COSC 341 Human Computer Interaction. Dr. Bowen Hui University of British Columbia Okanagan
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1 COSC 341 Human Computer Interaction Dr. Bowen Hui University of British Columbia Okanagan 1
2 Last Class Introduced hypothesis testing Core logic behind it Determining results significance in scenario when: Comparison distribution has known mean and standard deviation (e.g., from literature) Test sample has 1 individual Next step: extend test sample to more than one individual Key concept: distribution of means 2
3 Previously Determining results significance in scenario when: Test sample has 1 individual Comparison distribution is a distribution of individual scores Compare single score to distribution of individual scores 3
4 Now Determining results significance in scenario when: Test sample has more than 1 individual Use mean of test sample Comparison distribution is? Cannot compare mean score to distribution of individual scores Need to compare: mean score to distribution of means 4
5 Distribution of Means A distribution of means is a distribution of means with lots and lots of samples of the same size Each sample randomly taken from the same population Also known as sampling distribution Need to revise Step 2 of hypothesis testing First, understand concept of distribution of means Then, revise Step 2 5
6 How to Build a Distribution of Means 90,000 pieces of corn in tub 6
7 How to Build a Distribution of Means 90,000 pieces of corn in tub labels of 1 on 10,000 corn, 2 on 10,000 corn,, 9 on 10,000 corn. tub = random sample of corn 7
8 How to Build a Distribution of Means 90,000 pieces of corn in tub labels of 1 on 10,000 corn, 2 on 10,000 corn,, 9 on 10,000 corn. tub = random sample of corn draw 2 corn, record labels (e.g., 2 and 9, mean 5.5) put corn back into tub 8
9 How to Build a Distribution of Means 90,000 pieces of corn in tub labels of 1 on 10,000 corn, 2 on 10,000 corn,, 9 on 10,000 corn. tub = random sample of corn draw 2 corn, record labels (e.g., 2 and 9, mean 5.5) put corn back into tub draw 2 corn, mean 4 draw 2 corn, mean 4.5 repeat build histogram with these means 9
10 How to Build a Distribution of Means Growing distribution, as N increases 10
11 Determining Characteristics of a Distribution of Means Mean of distribution: Same as mean of distribution of individuals Equal when we take an infinite number of samples Variance of distribution: Less spread out than distribution of individuals Equal to the variance of distribution of individuals divided by number of individuals in each sample Standard deviation of distribution: Equals to the square root of variance of distribution by definition Shape of distribution: Approximately normal if either (a) each sample is 30+ individuals or (b) distribution of individuals is normal 11
12 Determining Characteristics of a Distribution of Means Mean of distribution: Same as mean of distribution of individuals Equal when we take an infinite number of samples Variance of distribution: Less spread out than distribution of individuals Equal to the variance of distribution of individuals divided by number of individuals in each sample Standard deviation of distribution: Equals to the square root of variance of distribution by definition Shape of distribution: Approximately normal if either (a) each sample is 30+ individuals or (b) distribution of individuals is normal 12
13 Determining Characteristics of a Distribution of Means Mean of distribution: Same as mean of distribution of individuals Equal when we take an infinite number of samples Variance of distribution: Less spread out than distribution of individuals Equal to the variance of distribution of individuals divided by number of individuals in each sample Standard deviation of distribution: Equals to the square root of variance of distribution by definition Shape of distribution: Approximately normal if either (a) each sample is 30+ individuals or (b) distribution of individuals is normal 13
14 Determining Characteristics of a Distribution of Means Mean of distribution: Same as mean of distribution of individuals Equal when we take an infinite number of samples Variance of distribution: Less spread out than distribution of individuals Equal to the variance of distribution of individuals divided by number of individuals in each sample Standard deviation of distribution: Equals to the square root of variance of distribution by definition Shape of distribution: Approximately normal if either (a) each sample is 30+ individuals or (b) distribution of individuals is normal 14
15 Example Scores of students who took GRE: distribution is approximately normal, mean = 500, standard deviation = 100, what will be the characteristics of the distribution of means for samples of 50 students? Given: μ = 500 σ = 100 N = 50 15
16 Example Scores of students who took GRE: distribution is approximately normal, mean = 500, standard deviation = 100, what will be the characteristics of the distribution of means for samples of 50 students? Given: μ = 500 σ = 100 N = 50 Comparison distribution: μ M = μ = 500 σ 2 M = σ 2 /N = /50 = 200 Shape: normal since N > 30 16
17 Revised Calculations for Distribution of Means (Z test) Recall the 5 steps to hypothesis testing: 1. Identify Population 1, Population 2, H 0, H R 2. Determine characteristics of comparison distribution 3. Determine cutoff score for rejecting H 0 4. Determine the sample score on the comparison distribution 5. Decide to reject H 0 or not 17
18 Revised Calculations for Distribution of Means (Z test) Recall the 5 steps to hypothesis testing: 1. Identify Population 1, Population 2, H 0, H R 2. Determine characteristics of comparison distribution 3. Determine cutoff score for rejecting H 0 4. Determine the sample score on the comparison distribution 5. Decide to reject H 0 or not use distribution of means as the comparison distribution 18
19 Revised Calculations for Distribution of Means (Z test) Recall the 5 steps to hypothesis testing: 1. Identify Population 1, Population 2, H 0, H R 2. Determine characteristics of comparison distribution 3. Determine cutoff score for rejecting H 0 4. Determine the sample score on the comparison distribution 5. Decide to reject H 0 or not previously: Z = (X μ)/σ now: Z = (M μ M )/σ M 19
20 Worked Out Example Does being told a person with positive personality increases ratings of that person s attractiveness? Study: Asks 64 randomly selected students to rate attractiveness of individual in photo Photo description: positive qualities (kindness, warmth, sense of humour, intelligence) Rating: 0 (least attractive) to 400 (most attractive) Sample mean =
21 Worked Out Example Prior research of attractiveness with no description 21
22 1. Setting up Hypotheses Population 1: Population 2: H R : H 0 : 22
23 1. Setting up Hypotheses Population 1: Students who are told person in photo has positive personality. Population 2: Students who are told nothing about person in photo. H R : H 0 : Which is the control group? 23
24 1. Setting up Hypotheses Population 1: Students who are told person in photo has positive personality. Population 2: Students who are told nothing about person in photo. H R : People with positive personalities are found to be more attractive than people we know nothing about. H 0 : People with positive personalities are found to be just as attractive or less than people we know nothing about. Is this a one- tailed or two- tailed test? 24
25 2. Characteristics of the Comparison Distribution Given, about test sample: N = 64 Given, about population 2: μ = 200 σ 2 = 2,304 σ = 48 25
26 2. Characteristics of the Comparison Distribution Given, about test sample: N = 64 Given, about population 2: μ = 200 σ 2 = 2,304 σ = 48 Comparison distr.: μ M = μ = 200 σ 2 M = σ 2 /N = 2,304/64 = 36 shape: normal since N > 30 26
27 3. Cutoff Sample Score on Comparison Distribution Use 5% (not sensitive data) Is this a one- tailed or two- tailed test? Recall: H R : People with positive personalities are found to be more attractive than people we know nothing about. Cutoff Z score is? 27
28 3. Cutoff Sample Score on Comparison Distribution Use 5% (not sensitive data) Is this a one- tailed or two- tailed test? Recall: H R : People with positive personalities are found to be more attractive than people we know nothing about. Cutoff Z score is
29 Comparison distribution of means unshaded region: 95% too extreme to be chance 29
30 4. Determine Sample Score on Comparison Distribution M = 220 (test sample s mean from 64 people) Compute Z score for M = 220 Z = (M μ M )/σ M = ( )/6 = 20/6 =
31 Comparison distribution of means Distribution of our sample 31
32 5. Decide to Reject H 0 or Not Is data s Z score more extreme than cutoff? Yes: 3.33 > 1.64 Thus, result of the Z test is statistically significance at p < 0.05 level Interpretation: Being told someone has positive personality increases attractiveness rating of that person 32
33 Summary Distribution of means When it is needed How to build one (from scratch) Determining the characteristics of one Z- test Revised hypothesis testing procedure with: Sample of 1+ individuals A distribution of means as comparison distribution Revised formula: Z = (M μ M )/σ M 33
COSC 341 Human Computer Interaction. Dr. Bowen Hui University of British Columbia Okanagan
COSC 341 Human Computer Interaction Dr. Bowen Hui University of British Columbia Okanagan 1 Last Topic Distribution of means When it is needed How to build one (from scratch) Determining the characteristics
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