Probability, Random Variables, and Probability Distributions

Standard deviation always decreases as predictor correlations increase.

The correlation guarantees higher R-squared values improving model fit.

Multicollinearity can inflate standard errors leading to less reliable estimates.

Predictor variable coefficients become more stable with higher correlation levels.

Nonparametric tests analyzing rank order differences in standardized test scores among income brackets within educational categories.

Point-biserial correlation coefficients calculated separately within each parental education category against income groups.

A series of independent samples t-tests comparing test scores across different household incomes stratified by parent education levels.

Moderated multiple regression including income, parents' education level, and their interaction term.

Relying on cross-sectional surveys showing higher self-reported rates of exercising among people with fewer cardiac events suggests correlational not causal links.

Citing anecdotal instances where individuals with high levels of physical activity experience heart issues casts doubt on any causal link when anecdotal evidence cannot establish causal relations.

Presenting case studies of patients with little to no physical exercise having low rates of heart disease contests the possible connection between exercise and cardiac health.

Channeling effects through multiple well-controlled observational studies comparing similar populations with varying levels of physical activity while accounting for known confounders strengthens causal inference..

Dot plot

Two-way table

Stem-and-leaf plot

Boxplot

The sample proportion must be equal to the hypothesized population proportion $p_0$ under $H_0$.

There must be at least 30 individuals in the population from which we are sampling.

The sample size $n$ should be large enough so that $np$ and $n(1-p)$ are both greater than 10.

The standard deviation $\sigma_{\hat{p}}$ must be known and remain constant across samples.

Y-intercept of least squares regression line.

Residual sum of squares for prediction errors.

Slope of least squares regression line.

Correlation coefficient between two variables.

An experiment where participants randomly assigned to eat chocolate show significant cognitive improvements compared to non-chocolate eaters supports causality directly rather than indirectly weakening it.

A low correlation coefficient between chocolate intake frequency and measures of cognitive ability suggests a strong linear relationship exists.

A large p-value for the slope coefficient in a regression analysis predicting cognitive function from chocolate consumption indicates weak evidence against the null hypothesis.

Longitudinal studies showing consistent improvement in cognition among individuals who consume chocolate frequently reinforce potential causality rather than undermining it.

Binomial coefficient formula $C(8,2)$ divided by $2^8$

Binomial coefficient formula $C(4,2)$ divided by $2^4$

Binomial coefficient formula $C(8,6)$ divided by $16^8$

Binomial coefficient formula $C(8,4)$ divided by $16^8$

A pattern in the residuals plot.

No outliers in the residuals plot.

Homoscedasticity of residuals.

Normally distributed residuals.

A positive test statistic indicating the difference significant enough to suggest a sample mean greater than pop mean

A zero test statistic suggesting no difference between sample means and hypothesized pop mean

A negative test statistic indicating the difference not significant and sample mean likely close to pop mean

An undetermined test statistic since we haven’t discussed the population standard deviation