We assume a linear relationship:

\[Sales_i = b_0 + b_1(Temp_i) + e_i\]

Data

temp <- seq(60,100,length.out=30)
sales <- 200 + 15*temp + rnorm(30,0,80)
day <- 1:30
data <- data.frame(day,temp,sales)
head(data)

##   day     temp    sales
## 1   1 60.00000 1055.162
## 2   2 61.37931 1102.275
## 3   3 62.75862 1266.076
## 4   4 64.13793 1167.710
## 5   5 65.51724 1193.102
## 6   6 66.89655 1340.653

Fit Model

model <- lm(sales ~ temp, data=data)
summary(model)

## 
## Call:
## lm(formula = sales ~ temp, data = data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -148.64  -56.21   -9.89   45.07  147.71 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  310.386     96.361   3.221  0.00323 ** 
## temp          13.573      1.191  11.393 5.02e-12 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 77.9 on 28 degrees of freedom
## Multiple R-squared:  0.8226, Adjusted R-squared:  0.8162 
## F-statistic: 129.8 on 1 and 28 DF,  p-value: 5.015e-12

Plot 1

ggplot(data,aes(temp,sales)) +
  geom_point(color="blue") +
  geom_smooth(method="lm", color="red") +
  labs(x="Temperature (F)", y="Sales ($)",
       title="Ice Cream Sales vs Temperature")

Plot 2

plot(model$fitted.values, resid(model),
     xlab="Fitted", ylab="Residuals",
     main="Residuals vs Fitted")
abline(h=0,lty=2)

Plot 3

plot_ly(data, x=~temp, y=~day, z=~sales,
        type="scatter3d", mode="markers",
        marker=list(color="orange", size=4)) %>%
  layout(title="Sales vs Temperature vs Day",
         scene=list(
           xaxis=list(title="Temperature"),
           yaxis=list(title="Day"),
           zaxis=list(title="Sales ($)")
         ))

Inference TEST: temperature predicts sales:

\[t = \frac{b_1 - 0}{SE(b_1)}\]

with this we can determine that temperature does affect sales.

Prediction

predict(model, data.frame(temp=85), interval="prediction")

##        fit      lwr      upr
## 1 1464.097 1301.428 1626.766

Conclusion

• Higher temps will result in higher sales
  
• Model is good and meets assumptions
  
• Useful for predicting summer demands