Maths · Statistics and probability · chapter 13 of 14

Class 11 Maths Chapter 13: Statistics case study question with answers

One case-based question of the kind CBSE-pattern school exams ask: a passage and a table, then 3 parts worth 4 marks in all, with the answer and the marking scheme for each. Written from the NCERT chapter.

Case study · 4 marks

A transport officer in Indore compares two city bus routes. Over the same eight days she records how many minutes late the first bus of the morning reached its last stop. Both routes were, on average, 6 minutes late each day, so the mean alone cannot separate them. She decides to look at how far the daily delays spread around that average before deciding which route to recommend to office-goers who need to plan their day.

Day	1	2	3	4	5	6	7	8
Route X	2	14	0	9	1	12	3	7
Route Y	5	7	6	4	8	6	7	5

Part 1 · 1 mark

The range of the delays on Route X is

(a) 12 minutes(b) 6 minutes(c) 14 minutes(d) 7 minutes

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Answer: (c) 14 minutes

Range = 14 − 0 = 14 minutes.

Part 2 · 1 mark

The mean deviation about the mean of the delays on Route Y is

(a) 1.5 minutes(b) 1 minute(c) 0 minutes(d) 6 minutes

Show answer

Answer: (b) 1 minute

x̄ = 48/8 = 6; |x_i − 6| = 1, 1, 0, 2, 2, 0, 1, 1, sum 8; M.D.(x̄) = 8/8 = 1 minute.

Part 3 · 2 marks

Find the variance of the delays on Route Y, and say which route is more consistent.

Answer and marking scheme

Squared deviations from 6: 1, 1, 0, 4, 4, 0, 1, 1, sum 12; σ² = 12/8 = 1.5. Route Y's delays stay within 4 to 8 minutes and its spread is small, while Route X's run from 0 to 14: Route Y is more consistent.

Σ (x_i − 6)² = 12 and σ² = 1.5 (1)
Route Y more consistent, with reason (1)

Find the mean deviation about the mean of the delays on Route X, and compare it with that of Route Y.

Answer and marking scheme

|x_i − 6| = 4, 8, 6, 3, 5, 6, 3, 1, sum 36; M.D.(x̄) = 36/8 = 4.5 minutes, against 1 minute for Route Y, so Route X's delays are far more scattered about the mean.

Σ |x_i − 6| = 36 and M.D. = 4.5 (1)
comparison with Route Y's 1 minute (1)

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