ETX5900 · Business Statistics
Introduction to Correlation & Regression Analysis
Module 9 (Week 9) of ETX5900 Business Statistics at Monash University is the unit's first look at modelling the relationship between two numeric variables. You measure how tightly they move together with the Pearson correlation coefficient r, fit a least-squares regression line (slope b1 and intercept b0) to predict a response Y from an explanatory variable X, quantify the fit with R² = r², and test whether the relationship is statistically significant. It maps to Berenson (Australian 5th ed.) Chapter 12 and leads into the fuller regression inference of Module 10.
What this chapter covers
- 01Scatterplots and the population model Y = beta0 + beta1 X + epsilon vs the fitted line Y-hat = b0 + b1 X
- 02Pearson correlation r in [-1, +1]: strength and direction of a straight-line relationship
- 03Correlation is not causation, and r only detects LINEAR patterns
- 04Least-squares slope b1 = Sxy / Sxx and intercept b0 = y-bar - b1 x-bar
- 05Interpreting the slope in context (change in Y per one-unit change in X)
- 06Sums of squares: SST = SSR + SSE, and the coefficient of determination R^2 = SSR/SST = r^2
- 07Using the line to predict Y from X, and the danger of extrapolation beyond the data range
- 08The t-test on the slope: H0: beta1 = 0 vs H1: beta1 != 0, with df = n - 2
- 09Reading regression output (Excel / R): b0, b1, R^2 and the slope p-value
- 10Residuals e = y - y-hat as the raw material for model checking in Module 10
Correlation, the least-squares line, R-squared and a prediction
- +1Means. Sum of X = 15 so x-bar = 3; sum of Y = 70 so y-bar = 14.
- +1Deviation sums. x - x-bar = -2, -1, 0, 1, 2 and y - y-bar = -5, -4, 0, 2, 7. So Sxx = 4+1+0+1+4 = 10; Syy = 25+16+0+4+49 = 94; Sxy = 10+4+0+2+14 = 30.
- +1Least-squares line (slope first). b1 = Sxy/Sxx = 30/10 = 3.0; b0 = y-bar - b1 x-bar = 14 - 3.0 x 3 = 5.0. So Y-hat = 5 + 3X.
- +1Correlation. r = Sxy / sqrt(Sxx x Syy) = 30 / sqrt(10 x 94) = 30 / sqrt(940) = 30/30.66 = +0.978 (very strong, positive).
- +1Coefficient of determination. SST = Syy = 94; SSR = b1 x Sxy = 3.0 x 30 = 90; R^2 = SSR/SST = 90/94 = 0.957 = (0.978)^2 = r^2. About 95.7% of the variation in sales is explained by advertising spend.
- +1Prediction (in range). X = 2.5 lies inside the observed range 1 to 5, so Y-hat = 5 + 3 x 2.5 = 12.5, i.e. about $12,500 of weekly sales.
Key terms
- Pearson correlation coefficient (r)
- A unit-free measure of the strength and direction of the straight-line relationship between two numeric variables, always between -1 and +1. Its sign matches the slope b1; r = 0 means no LINEAR relationship.
- Least-squares regression line
- The line Y-hat = b0 + b1 X that minimises the sum of squared vertical residuals; it always passes through the centroid (x-bar, y-bar).
- Slope (b1)
- The estimated change in the response Y for a one-unit increase in the explanatory variable X, computed as b1 = Sxy / Sxx.
- Intercept (b0)
- The fitted value of Y when X = 0, computed as b0 = y-bar - b1 x-bar; meaningful only if X = 0 is inside or near the data range.
- Coefficient of determination (R^2)
- The proportion of the total variation in Y explained by the model, R^2 = SSR/SST = 1 - SSE/SST, between 0 and 1. In simple linear regression R^2 = r^2.
- Residual (e)
- The vertical gap between an observed value and the fitted line, e = y - y-hat. Least squares minimises the sum of the squared residuals (SSE).
- Extrapolation
- Using the fitted line to predict Y for an X far outside the observed data range. It is unreliable because the linear pattern is only supported within the data.
- Slope t-test
- The hypothesis test H0: beta1 = 0 vs H1: beta1 != 0 using t = b1 / s(b1) with df = n - 2; rejecting H0 signals a statistically significant linear relationship.
Introduction to Correlation & Regression Analysis FAQ
What is the difference between correlation and regression?
Correlation summarises the relationship in a single number, the Pearson coefficient r, which measures how strong and in which direction two numeric variables move together on a straight line (r between -1 and +1). Regression goes further and fits an actual equation, Y-hat = b0 + b1 X, so you can interpret the slope, predict Y from X, and quantify the fit with R^2. In simple linear regression the two are linked: the sign of r equals the sign of the slope, and R^2 = r^2.
Does a high correlation mean X causes Y?
No. A high r only tells you the points cluster tightly around a straight line, not that changing X would change Y. The relationship could be driven by a lurking third variable, or the direction of cause could run the other way. The unit's data are observational, so you describe the relationship as an association ('associated with'), and you always plot the scatter first because r only detects linear patterns and is sensitive to outliers.
Can AI help me with correlation and regression in ETX5900?
Yes — as a study aid. Sia (AskSia) explains step by step how to build the sums Sxx, Syy and Sxy, compute the slope and intercept, read r and R², and set up and decide the slope t-test, so you can interpret Excel or R output yourself. It also generates fresh practice problems at the level you need and checks your own working line by line. It never hands over answers, sits an assessment in your place, or guarantees a grade — always follow Monash's academic-integrity rules and confirm what is permitted on Moodle.
Exam move
Correlation and regression is a method-selection topic: the exam supplies a formula sheet and statistical tables, so marks come from choosing and applying the right steps, not from recall. Drill the routine until it is automatic - build the deviation sums Sxx, Syy and Sxy, get b1 = Sxy/Sxx then b0 = y-bar - b1 x-bar, read r and R^2 (remembering R^2 = r^2 in simple regression), interpret the slope in the units of the problem, and run the slope t-test with df = n - 2. Practise reading b0, b1, R^2 and the slope p-value off Excel or R output, always plot the scatter first, and never predict far outside the data or claim causation from correlation. Because the Final Examination is worth 50% and its duration is not stated in the unit materials, plan by marks rather than minutes: spend time on each question in proportion to its marks and show the working before the number so you earn the method marks. Confirm the exam date, length and format on Moodle / my.Monash.
Working through Introduction to Correlation & Regression Analysis in ETX5900? Sia is AskSia’s AI Statistics tutor — ask any ETX5900 Introduction to Correlation & Regression Analysis question and get a clear, step-by-step explanation grounded in how ETX5900 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
