Lesson 78 — Correlation and simple linear regression

$X = (1, 2, 3, 4)$, $Y = (2, 4, 6, 8)$. Calculate r without using a calculator and justify the result.

$X = (1, 2, 3, 4)$, $Y = (8, 6, 4, 2)$. Calculate r and identify the expected sign before computing.

$X = (1, 2, 3)$, $Y = (1, 4, 9)$. Calculate r and discuss if the relationship is linear.

If $U = X + 5$ and $V = 2Y$, what is the relationship between $r(U, V)$ and $r(X, Y)$? Justify with the definition.

Data with $n = 5$ pairs: $x = (1, 2, 3, 4, 5)$ and $y = (10, 7, 5, 4, 3)$. Calculate r.

$X = (1, 2, 3, 4, 5)$, $Y = (1, 4, 5, 9, 10)$. Calculate r and the covariance $s_{xy}$.

$r = 0.85$, $\bar x = 10$, $\bar y = 50$, $s_x = 3$, $s_y = 12$. Find the least-squares line.

Using the line from exercise 78.7 ($\hat y = 16 + 3.4x$), predict Y for $x = 15$ and for $x = 5$.

With $r = 0.85$ (exercise 78.7), calculate r² and interpret in terms of explained variance.

Using the line from 78.7, calculate the residual of the point $(10, 55)$.

What does $r = 0$ mean?

Ice cream sales correlate positively with drowning deaths ($r \approx 0.8$). The best explanation is:

With $r = 0.6$, $s_x = 2$, $s_y = 5$, calculate the slopes of the two regression lines: Y on X and X on Y. Do the lines coincide?

A regression model explains 64% of the variance in spending as a function of income. What is $|r|$?

If $V = -Y$, what is the relationship between $r(X, V)$ and $r(X, Y)$?

Height ($X$) vs. weight ($Y$) relationship: $\bar x = 170$ cm, $\bar y = 70$ kg, $s_x = 8$ cm, $s_y = 12$ kg, $r = 0.75$. Line equation and prediction for a person of 175 cm.

A researcher found $r = 0.82$ between the Corruption Perception Index and GDP per capita in 120 countries. Interpret r² and discuss causal limitations.

A plot of residuals vs. fitted values shows a U-pattern (residuals first negative, then positive). What does this indicate about the linear model?

$n = 25$, $r = 0.45$. Test $H_0: \rho = 0$ vs. $H_1: \rho \neq 0$ at the 5% level.

$n = 50$, $r = 0.60$. Construct a 95% CI for $\rho$ using the Fisher transformation.

For each pair, identify if it is causal correlation, spurious, or reverse causality: (a) rain and umbrella sales; (b) number of police and crime per city.

Interpret $r^2 = 0.25$ in a study relating years of schooling to salary.

Explain the risk of extrapolating the regression line to x-values outside the sample range.

In finance, the "beta" of a stock is the regression coefficient of the stock return on the market return. Express beta in terms of $r$, $s_{r_i}$, and $s_{r_m}$.

An energy distributor has monthly data on average temperature (°C) and consumption (MWh) for the last 5 years. Describe the correlation and regression analysis flow to predict consumption.

The four Anscombe sets have $r \approx 0.82$ and the same regression line. Why is the linear model adequate for set I but not for the other three?

Why is Spearman correlation more suitable than Pearson for ordinal data (e.g., satisfaction 1 to 5) or with outliers?

Differentiate confounder, mediator, and moderator in an observational study.

$n = 22$ pairs; $r^2 = 0.64$; SST = 500. Calculate the Sum of Squared Errors (SSE) and the RMSE.

Why does R² never decrease when adding a variable to the model, and how does adjusted R² solve this problem?

What property defines the least-squares line (OLS)?

Prove that $-1 \leq r \leq 1$ using the Cauchy-Schwarz inequality.