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Statistics :fire: (Hypothesis testing (Step (2: Set criteria for decision,…

- - - - Null hypothesis (H0): no significant change/ difference/relationship in general population
      - Alternative hypothesis (H1): a significant change/ difference/relationship in general population
    - - Alpha level (level of significance): probability value to define “very unlikely”
        
        Commonly α = .05
      - Critical region: extreme sample values that are very unlikely
        
        Critical values of critical
        region determined by alpha level
      - p-value: actual probability of specific value of test statistic (lower is better)
    - - Compute sample statistic (e.g. z-score)
    - - Reject null hypothesis if sample data in critical region (p-value < significant level)
      - Fail to reject null hypothesis if sample data not in critical region (p-value > significant level)
        
        Statistically significant if very unlikely to occur (i.e. 5% probability)
    - - 1 - H0: μ special training = 65
        H1: μ special training ≠ 65
        2 - α = .05
        (Critical region: z beyond ±1.96)
        3 - zobtained = +1.67 < zcritical
        4 - Fail to reject H0
        
        z = M – μ / σM
        = (70 – 65) / 3
        = +1.67
      - APA format
        
        State significance
        
        Report value of test statistic
        
        Report alpha level or p-value
        
        A one-sample z-test was conducted to examine if a sample of 25 people who participated in a special training in reading skills programme show an improvement in reading skill scores.
        While reading skill scores are higher (M = 70) than the population in general (μ = 65, σ = 15), the difference is not statistically significant, z = 1.67, p > .05.
  - - - Wrongly rejects null hypothesis
      - Concluding a treatment has
        effect when it has none
      - Alpha level: probability of Type I error
        
        Lower probability by
        lowering alpha level (e.g. use α = .01)
    - - Fail to reject null hypothesis that
        is really false
      - Failed to detect real treatment effect
      - Lower probability by increasing alpha level
  - - - d = Mean difference ÷ Standard deviation
        d = (μ treatment - μ no treatment) / σ
      - Magnitude and evaluation
        
        Small effect: d = 0.2
        
        Medium effect: d = 0.5
        
        Large effect: d = 0.8
  - - - Flaw
        
        Requires more information than available
        
        Needs knowledge of pop. standard deviation (σ), but usually have only sample data
        
        Sample means (M) and standard deviations (SD) used to estimate pop. parameters
- - - - Affected by
        
        Size of sample
        
        Standard deviation of population or population variance
- - - - Example
        An independent-samples t-test showed that employees with normal reporting time (n=127) have statistically significant lesser sleeping time (M = 347.59 min, SD = 83.48) as compared to employees with delayed reporting time (n=130, M = 380.12 min, SD = 95.01), t (255) = -2.92, p = .004 (two-tailed), d = -0.36.
      - Numbers to 2 decimal places
        
        Initials to be italicized
        
        If p-value < .05, report exact figure
        
        p-values ⩾ .05, state p > .05
  - - - Same subjects used in all treatment conditions, two separate scores for each individual
      - Advantage
        
        Requires fewer subjects
        
        Study changes over time
        
        Reduces influence of individual differences
      - Disadvantage
        
        Extraneous factor may cause change in score
        
        Participation in first treatment may influence score in second treatment (order effects)
      - Assumption
        
        Observations within each treatment condition are independent
        
        Pop. distribution of different scores must be normal (ignore for large sample size when n > 30)
- - - - Sample - selected representation to generalise findings back to entire population
        
        Sampling error - discrepancy between sample statistic and population parameter
    - - Descriptive
        
        Procedure to summarise, organise, simplify and presenting
        
        E.g. mean and median or frequency distribution
      - Inferential
        
        Technique to study sample and make generalisation to population
        
        E.g. hypothesis-testing, correlation and one-way ANOVA
    - - Qualitative (non-numeric)
      - Quantitative (numeric)
        
        Discrete - certain values, separate categories, whole number from count (e.g. no. of cars/students/rooms)
        
        Continuous - divisible values within range, decimal from measuring (e.g. weight/GPA/air pressure)
      - Independent (IV, manipulated) and Dependent (DV, resultant)
        
        Control: extraneous
        
        Random assignment
        
        Matching pairs of subjects on similarities on variables
        
        Holding variables constant
    - - Nominal (assigned name) and ordinal (sequential ranking)
      - Interval (equidistant, no true zero) and ratio (absolute zero)
  - - - Frequency table
        
        Left column: categories (X value, highest to lowest)
        
        Right column: frequency (f)
        
        Computation
        
        Total no. of scores (by adding total no. of individuals)
        
        Σf = N
        
        Sum of scores (sigma X)
        
        ΣX
        
        E.g. ΣX = 10 + 10 + 9 + …
        
        Grouped (if range is too wide, ~10 class intervals)
      - Bar chart (gap between bars) for qualitative, histogram (no gap) for quantitative or frequency polygon (line connecting dots)
        
        Vertical Y-axis: frequency
        
        Horizontal X-axis: category
      - Graph
        
        Shape (curve)
        
        Normal (mean, median and mode all same at centre, "bell curve")
        
        Skewed: positive (scores clustered bottom of range, left peak, mode-median-mean, Mean–Median > 0) or negative (scores clustered top of range, right peak, mean-median-mode, Mean–Median < 0)
        
        Central tendency (centre of distribution)
        
        Mean
        
        Population mean = Sum of scores ÷ No. of pop. scores
        μ = ΣX / N
        
        Sample mean = Sum of scores ÷ No. of sample scores
        x̅ = ΣX / n
        
        Median
        
        50th percentile, 2 groups equal size
        
        Mode
        
        Greatest frequency, any level of measurement, correspond to an actual score in data and can be bimodal/no mode
        
        Variability
        
        Spread/Clustering of scores (wide/ narrow range)
        
        Measures how well an individual score represents whole distribution
        
        High/Narrow peak (little diff. between scores) = less variability Low/Broad peak (wider diff. between scores) = more variability
        
        Measure
        
        Range (distance covered by scores, lowest to highest)
        
        Range = URL Xmax - LRL Xmin
        
        Unreliability: easily affected by extreme scores
        
        Standard deviation (average distance from mean)
        
        Deviation = X - μ
        
        Sum of deviations = 0 because mean is the balance point, so mean deviation always = 0
        
        Pop. variance = mean squared deviation
        σ^2 = Σ(X - μ)^2 / N
        
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        Hence, pop. standard deviation = √ variance = √ (SS / N)
        
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        Variance
        
        Pop. variance = Sum of squared deviations (sum of squares) ÷ No. of pop. scores
        σ^2 = SS / N
        
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