temp = c(20, 30, 40, 50, 60, 70, 80, 90, 100, 110)
expansion = c(0.11, 0.15, 0.22, 0.28, 0.31, 0.39, 0.43, 0.50, 0.58, 0.63)

data <- data.frame(temp, expansion)
model = lm(expansion ~ temp, data = data)

## 
## Call:
## lm(formula = expansion ~ temp, data = data)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.020848 -0.005894  0.003303  0.007591  0.015939 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -0.0189697  0.0102101  -1.858      0.1    
## temp         0.0058303  0.0001437  40.580  1.5e-10 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.01305 on 8 degrees of freedom
## Multiple R-squared:  0.9952, Adjusted R-squared:  0.9946 
## F-statistic:  1647 on 1 and 8 DF,  p-value: 1.497e-10

# For scatter plot
ggplot(data, aes(x = temp, y = expansion)) +
  geom_point(size = 3) +
  geom_smooth(method = "lm", se = FALSE) +
  labs(
    title = "Rod Expansion vs Temperature",
    x = "Temperature (°C)",
    y = "Expansion (mm)"
  ) + theme_minimal()

# For Resi plot
data$residuals <- resid(model)

 ggplot(data, aes(x = temp, y = residuals)) +
  geom_point(size = 3) +
  geom_hline(yintercept = 0, linetype = "dashed") +
  labs(
    title = "Residual Plot",
    x = "Temperature (°C)",
    y = "Residual"
  ) +
  theme_minimal()

# For interactive plot
plot_ly(
  data = data,
  x = ~temp,
  y = ~expansion,
  type = "scatter",
  mode = "markers"
) %>%
  layout(
    title = "Interactive Scatterplot of Temperature vs Expansion",
    xaxis = list(title = "Temperature (°C)"),
    yaxis = list(title = "Expansion (mm)")
  )

A simple linear regression describes the relationship
between two variables.
The model is:
$$
y = \beta_0 + \beta_1 x + \varepsilon
$$
Where:

- $\beta_0$ = intercept  
- $\beta_1$ = slope  
- $x$ = predictor variable  
- $y$ = response variable  
- $\varepsilon$ = random error

The slope of the regression line is calculated using the 
least squares  formula:
$$
\beta_1 =
\frac{\sum (x_i - \bar{x})(y_i - \bar{y})}
{\sum (x_i - \bar{x})^2}
$$
The intercept is:
$$
\beta_0 = \bar{y} - \beta_1 \bar{x}
$$