STATS ALLTOGETHER 2

Bivariate Data

A dataset consisting of measurements or observations on two quantitative variables, (x, y).

Explanatory Variable (x)

The variable that may help explain or influence changes in a response variable.

Response Variable (y)

The variable that measures an outcome of a study.

Scatterplot

The primary graph used to show the relationship between two bivariate quantitative variables.

3 Things to look for in a Scatterplot

Direction (positive, negative, none),

Form (linear, non-linear/curved),

Strength (weak, moderate, strong).

Correlation

Exists between two variables when the value of one variable is associated with the values of the other variable.

Linear Correlation Coefficient (r)

A quantitative assessment of the strength and direction of a linear relationship.

Boundary Values of r

Always between -1 and 1.

Meaning of r = 1 and r = -1

1 = strong positive linear relationship

-1 = strong negative linear relationship.

Meaning of r = 0

Indicates a very weak linear relationship (or no linear correlation).

Units of r

It has no unit of measure and does not change if you change measurement units.

Swapping x and y on r

The value of r does not change if x and y are swapped.

Is r resistant to outliers?

No, correlation is not resistant to outliers or extreme values.

Correlation vs. Causation

Correlation does not imply causation. High correlation often happens because both variables are strongly related to a third variable.

Objective of Linear Regression

To find a straight line that is as close as possible to the points in the plot for predictive modeling.

Least-Squares Regression Line Equation

y-hat = a + bx (where y-hat is the predicted y value).

Slope (b) Meaning

The approximate amount by which y increases when x increases by 1 unit. (Slope = rise/run).

y-intercept (a) Meaning

The approximate height of the line when x = 0 (often has no real-world meaning).

Residual (Deviation)

Actual y - Predicted y-hat. It measures how far a point is from the line.

Extrapolation

Making predictions using x-values outside the range of the original dataset. Do not do it!

Probability

A number between 0 and 1 that reflects the likelihood or long-term proportion of an event occurring under short-term uncertainty.

Factor & Levels

Factor: A categorical or discrete variable.

Levels: The specific values a factor can take (e.g., Freshman, Sophomore).

Two-Way (Contingency) Table

A table that summarizes data on two qualitative (categorical) variables.

Complement of an Event (E^c)

All outcomes in the sample space that do not belong to event E

P(E) + P(E^c) = 1.

Disjoint (Mutually Exclusive) Events

Two events that cannot occur at the same time.

• P(A and B) = 0

Addition Rule for Disjoint Events:

• P(A or B) = P(A) + P(B).

Independent Events

Knowing that event A happened has no impact on the probability of event B happening.

P(B|A) = P(B)

Multiplication Rule for Independent Rule:

P(A and B) = P(A) x P(B).

Conditional Probability Rule ("Given")

P(A|B) = P(A and B)/ P(B)

Random Variable

A quantitative variable whose value depends on the outcome of a probability experiment.

Discrete Random Variable

Values are isolated points along the number line (think whole, countable numbers).

Continuous Random Variable

Values are all points within an interval (think decimals, fractions, or measurements).

Two Rules for Discrete Probabilities

1. Each individual probability must be between 0 & 1

2. The sum of all probabilities must equal 1

Probability at an Exact Point for a Continuous Variable

Always zero! P(X = c) = 0. Probabilities are found over intervals, like P(a < X < b).

Probability Density Function (PDF) Properties

1. Total area under the curve equals 1

2. Height of the curve must be ≥ = 0 everywhere.

Normal Distribution Properties

Continuous, symmetrical, bell-shaped, and unimodal curve written as X ~ N(μ, σ).

Standard Normal Distribution

A normal curve with a mean of zero and standard deviation of one: Z ~ N(0, 1). Uses z-scores.

Inflection Points on a Normal Curve

Where the curve changes direction, located exactly 1 standard deviation (σ) away from the mean (μ).

TI-84: Finding Probability/Area from a value

Use normalcdf(lower, upper, μ, σ).

TI-84: Finding a value from an Area/Percentile

Use invNorm(area_to_the_left, μ, σ). Always use the left side area!

Parameter vs. Statistic

• Parameter: Numerical trait of a population (fixed, usually unknown).

• Statistic: Numerical trait of a sample (varies from sample to sample).

Sampling Variability

The natural fact that the observed value of a statistic will vary from sample to sample.

Sample Proportion (p-hat) Formula

p-hat = x / n (where x = successes, n = sample size).

Sampling Distribution

The distribution formed by values of a sample statistic for every possible different sample of size n.

Center of Sampling Distribution of p-hat

μ p-hat = p (The mean of sample proportions equals the true population proportion).

Spread of Sampling Distribution of p-hat

σ p-hat = √(p(1-p)/n)

Large Counts Condition (Normality Check for p-hat)

The shape is approximately normal if both are true:

1. np ≥ 10

2. n(1-p) ≥ 10.

Effect of Increasing Sample Size (n) on Sampling Distributions

• Center stays the same

• Spread gets smaller (less variation)

• Shape becomes more normal.

When to use a Scatter Plot?

• Need 2 variables

• Quantitative variables

• Investigate whether there is a relationship between x and y? And if there is describe the relationship?

The 3 key features to look for when describing a scaterplot:

• Direction: Positive, Negative, or No relationship.

• Strength: Weak, Moderate, or Strong

• Form: Linear or Non-linear

Age of a car in years and its current value, relationship?

Negative Relationship

Weight of a person and calories consumed, relationship?

Positive Relationship

Height of a person and the person’s birth mom, relationship?

No relationship

Number of hours spent in safety training and the number of accidents that occur, relationship?

Negative Relationship

What is the Correlation Coefficient ( r ):

A quantitative measure of the strength and direction of a linear relationship between two quantitative variables.

Crucial Boundaries and values of ( r ):

• r is always between 1 and -1

• A value near the upper limit (1) indicates strong positive linear relationship.

• A value near the lower limit (-1) indicates a strong negative linear relationship

• A value near 0 indicates a very weak liner relationship (no correlation)

4 Properties of r (Units, Swapping, and Outliers):

• r itself has no unit of measurement

• Changing the units of x or y (ex: Inches to meters) does not change r

• Swapping x and y does not change r

• r is not resistant to outliers; extreme values can heavily alter it.

If r ≈ 0, does that mean there is no relationship at all?

No. It means there is no linear relationship. There could still be a very strong non-linear relationship.

The Golden Rule of Correlation:

Correlation does NOT imply causation. A strong association between two variables is often driven by a third variable that influences both.

Ex: Ice cream sales and lifeguard rescues are highly correlated because both increase due to a third variable: hot summer temp.

Least Squares Regression Line Equation:

ŷ= a+bx

ŷ (y-hat):

The predicted response variable

a:

The y-intercept (the predicted value of y when x=0)

b:

The slope (the predicted rate of change in y for every 1-unit increase in x)

How to find slope?

rise (y)/ run(x)

What is a Residual (Deviation)?

The vertical distance between an observed data point and the regression line. It represents the prediction error.

Formula: Residual = y-ŷ (actual y - predicted y)

n:

number of observations

The Principle of Least Squares

The method used to find the line of best fit by minimizing the sum of the squared residuals (𝝨 (y-y²)) Squaring ensures negative errors don’t cancel out positive errors.

Interpret the Slope (b) in context:

“We expect the response variable (y) to increase/decrease by [value of b] units for every 1 unit increase in the explanatory variable (x).”

Predicted height= 38.30+1.03 (foot length), We expect height to increase by 1.03 cm for every 1 cm increase in foot length.

N=

Normal Distribution

State Distribution:

Z~N (𝜇,𝜎)

Turn to percentage

__ x 100

Make percentage to a decimal:

__÷ 100

p-hat

sample proportion

p

population proportion/ true proportion

Descriptive statistics

summarize and describe the features of a specific dataset

Inferential Statistics

use sample data to make predictions, test hypotheses, or generalize conclusions about a broader population

x bar

sample mean

At Least

At Most