Psycstat exam #2

probability (p)

• likelihood of event occuring

• specific outcomes/total outcomes

• random sampling required

random sampling

equal chance of being selected

constant probability - sampling w/ replacement

role of probability in inferential statistics

determine if sample is likely or unlikely to occur by chance if null hypothesis is true

Unit Normal Table

used to find proportion (probability) corresponding to z-score

what makes a distribution of sample means normal

• population samples obtained from is normal

• sample size: n ≥ 30

sampling error

natural discreprenancy between sample statistic and population parameter

central limit theorem

rules for defining distributions for sample means

central limit theorem - shape

Nearly Normal (if n≥30 or population is normal)

central limit theorem - central tendency

Mean (μM​) is equal to the population mean (μ)

central limit theorem - variability

Measures the average distance between a sample mean (M) and μ. '

• Formula: σM​=σ/n​

alpha level (α)

significance level, defines “unlikely” (critical region) region

• maximum probability of Type I Error

outcome of hypothesis test by alpha level (α)

• p-val < α → reject H₀

• p-val > α → fail to reject H₁

• large α (.05) → lower standard → Higher chance of rejecting H₀

• small α (.01) → higher standard → Lower chance of rejecting H₀.

Type I Error

false positive, rejecting a true null hypothesis

• controlled by setting alpha value

• ex: telling a man he’s pregnant

Type II Error

False negative, failing to reject a false null hypothesis

• occurs when effect is too small to observe in small sample

• ex: telling a pregnant woman she’s not pregnant

p-value

probability of obtaining observed sample result (or more extreme) if H₀ is true

• if p-val ≤ α, Reject H₀

likelihood of committing Type I error when p = .05

5% when H₀ is rejected

why calculate effect size (Cohen’s d or r²)?

calculate effect size in addition to statistical significance bc significant result might be too small to have any practical importance

Cohen’s d

mean difference in standard deviation units

• influenced by SD (σ)

  • larger SD (σ) = more variability → decrease d

• no influence by sample size (n)

Statistical power

probability of correctly rejecting false Null Hypothesis

factors that influence statistical power

• effect size (larger effect = more power)

• sample size (larger n = more power)

• alpha level (larger α = more power)

• direction vs. non-directional hyp (Directional hyp. concentrates α on one side, increasing power)

when to use t-test vs. z-score

t-test: when population SD (σ) is UNKNOWN

• estimate σ using sample SD (s)

t-test

• use sample data to test hypothesis about diff between sample mean & pop. mean

• test hyp. for unknown pop. (both μ & σ UNKNOWN)

• requires sample & reasonable hyp. about μ

assumptions of t-test

1. interval or ratio scale

2. randomness → randomly sampled from pop.

3. homogeneity of variance → similar variability of data in each group

4. normality → sample pop. normally distributed

5. independence → independent observations

one-sample t-test

sample mean vs. pop. mean

• top = diff between M and hypothesized μ (SIGNALl)

• bottom = estimated standard error (NOSIE) → more variability

• df = n -1

influence of sample size & variance on t-test (one sample)

larger sample size (n) → more normal t-distribution (more power)

smaller sample variance (s²) → larger t-stat (more likely to reject null hyp.)

cohen’s d (effect size) - one sample

r²

percentage of variance accounted for by IV

SPSS output (one sample t-test)

Descriptive stats:

• N = sample size

• Sample Mean (M) - value compared against test value

• Std. Deviation (s) - variability of scores in sample

• Std. Error Mean (S_M) - denominator of t-ratio - avg. distance between sample mean and pop mean

Inferential stats:

• test value (μ0​) - Hypothesized Mean - compare to sample mean (M)

• t - t-statistic

• df - N-1 - find critical value

• Sig. (2-tailed) - p-val

• Mean diff. - M−μ₀​ - diff. between sample mean & test value - used to find Effect Size

compare p-val (sig.) from table 2 to alpha level:

• p-val ≤ α → Reject H_0​

• p-val > α → Fail to Reject H0_​

independent measures t-tests (independence samples)

two speerate & indepdent sample groups (ex: male vs. female)

determines whether sample mean diff indicates

• real diff between pop. means

• or obtained diff due to sampling error

repeated measures t-tests (dependent samples)

one group measured twice (ex: pre-test vs. post-test)

hypothesis for nondirectional two-tailed (independent-measures test)

H₀: μ1 = μ2

H₁: μ1 ≠ μ2

locating critical region (independent-measures)

1. df = (n1-1) + (n2-1)

2. look up corresponding val in t-distribution table

calculating t-stat (indepedent-measures)

making decision based on t-stat (indepdent-measures)

if t-value more extreme than critical val (i.e. p-val ≤ α)→ reject null hyp.

effect size (indepdent-measures)

assumptions of independent measures t-test

1. interval or ratio

2. randomness

3. homogeneity of variance: similar variance

4. normality

5. independence

SPSS output (ANOVA)