STS 1100 Exam 2 Study Guide: Correlation, Prediction, Probability, Normal Distributions, and Sample Means

Vocabulary flashcards covering correlation, linear regression, empirical and theoretical probability, significance, the empirical rule, z-scores, and the Central Limit Theorem.

Correlation coefficient ($r$)

A measure ranging from $-1$ to $1$ where values near $-1$ or $1$ indicate a strong linear relationship and values near $0$ indicate little linear relationship between two quantitative variables.

Scatterplot

A graphical display of two quantitative variables used to examine direction, strength, and whether the overall pattern is approximately linear.

$$R^2$$

The proportion of variation in the response variable explained by the linear relationship with the explanatory variable, equal to $r^2$ in simple linear regression.

Best-fit line

A straight line used to predict a response variable $y$ using an explanatory variable $x$, typically written in the form: predicted $y = \text{intercept} + \text{slope} \times x$.

Intercept (in simple linear regression)

The predicted value of the response variable $y$ when the explanatory variable $x = 0$.

Slope (in simple linear regression)

The predicted change in the response variable $y$ for a one-unit increase in the explanatory variable $x$.

Extrapolation

The practice of making predictions outside the observed range of the explanatory variable $x$, which may be unreliable.

Empirical probability

A probability calculated as the observed number of times an event occurs divided by the total number of observations.

Theoretical probability

A probability calculated as the number of favorable outcomes divided by all equally likely outcomes.

Complement of an event

The probability that an event does not occur, found using the relationship $P(\text{event does not occur}) = 1 - P(\text{event})$.

Law of large numbers

The principle that with many independent repetitions under the same conditions, the observed proportion tends to get closer to the probability, though it need not equal it exactly.

Expected value

The long-run average outcome per repetition, calculated by multiplying each possible outcome value by its probability and adding them: $\text{expected value} = \textstyle\binom{}{}\text{all outcomes} \rightarrow \text{sum} = \textstyle\binom{}{}\rightarrow \text{formula: } \text{sum}(\text{value} \times \text{probability})$.

Statistical significance

A determination of whether an observed result at least as extreme would be unlikely to occur purely by chance if there were no actual effect or difference.

Practical significance

An evaluation of whether the size of an observed difference or effect is large enough to matter in a real-world setting when considering scale, cost, and consequences.

Normal distribution

A symmetric, bell-shaped distribution whose mean describes the center, standard deviation describes the spread, and exactly half of observations fall below the mean and half above it.

Empirical rule

A rule stating that for a normal distribution, approximately $68\%$ of observations fall within $1$ standard deviation of the mean, approximately $95\%$ fall within $2$ standard deviations, and approximately $99.7\%$ fall within $3$ standard deviations.

$z$-score

The number of standard deviations a value lies from its mean, calculated as $z = \frac{\text{observed value} - \text{mean}}{\text{standard deviation}}$.

Outlier (course rule for normal distributions)

An observation located more than two standard deviations away from the mean, corresponding to a $z$-score of $z < -2$ or $z > +2$.

Sampling distribution of the sample mean

A distribution that describes the behavior and values of the sample means ($\bar{x}$) obtained from many random samples of the exact same size drawn from a population.

Central Limit Theorem (course rule of thumb)

The rule stating that the distribution of sample means is approximately normal if the population itself is approximately normal or if the sample size is at least $n = 30$.

Standard deviation of sample means

The measure of spread for the distribution of sample means, calculated as the population standard deviation divided by the square root of the sample size: $\frac{\sigma}{\sqrt{n}}$.