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Multiple Choice about R grammar

  1. What does the function seq(1, 10, by = 2) produce in R?

    • A) 1, 2, 3, 4, …, 10
    • B) 1, 3, 5, 7, 9
    • C) 1, 4, 7, 10
    • D) 2, 4, 6, 8, 10

  2. Which of the following is not a valid way to create a vector in R?

    • A) c(2, 5, 7)
    • B) 1:5
    • C) vector(3)
    • D) array(1, 3)

  3. What is the result of length(c(4, 9, 2, NA, 7))?

    • A) 4
    • B) 5
    • C) NA
    • D) Error

  4. Given x <- c(10, 20, 30), what is x * 2?

    • A) c(20, 40, 60)
    • B) c(10, 20, 30, 10, 20, 30)
    • C) c(12, 24, 36)
    • D) Error

  5. What does NA represent in R?

    • A) The string "NA"
    • B) Missing data / Not Available
    • C) A logical FALSE
    • D) Zero

  6. What is the output type of mean(c(1, 2, 3, 4))?

    • A) integer
    • B) numeric (double)
    • C) character
    • D) logical

  7. Which operator in R is used for elementwise logical AND?

    • A) &
    • B) &&
    • C) |
    • D) ||

  8. What is the difference between & and &&?

    • A) & works only with scalars, && is vectorized
    • B) & does elementwise logical AND, && only returns a single TRUE/FALSE (with short-circuit)
    • C) They are identical
    • D) && works only on numeric vectors

  9. What does the function is.na() do?

    • A) Tests if a value is not numeric
    • B) Tests if a value is NA (missing)
    • C) Tests if a value is NaN
    • D) Converts a value to NA

  10. Suppose x <- c(1, 2, NA, 4). What is the result of sum(x)?

    • A) 7
    • B) NA
    • C) 8
    • D) Error

  11. How can you instruct sum() to ignore NA values?

    • A) sum(x, remove = TRUE)
    • B) sum(x, na.rm = TRUE)
    • C) sum(x, na = TRUE)
    • D) sum(x, ignore.na = TRUE)

  12. Which of the following is not a mode (or atomic type) in R?

    • A) numeric
    • B) logical
    • C) factor
    • D) character

  13. Suppose you have y <- c("a", "b", "c"). What is y[2]?

    • A) "a"
    • B) "b"
    • C) "c"
    • D) NA

  14. What happens if you access an index out of bounds, e.g. y[10] for a vector of length 3?

    • A) Error
    • B) NULL
    • C) NA
    • D) The vector recycles

  15. Which function gives the unique values of a vector?

    • A) unique()
    • B) distinct()
    • C) uniq()
    • D) levels()

  16. What does factor() do?

    • A) Converts a numeric vector to binary
    • B) Converts a vector into a factor (categorical) type
    • C) Returns factorials
    • D) Converts character to numeric

  17. What is the output of as.numeric(factor(c("a", "b", "a")))?

    • A) c(1, 2, 1)
    • B) c("a", "b", "a")
    • C) c(0, 1, 0)
    • D) c("1", "2", "1")

  18. What does data.frame() create?

    • A) A matrix
    • B) A list
    • C) A tabular structure with equal-length columns
    • D) A vector

  19. If df <- data.frame(a = c(1,2), b = c("x","y")), what is df$a?

    • A) A data frame
    • B) The column named “a” as a vector
    • C) The entire data frame
    • D) Error

  20. What does the str() function do when given an R object?

    • A) Prints only the names of the object
    • B) Gives the structure (internal representation) of the object
    • C) Summarizes statistical properties
    • D) Converts object to string

Open Questions in R Grammar

  1. Given this R snippet, what is the output? Explain step by step.
x <- c(5, NA, 10, 15)
mean_x <- mean(x)
total <- sum(x, na.rm = TRUE)
c(mean_x, total)
  1. Identify the bug / error in the following R code, and suggest a fix.
v <- c(2, 4, 6, 8)
w <- c(1, 2)
result <- v + w
print(result)

  1. Write a short program in R that takes a numeric vector v and returns a vector of the same length where each entry is TRUE if the corresponding entry in v`` is above the mean of v` (ignoring NAs), and FALSE otherwise.

  2. Given the following code, describe what it does (in plain English):

df <- data.frame(id = 1:5, score = c(10, 15, NA, 20, 18))
df$above_avg <- df$score > mean(df$score, na.rm = TRUE)
subset(df, above_avg == TRUE)
  1. What will be the result (or error) of this code? Explain.
x <- factor(c("low","medium","high","low"))
as.numeric(x) + 1
levels(x)

Open Questions in Matrix Operation

  1. Suppose
A <- matrix(c(1, 2, 3, 4), nrow = 2, byrow = TRUE)
B <- matrix(c(2, 0, 1, 2), nrow = 2, byrow = TRUE)
A %*% B

What is the result of A %*% B?

- A) matrix(c(4, 4, 10, 8), nrow = 2, byrow = TRUE)

- B) matrix(c(2, 2, 6, 8), nrow = 2, byrow = TRUE)

- C) matrix(c(4, 4, 8, 10), nrow = 2, byrow = TRUE)

- D) Error (dimensions do not match)

27/ Which operator is used for matrix multiplication in R?

- A) *

- B) %*%

- C) .

- D) crossprod()
  1. Given:
M <- matrix(1:6, nrow = 2, ncol = 3)
N <- matrix(1:6, nrow = 3, ncol = 2)
M %*% N

What is the dimension of the result?

- A) 2×2

- B) 3×3

- C) 2×3

- D) Error
  1. Suppose:
X <- matrix(c(1, 2, 3, 4), nrow = 2)
Y <- matrix(c(5, 6), nrow = 2)
X %*% Y

What happens?

- A) 2×1 matrix result

- B) 1×2 matrix result

- C) Error (non-conformable arguments)

- D) A scalar
  1. Write pseudo-code for multiplying two matrices A (m×n) and B (n×p) to produce matrix C (m×p). Clearly specify the loop structure you would use. You do not need to write actual R code, just outline the algorithm in clear steps.

Regresison Questions

  1. In the linear regression model $y = X\beta + \epsilon$, which of the following is the closed-form solution for $\hat\beta$?

    • A) $X^\top y$

    • B) $(X^\top X)^{-1} X^\top y$

    • C) $(X X^\top)^{-1} y$

    • D) $X (X^\top y)^{-1}$

  2. Under the standard linear regression assumptions, which of the following is true about bias and variance of $\hat\beta$?

    • A) $\hat\beta$ is unbiased, variance depends on $\sigma^2 (X^\top X)^{-1}$

    • B) $\hat\beta$ is biased, variance is always zero

    • C) $\hat\beta$ is unbiased, variance does not depend on $X$

    • D) $\hat\beta$ is biased, variance depends only on sample size

  3. In R, which of the following correctly computes the OLS estimate?

    • A) beta_hat <- solve(t(X) %% X) %% t(X) %*% y

    • B) beta_hat <- X %% t(X) %% y

    • C) beta_hat <- lm(X, y)

    • D) beta_hat <- solve(X) %*% y

  4. Explain in words what the bias-variance tradeoff means in the context of linear regression.

  5. Suppose you fit a regression model in R using:

model <- lm(y ~ x1 + x2, data = df)
summary(model)

Explain the following concepts:

The coefficient estimate of x1.

The p-value associated with the coefficient estimate of x1.

  1. The following is the output of a summary function:
Call:
lm(formula = y ~ x1 + x2, data = df)

Residuals:
    Min      1Q  Median      3Q     Max 
-2.345  -0.876  -0.123   0.754   2.567 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   1.2350     0.4321   2.86   0.005**  
x1            0.5678     0.0987   5.75  1.2e-07***
x2           -0.2345     0.1123  -2.09   0.039*   

Residual standard error: 1.05 on 96 degrees of freedom  
Multiple R-squared: 0.642, Adjusted R-squared: 0.631  
F-statistic: 58.4 on 2 and 96 DF,  p-value: < 2.2e-16

Questions:

- How would you interpret the coefficient of x1?

- Is x2 statistically significant at the 5% level? Why or why not?

- How do you interpret the F-test and its result?

Please revise the inclass practices as well.