- This presentation examines whether daily ice-cream sales increase as outdoor temperature rises using a simple linear regression model.
- The predictor \(x\) is Temperature (°F), and the response \(y\) is Daily Ice Cream Sales ($).
- The objective is to visualize the relationship, fit a straight-line model, interpret the slope and model fit, and communicate what the results imply for sales planning.
2025-10-20
Simple Linear Regression — Temperature vs Sales
What is Simple Linear Regression?
- The model treats the average sales as a linear function of temperature.
- It chooses the line that minimizes squared residuals, i.e., the squared gaps between actual and predicted sales.
- \(\hat\beta_0\) is the expected sales at \(x=0\) (line anchor); \(\hat\beta_1\) is the change in sales per 1°F.
- Predictions \(\hat y\) give expected sales for a given temperature; residuals show day-to-day factors not explained by temperature.
- Reasonable use here assumes the trend is roughly linear and residual spread is roughly constant across temperatures.
Regression Equation (LaTeX)
- The simple linear regression model is expressed as \(\,y_i = \beta_0 + \beta_1 x_i + \varepsilon_i,\; i=1,\dots,n.\)
- In this expression, \(\beta_0\) is the population intercept, \(\beta_1\) is the population slope, and \(\varepsilon_i\) is the random error for observation \(i\).
- Interpreted in context, the expected value of sales changes linearly with temperature, while the error term accounts for day-to-day factors not captured by temperature alone.
Least Squares Coefficients (LaTeX)
- The least squares estimates are \(\displaystyle \hat{\beta}_1=\frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sum (x_i-\bar{x})^2}\) and \(\displaystyle \hat{\beta}_0=\bar{y}-\hat{\beta}_1\bar{x}.\)
- These estimates minimize the total squared residuals \(\sum_{i=1}^n (y_i-\hat{y}_i)^2\) with \(\hat{y}_i=\hat{\beta}_0+\hat{\beta}_1 x_i.\)
- In this study, \(\hat{\beta}_1\) indicates how many dollars of sales are associated, on average, with a one-degree Fahrenheit increase in temperature, while \(\hat{\beta}_0\) positions the line when \(x=0\) (often mainly for anchoring if \(x=0\) lies outside the observed range).
Data Scatter (ggplot #1)
Fitted Line (ggplot #2)
Interactive 3D Plot (plotly)
R Code (shown)
# Data and model (duplicated so the code is visible on this slide) temperature <- c(60, 65, 70, 75, 80, 85, 90, 95) sales <- c(120,150,180,200,220,250,270,300) df <- tibble(temperature, sales) model <- lm(sales ~ temperature, data = df) # Show model summary (output appears below the code) summary(model)
Call:
lm(formula = sales ~ temperature, data = df)
Residuals:
Min 1Q Median 3Q Max
-4.167 -3.512 1.071 1.488 6.071
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -174.4048 9.2013 -18.95 1.39e-06 ***
temperature 4.9762 0.1174 42.37 1.16e-08 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 3.806 on 6 degrees of freedom
Multiple R-squared: 0.9967, Adjusted R-squared: 0.9961
F-statistic: 1795 on 1 and 6 DF, p-value: 1.157e-08Interpretation & Conclusion
- The fitted regression line is \(\hat{y} = \hat{\beta}_0 + \hat{\beta}_1 x,\) which summarizes the average change in sales as temperature varies.
- Based on the fitted model, each additional \(1^\circ\)F of temperature is associated with an average increase of 4.98 dollars in daily sales.
- The model explains a meaningful portion of the variability in sales, with an \(R^2\) of approximately 0.997, indicating how much of the variation is captured by temperature alone.
- In practical terms, warmer days tend to produce higher sales, so temperature is a useful predictor for basic planning and forecasting.
- Because the dataset is small and includes only one predictor, conclusions should avoid extrapolating far beyond the observed temperature range and should acknowledge that other factors can also influence sales.