Practice Questions with Solutions

• v[-2] excludes the 2nd element. R has no negative-from-the-end indexing; tail(v, 1) gives the last element.

• c(TRUE, FALSE) is recycled to length 5 as T F T F T, keeping positions 1, 3 and 5.

• One element removed from five leaves four.

Any arithmetic involving NA yields NA — hence the first. na.rm = TRUE drops it, leaving mean(1,2,3,5) = 11/4 = 2.75. And == NA compares with an unknown value, so every result is unknown. Use is.na(x).

10 is a double; 10L forces an integer. The third recycles c(10,20) to 10 20 10, giving 11, 22, 13 — and warns because 3 is not a multiple of 2. That warning is a bug signal; check your lengths when you see it.

df$grade <- if (df$marks >= 40) "Pass" else "Fail"

Error: if takes a single condition; df$marks is a whole column. Modern R errors; older R silently used only the first row. Fix: df$grade <- ifelse(df$marks >= 40, "Pass", "Fail")

library(dplyr)
result <- df %>% filter(marks > 50) + select(name, marks)

Error: + between dplyr verbs. + combines ggplot2 layers; %>% chains data operations. Fix: df %>% filter(marks > 50) %>% select(name, marks)

km <- kmeans(customers, centers = 3)

Errors: no scaling and no nstart. Fix: km <- kmeans(scale(customers), centers = 3, nstart = 25)

Without scale() a variable in rupees swamps one in years — the variance ratio in the lab data is about 3.5 billion to 1 (corrected from 4 billion: R gives 3.48 billion). Without nstart you get whatever single random start produced.

f <- factor(c("10", "20", "30"))
values <- as.numeric(f)

Error: returns the level codes 1 2 3, not 10 20 30. Fix: as.numeric(as.character(f))

students %>%
  filter(!is.na(marks)) %>%
  group_by(section) %>%
  summarise(n = n(), avg = mean(marks), .groups = "drop") %>%
  arrange(desc(avg)) %>%
  slice_head(n = 3)

The same six operations as the SQL you know from Database Management Systems: SELECT section, AVG(marks) … GROUP BY section ORDER BY … LIMIT 3.

impute_median <- function(x, max_missing = 0.3) {
  pct <- mean(is.na(x))
  if (pct > max_missing) {
    warning(sprintf("%.0f%% missing -- too much to impute", pct * 100))
    return(x)
  }
  ifelse(is.na(x), median(x, na.rm = TRUE), x)
}

Median, not mean, because an outlier has already distorted the mean — the lab data shows mean 79.35 against median 69.00 with one 250 present (corrected from 79.06 and 68.00, which neither R nor the Python version gives). And the guard matters: imputing a column that is 60% missing invents most of it.

ggplot(students, aes(x = section, y = marks, fill = section)) +
  geom_boxplot(alpha = 0.7, outlier.colour = "red") +
  facet_wrap(~ gender) +
  labs(title = "Marks by section", x = "Section", y = "Marks") +
  theme_minimal() +
  theme(legend.position = "none")

fill (not colour) for the box interior, and the legend removed because the x-axis already labels the sections.

Given TP = 80, FP = 20, FN = 40, TN = 860:

The verdict: 94% is misleading. 88% of these cases are negative, so "always predict negative" already scores 0.88 — the model adds only 0.06. And recall of 0.667 means it misses 40 of 120 real cases.

For disease screening that is unacceptable: lower the threshold, accept more false positives, raise recall. Report the confusion matrix and per-class recall, never accuracy alone.

(These are the numbers in 14_evaluation.py, which asserts them.)

1. Plot it. Growing seasonal swing → multiplicative → take logs.
2. Decompose — decompose() or stl() to see trend, seasonal, remainder.

3. Test stationarity — adf.test() (H₀ = non-stationary) and kpss.test() (H₀ = stationary). Opposite nulls; write down which you are using.

4. Difference — diff() for the trend, diff(lag = 12) for seasonality. ndiffs()/nsdiffs() say how many.

5. Identify orders — ACF and PACF of the differenced series. PACF gives p, ACF gives q.

6. Fit — auto.arima() searches by AIC.

7. Check residuals — checkresiduals(). They must be white noise; you want a large Ljung-Box p-value.

8. Forecast — forecast(fit, h = 24), back-transform with exp().

A point worth making that most answers miss: step 5 must come after step

4. On the raw airline series the ACF decays slowly — the trend swamps everything — and the 12-month seasonality is only a ripple on that decay (0.66 at lag 8, back up to 0.76 at lag 12). Only after differencing does the seasonality stand out: after one difference the ACF peaks at lag 12, at 0.83. 16_arima.py shows the same on a series of its own, where the raw ACF decays monotonically and the differenced one troughs at lag 6 and peaks at lag 12, and asserts both patterns.

Corrected: this said the airline series' raw ACF decays monotonically, with the seasonality invisible, and that the differenced ACF troughs at lag 6. Those hold for the Python version's series; R on the airline data gives the figures above.