Stats 1770 in R

The basics

Symmetrical plots: left skewed and right skewed

• Population size = N
• Central tendency is where the data gathers around
• mean is the average
• median the middle value
• mode most frequent occurring data item
• range largest data item - smallest item
• variance average distance from the mean, \(\sigma^2\)
• standard deviation square root of variance
• proportion (p) the ratio of data items divided by total
• InterQuartile Range (IQR) 3rd quartile minus the 1st quartile
• sample variance \(s^2\)
• sample mean \(x^-\) x with a bar on top

simple stats

sample.pop = 1:20

mean(sample.pop)

[1] 10.5

var(sample.pop)

[1] 35

sd(sample.pop)

[1] 5.91608

Central Limit Theorem = the sample size n taken from population with average \(\mu\) and variance \(\sigma^2\) . If n > 30 then the sampling is approximately Normal Distributed (\(\sigma^2\) / N)

Confidence Interval

• confidence level (90%, 95%, 98%)
• define alpha \(\alpha\) value, [1 - 0.95] \(\alpha\) = 0.05
• find z value \(\alpha\) / 2

IF n > 30 use z-table

• if population is normal and std is known, use z-table
• if population is normal and std is not known, use t-table

IF n < 30, use Degrees of Freedom and t-test

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Ex. weights of Canadian adults in kg, normal distribution. Sample size is 25, sample mean is 80.4, sample variance is 20.25. Find 98% confidence interval for population mean.

sample population < 30, use DoF, df= n-1, df= 24 alpha/2 = .02/2 = .010 t-table = 2.492

the average confidence interval (80.4 - 2.492) < \(\mu\) < (80.4 + 2.492)

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4 bags of chips of 220g, with a sample mean 217.25, sample variance 6.25. Normal distribution. Alpha level is 0.05.

1. Null: \(\mu\) = 220. Alternative: \(\mu\) < 220.
2. Confidence Interval: \(\alpha\) = 0.05
3. sample size: 4 < 30, use t-table
4. df= 4 - 1 = 3
5. t-table value: (df and alpha) 2.353.

• interested in 1 tail, left side (less chips), t= -2.353

6. reject null if t falls in rejection region
7. t-test = (217.2 -220/ 2.5/ sqrt(4)) = -2.2
8. fail to reject null

Descriptive Stats

Descriptive stats

• central tendency (mean, median, mode)
• variability (range, std)

Inferential Stats

there are 2 main uses

• significance testing, determine if differences or relationship between groups is statistically significant (occurrence is not just pure chance)
• estimate population parameters from sample stats

decision

Between Groups: select a sample, assign each person to 1 of the binary label of Indep. Var

Within Groups: each person is assigned to all labels, same group gets diff levels, groups are matched related to the dependent variable

Hypothesis Testing

The p-value is the probability of a result happening by pure randomness alone. The alpha \(\alpha\) = 0.05 (95% confidence level, 5% chance of making type 1 error)

IF p-value <= \(\alpha\) THEN reject the null, results are significant
IF p-value > \(\alpha\) THEN accept the null

Errors

• type 1 is rejecting a null when shouldn’t
• type 2 is accept null when shouldn’t

Effect size

Effect size indicates how strong the relationship between variables or differences are between groups

Interpretation of strength of a relationship:

• for r and Phi
  • very strong >= .70
  • large = .50
  • medium = .36
  • small = .10
• for eta
  • very strong >= .45
  • large = .37
  • medium = .24
  • small = .10

Correlation tests of coefficients

Pearson r and bivariate regression (parametric tests) Correlation coefficients -1 to +1, closer to -1 or +1 the stronger the relationship

-1 strong <——– weak ——-> strong +1
- -0.7 -0.5 -0.3 -0.1 <> 0.1 0.3 0.5 0.7

Chi square test

This non-parametric test can be used for difference questions with nominal or dichotomous variables. Classes: 1 or 2, Condition: on/off

Phi and Cramer’s V

these are tests for association between 2 nominal or dichotomous variables.

• Phi is used when both variables are dichotomous
• Cramer’s V when 1+ variable has multiple levels/groupings

Pearson r

this test is for strength of association between 2 scale variables. Ex. GPA vs Hours of study

Wilcoxon and McNemar to compare groups

ANOVA 1-way, F-score

analysis of variance is used to compare 3+ unrelated groups on an independent variable, determines whether there are significant variance between groups. The value is F-value

Post-hoc tests

the ANOVA just tells you that a significant difference exists

• do a Tukey’s HSD test if the p-value is not stat significant
• do a Games-Howell test if the p-value is stat significant

ANOVA 2-way

Independent variables each have 2+ levels, samples must be indep. and each participant must receive only 1 level of each indep. var. Variances must be equal. There is 3 null hypotheses: there is no diff between various levels (no main effect), there is no diff between various levels of 2nd Indep. var, and there is no interaction between indep. vars