT Nominal Ordinal Interval IV Ratio Manipulated Treatment Instructions Depression Alcohol Consumption Levels Experiment / Control Clear / Mod / Vague Induced Score on Beck Depression Inventory Amount of ml consumed per kg (ml/kg) Gender Age Intro-/Extroversion Reaction Time F/M/T/NB/2S 6- / 12- / 24Months Measured Levels Chapter 01 Recorded as time to Self Report Score respond in ms PSYC 300B — Statistical Methods in Psychology II — © 2024 David A. Medler 14 Implications of Scales of Measurement for Independent Variables Type of Graphic Figure used ‣ If IV is measured on… ‣ Nominal or Ordinal scale ‣ Bar Graph. ‣ Interval or Ratio scale ‣ Histogram, Frequency Polygon, Number of Bar Presses Number of Bar Presses or Line Graph. 25 20 15 10 5 0 Food Water Alcohol 25 20 15 10 5 0 0 mg Type of Reward Chapter 01 1 mg 2 mg 3 mg Amount of Food PSYC 300B — Statistical Methods in Psychology II — © 2024 David A. Medler 15 Lecture Outline ‣ Making a Science ‣ Defining the Independent Variable ‣ Defining the Dependent Variable Chapter 01 PSYC 300B — Statistical Methods in Psychology II — © 2024 David A. Medler 16 Dependent Variable (DV): Criterion ‣ Data you collect; ‣ Behaviour being measured; ‣ Dependent on Participants performance; ‣ Experimenter observes, measures, records response. ‣ Also known as the criterion. Chapter 01 PSYC 300B — Statistical Methods in Psychology II — © 2024 David A. Medler 17 Two Forms of DV Qualitative DV Quantitative DV ‣ Nominal scale only. ‣ Scales: Ordinal, Interval, Ratio. ‣ Data ‣ Data ‣ Tally or Count for each Participant’s response. ‣ Ranks (Ordinal scale). ‣ Scores (Interval or Ratio scale). ‣ Each Participant contributes only 1 tally to data set! ‣ Each Participant may contribute 1 or more scores to the data set. Chapter 01 PSYC 300B — Statistical Methods in Psychology II — © 2024 David A. Medler 18 Dependent Variable (DV): Ordinal Scale Ordinal Scale: Ranks or Preferences ‣ Rank (1-5) most preferred to least preferred. ‣ Rank (1-10) runners as they cross finish line. ‣ “How often to do you engage in…” Never 1 Almost Never Almost Always 2 3 Always 4 ‣ Be very careful not to confuse a ranking with an interval scale. ‣ “How often to do you engage in…” Never 1 Chapter 01 Almost Never 2 Sometimes 2.5? Almost Always 3 Always 4 PSYC 300B — Statistical Methods in Psychology II — © 2024 David A. Medler 19 Dependent Variable (DV): Interval Scale Interval Scale: Well Standardized Scales, Physiological Measures ‣ Well Standardized Scales ‣ Self Report Scales ‣ e.g., anxiety, depression, happiness, ‣ Physiological Measures ‣ e.g., body temperature, galvanic skin response, EEG stress ‣ Behavioural Rating Scales ‣ e.g., Child Behaviour Problems Index ‣ Intellectual Rating Scales ‣ e.g., various forms of IQ tests Chapter 01 ‣ For score data, each Participant may contribute many units to each value. PSYC 300B — Statistical Methods in Psychology II — © 2024 David A. Medler 20 Dependent Variable (DV): Ratio Scale Ratio Scale: Frequency ‣ How often or how much in terms of behaviour. Ratio Scale: Duration or Size ‣ Measurements of ‣ Number correct ‣ Percentage correct ‣ Proportion missed Chapter 01 continuous variables ‣ Time Measures (RTs, Age). ‣ Length Measures ‣ Weight Measures ‣ Some Physiological Measures PSYC 300B — Statistical Methods in Psychology II — © 2024 David A. Medler 21 Measurement Scales & DVs Scale Nominal Ordinal Data Frequency, Tally, Ranks Count Example # who say yes Likert Question Preferences Ranks e.g., 1st, 2nd, … Chapter 01 Interval Ratio Scores Scores Self-report scales % Correct Behavioural # wrong measures Reaction Time e.g., Self-esteem; Height; Weight BDI; IQ; Intro-/ Extroversion PSYC 300B — Statistical Methods in Psychology II — © 2024 David A. Medler 22 Implications of Scale of Measurement for DV ‣ Statistical Analysis is applied to data. ‣ Type of Descriptive measures used ‣ Choice of Statistical Test used to analyze data ‣ Interpretation of outcome ‣ Qualitative DV: Describe relationship between IV & DV. ‣ Quantitative DV: Association between IV & DV. ‣ Cause & Effect: IV causes a change in DV (behaviour). ‣ Note: Three Criteria for Interpretation of Cause & Effect: 1. IV is manipulated, 2. Data are scores, and 3. Participants are randomly sampled/assigned to levels of the manipulated IV. Chapter 01 PSYC 300B — Statistical Methods in Psychology II — © 2024 David A. Medler 23 Goals of Research 1. Examine whether there is a systematic relationship or association between two variables (IV & DV). 2. Control bias in data to better understand what caused the relationship between variables or Participants’ change in behaviour. 3. Establish generality from sample to population. Chapter 01 PSYC 300B — Statistical Methods in Psychology II — © 2024 David A. Medler 24 How Well are the Goals Met? ‣ All researchers manage #1 relatively well. ‣ Most researchers try to control for bias (#2) ‣ many fail. ‣ Generalizing from the sample to the population is the hardest to attain ‣ Only a few truly seek generality to the population in the strictest sense by randomly sampling from the population to generate the sample. (In fact, it could be argued that true random sampling for human is impossible!) Chapter 01 PSYC 300B — Statistical Methods in Psychology II — © 2024 David A. Medler 25 Type of Research & Interpretation Sample Created by Survey Quasi-Experiment True Experiment Random assignment may Random assignment of at Random assignment of or may not occur least 1 (manipulated) IV all IVs Independent Variable Measured Manipulated &/or Measured Manipulated only Dependent Variable Qualitative (tallies) Quantitative (scores or ranks) Quantitative (scores) Interpretation Descriptive Association or Correlation Cause & Effect Example Independent Variable Political Party & Gender Gender & Training Method Dependent Variable Count (tally) % Correct on Test Chapter 01 Drug level (mg) & type memory (visual/auditory) Reaction Time PSYC 300B — Statistical Methods in Psychology II — © 2024 David A. Medler 26 PSYC 300B — Homework Assignment #1 m PSYCHOLOGY 300B Homework Assignment 1: Independent & Dependent Variables 1. After years of teaching statistics, I have anecdotally noticed two things; (1) students appear to be sleep deprived, and (2) students seem to consume a lot of caffeine in order to stay awake during my lectures (OK… it is just not students, but myself as well). In September 2021, Stepan, Altmann, & Fenn published an article in the Journal of Experimental Psychology: Learning, Memory, and Cognition on how caffeine selectively affects cognitive performance in sleep deprived individuals. Ah ha! I thought… is caffeine the cure all? Participants in their study first completed two tasks, a placekeeping task [UNRAVEL] and a Psychomotor Vigilance Task [PVT] and then were randomly assigned to either stay up all night in the lab, or go home to sleep. The next morning, the sleep group returned to the lab and all participants were randomly assigned to consume either a placebo or 200 mg of caffeine in capsule form. Participants then completed the two tasks again. Findings showed that sleep deprivation impaired performance on both tasks, but caffeine improved performance in the PVT but not the UNRAVEL task. Based only on information provided above and the graphic figure below… a. Identify the two independent variables (IV), providing either the name of the concept being examined and/or the research definition of that concept. b. State specifically how each IV is operationalized. c. Indicate whether each IV is a manipulated or measured variable. Page 1 of 2 © 2024 David A. Medler >
PSYC 300B — Homework Assignment #1
d. Identify the scale of measurement associated with each IV.
e. Identify the two dependent variables (DV) by giving its verbal name (concept tested or
research definition) and state
f. State specifically how the DVs are being operationalized (it is the same for both).
g. Identify the scale of measurement for the dependent variable.
h. Is the graphic figure appropriate for the scale of measurement of the variable? Justify
your response. Indicate which variable (DV or IV) you referred to when answering this
question.
i. Finally, indicate what type of research study this is and be able to justify your response:
(S) Survey, (Q) Quasi-experiment or (E) experiment
Stepan, M. E., Altmann, E. M., & Fenn, K. M. (2021). Caffeine selectively mitigates cognitive
deficits caused by sleep deprivation. Journal of Experimental Psychology: Learning, Memory,
and Cognition, 47(9), 1371–1382. https://doi.org/10.1037/xlm0001023
Page 2 of 2
© 2024 David A. Medler
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Statistical Analysis
Descriptive Statistics
Hypothesis Testing for a Related
Samples Research Design
Estimation
Interval Estimates
(confidence intervals)
Random Sampling
Score
Design: 1 Participant
You are here
PSYC 300B — Statistical Methods in Psychology II — © 2024 David A. Medler
Statistical Theory
Hypothesis Testing
Point Estimates
Chapter 02
Chapter 02
Inferential Statistics
Bayesian
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Lecture Outline
Chapter 02
Random Assignment
Ranks
Tally
Score
Tally
rS
χ2
Randomization
Test
Binomial Exact
Test
z-test
1-Sample
z-test; t-test; r
2-Sample
t-test
χ2
k>2
F-test (ANOVA)
χ2 goodness of fit
Factorial
F-test (ANOVA)
χ2 contingency table
Fisher’s Exact
Test
PSYC 300B — Statistical Methods in Psychology II — © 2024 David A. Medler
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Problems with the Single Sample Research Design
‣ The Related Samples Design
‣ Distributions for the Related Samples Design
‣ Example for the Related Samples Design
‣ Using R for Related Samples Design
‣ The Single Sample Research Design is based on comparing the
mean of the Sample (X), to the mean of the Sampling Distribution
(μX), which is directly related to the mean of the Population (μ).
‣ The problem is that we are comparing an experimental group to an
assumed control, without knowing what the actual control is…
‣ In other words, we are lacking a baseline measure for our experimental group
‣ The solution to this problem is to establish a baseline control group
‣ The most logical baseline group is to measure our experimental group in both
a baseline condition, and an experimental condition.
Chapter 02
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Chapter 02
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Choice of Tests:
Random Sampling Model of Hypothesis Testing
The Related Samples Research Design
‣ Experimental Design Involves…
‣ If the value for σ known, use…
‣ One group of Participants tested over 2 (or more) occasions or conditions.
‣ For now, we will focus on when each Participant provides 2 data points.
‣ z-test for related samples.
‣ If value for σ unknown (must estimate σ from the sample)
‣ We will represent the first (control/Null) condition as X0 and the second (experimental/
Alternative) condition as X1
‣ t-test for related samples…
‣ The difference between these two data points are then subtracted to produce a single Difference
… as long as no major assumptions are violated.
score (D), such that D = X1 − X0
‣ If major assumptions are violated…
‣ Also called: Repeated Measures; Dependent Measures; Within-Groups;
Paired-Samples
… do data transformations or switch to another form of hypothesis
testing.
‣ When same Participants tested over 2 occasions & data are scores…
‣ Multiple choices for tests of inference (1 major, 3 minor).
Chapter 02
PSYC 300B — Statistical Methods in Psychology II — © 2024 David A. Medler
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Choice of Tests: Other Approaches
Chapter 02
‣ Requires fewer Participants in comparison to an Independent Samples
design.
‣ This model makes no assumptions about population values; the focus is
only on behaviour of Sample Participants.
‣ Fewer external influences on behaviour (confounding variables).
‣ More Powerful (easier to Reject the Null):
‣ Apply a Randomization test to data.
‣ Less variability from individual differences & other influences.
‣ Smaller value for unexplained variability which results in a smaller denominator of
‣ Can we do Estimation/Confidence Intervals?
‣ Can be done but is a bit tricky for this design.
t-ratio due to smaller value for Estimated Standard Error.
‣ Bayesian Statistics with Related Samples
‣ Smaller value for SE produces a larger tobs value so it is easier to reject the null.
‣ Has a baseline score for Participant & measures the change…
‣ There have been many advances using this approach recently, but is beyond
this course.
PSYC 300B — Statistical Methods in Psychology II — © 2024 David A. Medler
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Advantages of Related Samples Design
‣ Random Assignment Model of Hypothesis Testing
Chapter 02
PSYC 300B — Statistical Methods in Psychology II — © 2024 David A. Medler
‣ we are testing the change in score, not where they are in the original distribution.
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Chapter 02
PSYC 300B — Statistical Methods in Psychology II — © 2024 David A. Medler
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Lecture Outline
Disadvantages of the Related Samples Design
‣ Carry-over or practice effects confound interpretation.
‣ The Related Samples Design
‣ Distributions for the Related Samples Design
‣ Example for the Related Samples Design
‣ Using R for Related Samples Design
‣ Are the changes simply due to exposure to the same test twice?
‣ Loss of Participant through attrition.
‣ If you lose a participant at the second time point, then you must discard
their data from the first time point.
‣ Cannot determine cause & effect if it is a time-based study
‣ Why might this be?
Chapter 02
PSYC 300B — Statistical Methods in Psychology II — © 2024 David A. Medler
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Chapter 02
PSYC 300B — Statistical Methods in Psychology II — © 2024 David A. Medler
Random Sampling Model of Hypothesis Testing:
The Three Distributions
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Difference = After – Before
Before (X0)
Population
‣ A single distribution of Difference scores (D = X1 − X0)…
After (X1)
‣ Not single values for X, but the difference between two related scores, X1 and X0
‣ Will have a mean of μD… then mean of the single distribution of Difference scores.
Sampling Distribution
‣ Distribution of all possible Mean Difference scores, D, based on a Sample size, n.
P3B
P1B
P2B
-1.5
0.5
2.25
Difference
P3A P2A
P1A
-1.5 -0.75
2.0
Sample
‣ Distribution of Difference scores (D) that has a mean difference score (D).
‣ Computed from 2 scores per participant: D = X1 − X0.
Chapter 02
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Hypothesis Testing of the
Related Samples Design
Chapter 02
-3.0
0.0
1.5
P2D
P3D
P1D
PSYC 300B — Statistical Methods in Psychology II — © 2024 David A. Medler
Characteristics of the Population Distribution
‣ Critical Value
‣ Data Represent
‣ Distribution of Difference scores (D = X1 − X0).
‣ Random behaviour as if Null Hypothesis is true.
‣ Determined from t-Table of Critical values.
‣ Based on values for df, α, and directionality of hypothesis
‣ Test Ratio for Related-samples t-test
‣ Descriptive Characteristics
‣ The statistic for analysis of the related samples design is D
□ − μ□
D − μD
Mean Difference
t□ =
→ tD =
=
s□
sD
Estimated Standard Error
‣ Variability: σD is unknown (estimated by sD from sample).
‣ Shape: Usually unknown, but is assumed to be a normal distribution.
Chapter 02
PSYC 300B — Statistical Methods in Psychology II — © 2024 David A. Medler
‣ Central Tendency: μD = 0
‣ Assumes “no difference” between two trials or conditions; there is “no change”.
‣ If Sample size n ≥ 25 − 30, shape of Population is not relevant*.
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Chapter 02
PSYC 300B — Statistical Methods in Psychology II — © 2024 David A. Medler
Sampling Distribution of Mean Difference Scores
(when σ is not known)
Sampling Distribution of Mean Difference Scores
(when σ is not known)
‣ Data for Sampling distribution represent
‣ Descriptive Characteristics
‣ True value for σD is unknown, so σD is unknown
‣ We use sD to estimate σD so sD can be computed.
‣ sD: “unbiased estimate of of the distribution of mean difference scores”, or…
‣ Descriptive Characteristics
“Estimated Standard Error of mean difference scores”
‣ Shape of Distribution
‣ Central Tendency: μD = 0 (same as Population)
‣ Unimodal & symmetrical
‣ family of curves that varies as a function of df.
‣ Represents the mean of the mean difference scores.
‣ Is equal to the mean of the population of difference scores.
PSYC 300B — Statistical Methods in Psychology II — © 2024 David A. Medler
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‣ Variability: sD = sD / n
‣ Distribution of Mean Difference scores.
‣ Random behaviour or “no difference”.
‣ Is called the “distribution of mean difference scores”.
Chapter 02
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Determined from the Distribution of the t-statistic / t-scores.
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Chapter 02
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Defining the Sample Distribution
Create the Sample Distribution
‣ Sample Distribution is created in four steps
‣ The sample is based on the difference between two scores, X0 & X1,
for each Participant, such that D = X1 − X0
‣ Step #1: Take measurement of the first (Control/Null) condition or time
(scores for X0).
‣ These scores are collected in one of two ways:
‣ Step #2: Manipulate or Measure the IV
‣ Step #3: Take measurement of the second (Experimental/Alternative)
‣ Time-based studies where the same condition is tested over time
(cause & effect cannot be established… why?)
condition or time (scores for X1 ).
‣ e.g., Studies where drugs are given over time
‣ Step #4: Compute a column of difference scores for each pair of scores:
‣ Condition-based studies where two conditions are presented within the same
time-frame
(cause & effect can be established if conditions are met)
This creates a Sample of Difference Scores D = (X1 − X0)
‣ It is essential that the scores from each Participant are lined up properly to
‣ e.g., Word vs. Non-word Discrimination
Chapter 02
PSYC 300B — Statistical Methods in Psychology II — © 2024 David A. Medler
compute the difference.
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Chapter 02
PSYC 300B — Statistical Methods in Psychology II — © 2024 David A. Medler
Computing Statistics for the
Related Samples Design
Assumptions of the Related Samples t-test
‣ Compute Mean for the control (X0) and experimental condition (X1)
‣ Compute Mean D, SSD, sD on difference scores
‣ Compute sD using df adjustment (n − 1).
‣ Data (dependent variable) are scores.
‣ Distribution of the Population of Difference scores is normally
distributed.
n (# of scores in Difference column) −1 because one population
parameter, σD, is estimated.
‣ Participants were randomly sampled so that the Sample is
‣ Statistic for analysis: D (Mean of Difference scores)
‣ n ≥ 7 (for the study)
representative of the Population.
‣ Shape of the distribution of values is defined by data.
‣ Remember, each Participant is in both conditions/time points, so each
‣ It is important that the shape of the distribution is not severely skewed.
Chapter 02
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group will have the same number of Participant (which means at least 7
scores in each group).
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Chapter 02
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Application of Hypothesis Testing to
Related Samples Research Design
Lecture Outline
‣ The Related Samples Design
‣ Distributions for the Related Samples Design
‣ Example for the Related Samples Design
‣ Using R for Related Samples Design
‣ The Problem: Vancouver Island Marmot population is in decline
‣ The Solution: Increase mating behaviour
‣ Previous observations have shown that Viagra® has been successfully used in
captive breeding programs.
‣ The Study:
Ps: 9 Vancouver Island Marmots (Alex, Bailey, Cameron, Devon, Emery, Finley,
Grayson, Harper, Indy) randomly sampled from the remaining population
IV: Effect of 10 mg of Viagra® ingested daily for one month
DV: Mean difference in the # of attempted matings in 1 week following ingestion
of 10 mg of Viagra® daily for 1 month.
Chapter 02
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Chapter 02
Steps in Hypothesis Testing:
Step 1: State Alternative & Null Hypothesis
‣ Step 2: Select Sampling Distribution
Model of HT:
Research Design:
Type of Data:
Statistic for Analysis:
Population Parameters:
Sample Size
‣ Step 1: State H1 & H0
H1 : μD ≠ 0
‣ There will be a change in the mean number of weekly matings for V.I.
Marmots following daily ingestion of 10 mg Viagra after 1 month of treatment.
‣ H0 : D = μD or
H0 : μD = 0
‣ “Distribution of Mean Difference scores when σ unknown for n = 9”
‣ “Distribution of Mean Difference scores for t-scores when df = 8”
Marmots following daily ingestion of 10 mg Viagra after 1 month of treatment.
PSYC 300B — Statistical Methods in Psychology II — © 2024 David A. Medler
Random Sampling Model of Hypothesis Testing
Related Samples Research Design
Scores (Mean change in Matings)
Mean Difference Score (D)
μD = 0 , σD is unknown and shape unknown
n=9
‣ Choice of Sampling Distribution
‣ There will be no change in the mean number of weekly matings for V.I.
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Steps in Hypothesis Testing:
Step 2: Select the Sampling Distribution
Steps in Hypothesis Testing
‣ H1 : D ≠ μD or
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Chapter 02
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Steps in Hypothesis Testing
Step 3: Set the A Priori Criterion
Critical Values of the t-Distribution for df = 8
‣ Step 3: Set the a priori criterion for p(α) = .052−tailed
‣ t-scores are standardized scores based on α and df
∞
10
5
2
-4
-3
-2
-1
0
1
2
3
4
-5
df
5
7
8
9
‣ p(α) = .052−tailed and df = 8, so t(8)crit = ± 2.3060
Chapter 02
∞
PSYC 300B — Statistical Methods in Psychology II — © 2024 David A. Medler
51
.20
.50
0.7111
0.7064
0.7027
0.6745
.40
0.8960
0.8889
0.8834
0.8416
Chapter 02
Chapter 02
X0 − X0
(X0 − X0)
X1
9 – 4.0000 = 5.0000
25.0000
10
3 – 4.0000 = -1.0000
1.0000
8
4 – 4.0000 = 0.0000
0.0000
9
3 – 4.0000 = -1.0000
1.0000
4
2 – 4.0000 = -2.0000
4.0000
3
3 – 4.0000 = -1.0000
1.0000
2
1 – 4.0000 = -3.0000
9.0000
4
7 – 4.0000 = 3.0000
9.0000
9
4 – 4.0000 = 0.0000
0.0000
5
0.0000 SS = 50.0000 ΣX1 =
54
s 2 = 6.2500 X1 = 6.0000
s = 2.5000
X0
Marmot
A
9
B
3
C
4
D
3
E
2
F
3
G
1
H
7
I
4
ΣX0 =
36
X0 = 4.0000
X1 − X1
(X1 − X1)
10 – 6.0000 = 4.0000
16.0000
8 – 6.0000 = 2.0000
4.0000
9 – 6.0000 = 3.0000
9.0000
4 – 6.0000 = -2.0000
4.0000
3 – 6.0000 = -3.0000
9.0000
2 – 6.0000 = -4.0000
16.0000
4 – 6.0000 = -2.0000
4.0000
9 – 6.0000 = 3.0000
9.0000
5 – 6.0000 = -1.0000
1.0000
0.0000 SS = 72.0000
s 2 = 9.0000
s = 3.0000
53
t-Statistic: tD =
n
2.0000
9
Chapter 02
0
1
2
3
4
Level of Significance for One-Tailed Test
.10
.05
.025
.01
Level of Significance for Two-Tailed Test
.30
.20
.10
.05
.02
1.1192
1.4149
1.8946
2.3646
2.9980
1.1081
1.3968
1.8595
2.3060
2.8965
1.0997
1.3830
1.8331
2.2622
2.8214
1.0364
1.2816
1.6449
1.9600
2.3263
.15
5
.005
.0025
.0005
.01
3.4995
3.3554
3.2498
2.5758
.005
4.0293
3.8325
3.6897
2.8070
.001
5.4079
5.0413
4.7809
3.2905
52
10
8
9
4
3
2
4
9
5
ΣX1 =
54
X1 = 6.0000
D = (X1 − X0)
1
5
5
1
1
-1
3
2
1
18
ΣD =
D = 2.0000
D−D
1 – (2.0000) = -1.0000
5 – (2.0000) = 3.0000
5 – (2.0000) = 3.0000
1 – (2.0000) = -1.0000
1 – (2.0000) = -1.0000
-1 – (2.0000) = -3.0000
3 – (2.0000) = 1.0000
2 – (2.0000) = 0.0000
1 – (2.0000) = -1.0000
0.0000
(D − D)2
1.0000
9.0000
9.0000
1.0000
1.0000
9.0000
1.0000
0.0000
1.0000
SSD = 32.0000
2
sD = 4.0000
sD = 2.0000
PSYC 300B — Statistical Methods in Psychology II — © 2024 David A. Medler
54
‣ t(8) = 3.0000 > t(8)crit = ± 2.3060
‣ Statistical Decision: Reject the Null!
= 0.6667
‣ Determine the observed p-value:
D − μD 2.0000 − 0.0000
=
= 3.0000
sD
0.6667
‣ Observed t-value is between ± 2.8965 and ± 3.3554, so p < .02* df X0 = 4.00, s0 = 2.50, X1 = 6.00, s1 = 3.00, D = 2.00, sD = 2.00 SE = 0.67, t(8) = − 3.00 PSYC 300B — Statistical Methods in Psychology II — © 2024 David A. Medler X1 ‣ Step 5: Compare tobs to tcrit and make a statistical decision ‣ Reported statistics: Chapter 02 -1 Steps in Hypothesis Testing: Step 5: Compare Observed & Critical Values ‣ Compute the Standard Error and the test statistic = -2 PSYC 300B — Statistical Methods in Psychology II — © 2024 David A. Medler Steps in Hypothesis Testing Step 4: Gather Data and Analyze Standard Error: sD = -3 ‣ Step 4: Compute Descriptive Statistics for the Differences 2 PSYC 300B — Statistical Methods in Psychology II — © 2024 David A. Medler sD p = .001 t = + 5.0413 Steps in Hypothesis Testing Step 4: Gather Data and Analyze ‣ Step 4: Compute Descriptive Statistics for the two time points X0 Marmot A 9 B 3 C 4 D 3 E 2 F 3 G 1 H 7 I 4 ΣX0 = 36 X0 = 4.0000 -4 .2