M303 Correlation and Simple Regression I

FALL 2026 · LECTURE 18
M303
Correlation & Simple Regression I
Measure association. Build a prediction. Know the difference.

Where we are

01Problem formulation

Management problem → research objectives → research questions → hypotheses

02Research design

Exploratory · descriptive · causal

03Data collection

Secondary · observation · qualitative · survey · experiment · sampling

04Data analysis & recommendation

Preparation · descriptives · hypothesis testing · correlation · regression

Learning Objectives

01Define

The correlation coefficient.

02Interpret

The direction and strength of correlation.

03Calculate

Correlation with the course Web Tools.

04Explain

The logic of simple regression.

05Use notation

Distinguish Y, X, a, and b.

Previous class review

01

One continuous mean compared with a benchmark

02

Two continuous ratings from the same respondent

03

One continuous outcome across three independent groups

04

Association between two categorical variables

Choose the analysis from the variables

ONE VARIABLE
ContinuousOne-sample t-test
TWO VARIABLES
Both continuousPaired t-test to compare meansCorrelation to measure association
Continuous DV + categorical IVIndependent-samples t-test · ANOVA
Continuous DV + continuous IVSimple regression
Both categoricalChi-square
THREE+ VARIABLES
Continuous DV + categorical and/or continuous IVsMultiple regression after dummy coding categorical IVs

A motivating example: Best Buy

BB
MANAGERIAL QUESTION

How much store traffic results from advertising?

Best Buy records advertising spending and store traffic for 20 months across stores in the Indianapolis area.

X · advertising spendingY · store traffic

Best Buy: raw data and scatter plot

Month
Ad spend
Traffic
Month
Ad spend
Traffic
140,0009,00011480,000100,000 275,00012,50012500,000117,000 3100,00032,00013520,000120,000 4110,00020,00014550,000150,000 5190,00060,00015560,000100,000 6200,00045,00016580,00090,000 7300,00040,00017690,00070,000 8310,00070,00018700,000100,000 9380,00080,00019710,000130,000 10410,00081,00020800,000135,000
Advertising spendingStore traffic

What pattern do you see?

Two questions, two tools

ASSOCIATIONDo the variables move together?

Two continuous variables → correlation coefficient

→
PREDICTIONHow much traffic should we expect?

Use advertising spending to predict traffic → simple regression

What is correlation?

r

Correlation is a statistical measure of the extent to which two continuous variables move together.

−1.00+1.0
r = −1.0

Perfect negative relation

r = 0

No linear relation

r = +1.0

Perfect positive relation

Positive correlation

DIRECTIONAs X increases, Y tends to increase.

The closer the points are to an upward-sloping line, the closer r is to +1.

No linear correlation

DIRECTIONChanges in X do not align with a linear change in Y.

The coefficient is near 0.

Negative correlation

DIRECTIONAs X increases, Y tends to decrease.

The closer the points are to a downward-sloping line, the closer r is to −1.

Calculate correlation with Web Tools

OPENM303 Web Toolskoraycosguner.github.io/tools.html
01

Load the Best Buy file or paste the two columns.

02

Select AdSpending and StoreTraffic.

03

Choose Correlation and run the analysis.

04

Read the direction and strength from r.

Data file · BestBuy – Ad Spending versus Store Traffic.xlsx

Read the correlation output

Ad spendingStore traffic Ad spending1 Store traffic0.861
0.00–0.32Weak
0.33–0.65Moderate
0.66–1.00Strong

r = 0.86 → strong positive association.

In-class exercise: correlation

KROGER · COLLEGE MALL

Are temperature and category sales associated?

Kroger records Bloomington’s daily temperature and daily unit sales of ice cream and soft drinks during June 2015.

WeatherVersusSoftDrinksandIceCreamSales.xlsx
01

Calculate r between temperature and ice-cream sales.

02

Calculate r between temperature and soft-drink sales.

03

Interpret each direction and strength in plain language.

From association to prediction

ONE VARIABLE
ContinuousOne-sample t-test
TWO VARIABLES
Both continuousPaired t-test to compare meansCorrelation to measure association
Continuous DV + categorical IVIndependent-samples t-test · ANOVA
Continuous DV + continuous IVSimple regression to predict Y from X
Both categoricalChi-square
THREE+ VARIABLES
Continuous DV + categorical and/or continuous IVsMultiple regression after dummy coding categorical IVs

What is simple regression?

Simple regression predicts the level of one continuous dependent variable from one continuous independent variable.
YDependent variable

The continuous outcome to predict—for example, store traffic.

XIndependent variable

The continuous predictor used to model changes in Y—for example, advertising spending.

Regressing Y on X lets us make inferences about Y for units with a given value of X.
What will store traffic be if Best Buy spends $500,000 on advertising?

Fit a line to the observed pattern

MODELYᵢ = a + bXᵢ

a · intercept    b · slope

Y · store trafficX · advertising
Regression fits a line to the pattern of observed data.

To sum up

01

Correlation describes the direction and strength of linear association between two continuous variables.

02

The sign of r gives direction; distance from zero gives strength.

03

Simple regression assigns roles: X predicts a continuous Y.

04

The fitted line is Yᵢ = a + bXᵢ; the intercept and slope make the prediction operational.

Next class

NEXT TOPIC
Estimate and interpret simple and multiple regression
BE READY TO
  1. Read R² and overall model significance.
  2. Interpret intercepts, slopes, and coefficient p-values.
  3. Use an estimated equation to make predictions.