PSYCH 201 Statistics Test 2

The true nature of most variables is to distribute ___

normally

___ ___ usually gets a distribution closer to distributing normally

More data

Ways to describe types of curves:

• Skewness

• Kurtosis

• Modality

Skewness

What direction the tail is pulled; where the outliers are

• Positive skew

• Negative skew

Kurtosis

Describes the observed data around the mean; the “tailedness” of the distribution

• Leptokurtic: tall; tails are “fatter” (more tail relative to the center)

• Mesokurtic: normal

• Platykurtic: flat, even distribution; tails are “skinnier” (less tail relative to the center)

Modality

The number of “peaks” (modes) in the distribution

• Uniform (0)

• Unimodal

• Bimodal

• Multimodal

Population vs. sample

Population

• Parameters

• μ (mu; mean)

• σ (sigma; standard deviation)

Sample

• Statistics

• x̄ (mean)

• S (standard deviation)

Z-score

The number of standard deviations a score is from the mean; standardizes units

• Allows for comparison of extreme scores across populations

• Allows for comparison of extreme scores across measurement scales

• Determines percentiles and outliers

What is considered an extreme z-score?

Around -2, around +2

What is considered an outlier, assuming a normal distribution and a p-value of 0.05?

z = -1.96 (or less), z = +1.96 (or more)

• The sign signifies direction

• Falls into the last 2.5% (5%) total; tail ends

  • So this is a 95% confidence interval

z-score equation

z = (x-μ) / σ

• z = z-score

• x = sample mean

• μ = population mean

• σ = population standard deviation

Steps of hypothesis testing:

1. State null and alternative hypotheses

2. Set decision rule (Where is the cutoff for significance?)

   1. One-tailed test

   2. Two-tailed test (gen. practice)

3. Calculate statistic

Null hypothesis

The effect being studied does not exist; no statistical significance exists

• y = x

Alternative hypothesis

A direct contradiction of the null hypothesis; statistical significance exists

• y > x

• y < x

z-test

Continuous variable compared to a population with a known standard deviation

z-test equation

z = (x̄ -μ) / σx̄

• σx̄ = σ / √N

  • σx̄: population standard error of the mean

With a sample size under ___, you cannot…

30, assume a normal distribution

Critical value

Depending on the sample size, the value that indicates the threshold of significance; if z or t are beyond it, then it is statistically significant

A smaller sample size (N) will…

increase the critical value (CV)

One-sample t-test

Use when comparing one sample to the population mean, and if the population mean is estimated and the standard deviation is unknown

One-sample t-test formula

t = (x̄ -μ) / Sx̄

• Sx̄ = S / √N

  • Sx̄: standard error of the sample mean

Tests of statistical significance:

• z score

• z test

• One sample t-test

• Independent sample t-test

• Paired sample t-test

Z score tests…

a single score against a sample

Z test tests…

a sample against a population (with a known standard deviation)

One sample t-test tests…

a sample against a population (with an unknown standard deviation)

Independent sample t-test tests…

two independent samples against each other

Paired sample t-test tests…

paired samples against each other

Between-subjects design

Group A vs. Group B; every participant experiences only one condition, and you compare group differences between participants in various condition

• Participant effects

• Independent samples t-test

Within-subjects design

Group A.1 vs. Group A.2; every participant experiences every condition (at diff. times)

• Order effects

• Paired samples t-test

Degrees of freedom (df)

The maximum number of logically independent values

• Formula: N - 1 (per group)

  • E.g., two separate samples of 20; df=18

  • E.g., one sample of 30; df=29

Independent sample t-test formula

t = (x̄1 - x̄2) / (Sx̄1 - Sx̄2)

• Sx̄1 - Sx̄2 = √(S1² / N1) + (S2² / N2)

  • Sx̄1 - Sx̄2: standard error of the difference in means

Use an independent samples t-test if…

• Comparing one sample to another sample

• Samples are independent

  • IV is categorical

  • DV is continuous

When estimating t-value, look at…

• Sample size

• Means

• Variance

Paired sample t-test formula

t = x̄_D - μ_D / S_Dˉ

• S_Dˉ = S_D / √N

  • x̄_{D =} Σ(x₁ - x₂) / N

  • S_D = √Σ((x₁ - x₂) - x_D) ² / df

Use a paired samples t-test if…

• Comparing one sample to another sample

• The samples are paired (same subjects, different conditions)

  • IV is categorical

  • DV is continuous

One-tailed hypothesis

When the hypothesis has ONE direction of interest (EITHER greater than the null or less than the null)

One-tailed test significance (p = 0.05)

5% error in one direction (one tail)

• This may make it easier to attain significance, as the margin is entirely concentrated on one side

Two-tailed hypothesis/test

When the hypothesis has no particular direction of interest (greater than OR less than the null)

Two-tailed test significance (p = 0.05)

2.5% error in BOTH directions (both tails)

• This may make it more difficult to attain significance, as there is a smaller margin

How effect size influences t-value

Larger effect size leads to higher t-value

How power influences t-value

Higher power increases t-value

How variability influences t-value

More variability decreases t-value