To perform multiple linear regression and estimate the end-semester marks for a student based on midterm marks and missed classes, as well as to create the scatter plot and regression plane in R, we can follow the steps below:

R Code

RCopy code
# Data
y <- c(85, 74, 76, 90, 85, 87, 94, 98, 81, 91, 76, 74)
x1 <- c(65, 50, 55, 65, 66, 70, 65, 70, 55, 70, 50, 55)
x2 <- c(1, 7, 5, 2, 6, 3, 2, 5, 4, 3, 7, 4)

# Combine data into a dataframe
data <- data.frame(y, x1, x2)

# Fit the multiple linear regression model
model <- lm(y ~ x1 + x2, data = data)

# Display the model summary to see coefficients
summary(model)

# Estimate y for a student with midterm marks (x1) = 60 and missed classes (x2) = 4
new_student <- data.frame(x1 = 60, x2 = 4)
predicted_y <- predict(model, new_student)
cat("Estimated end-semester marks (y) for x1=60 and x2=4:", predicted_y, "\n")

# Visualization
library(ggplot2)
library(plotly)

# Scatter plot with regression plane
plot_ly(data, x = ~x1, y = ~x2, z = ~y, type = "scatter3d", mode = "markers") %>%
  add_trace(type = 'mesh3d', x = ~x1, y = ~x2, z = fitted(model), opacity = 0.5) %>%
  layout(scene = list(
    xaxis = list(title = "Midterm Marks (x1)"),
    yaxis = list(title = "Classes Missed (x2)"),
    zaxis = list(title = "End Semester Marks (y)")
  ))

Explanation of Code

1. Data Preparation: We define y, x1, and x2 as vectors, representing end semester marks, midterm marks, and classes missed, respectively. We then combine these into a data frame.
2. Model Fitting: We fit a linear regression model of the form $y = a + b \cdot x1 + c \cdot x2$ using the lm function.
3. Prediction: We predict the end-semester marks for a student with midterm marks (x1) of 60 and missed classes (x2) of 4 using the predict function.
4. Visualization: We use the plotly package to create a 3D scatter plot and overlay the regression plane.

This will display the summary of the regression coefficients and an estimated mark for the specified conditions. The 3D plot helps visualize the relationship between the variables.

Would you like a breakdown of the output or need help with any specific part?

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1. How can we interpret the coefficients from the linear regression model?
2. What does each coefficient (a, b, c) represent in the context of this problem?
3. How reliable is the model prediction for students who deviate from this sample range?
4. How would we assess the accuracy of the regression model?
5. Can we perform similar analysis for a larger data set with additional predictors?

Tip: In R, the summary() function is useful for understanding model fit through metrics like R-squared and p-values for each predictor.