Unit 3: Sampling distributions, estimation, index numbers and time series
Business Research Methods notes · PTU syllabus (MCOP203-18)
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Unit summary
From samples we estimate population values, and from index numbers and time series we read change over time. This unit covers sampling distributions of means and proportions, the Central Limit Theorem, point and interval estimation, confidence intervals and sample size determination, construction and tests of index numbers, base shifting, splicing and deflating, and time series components and least-squares trend.
After this unit you can
- Explain sampling distributions and the Central Limit Theorem
- Construct confidence intervals and determine sample size
- Construct index numbers and apply consistency tests, base shifting, splicing and deflating
- Identify time series components and fit a least-squares trend
PTU syllabus topics
- Sampling distribution of means and proportions
- Central Limit Theorem
- point and interval estimation
- confidence intervals
- sample size determination
- index number construction and consistency tests
- base shifting/splicing/deflation
- time series components and trend analysis via least squares
Standard error of mean
σ / √n
Confidence interval (95%)
x̄ ± 1.96 × σ / √n
Sample size
n = (Z σ / E)²
Laspeyres index
Σp1q0 / Σp0q0 × 100
Fisher's ideal index
√(Laspeyres × Paasche)
Linear trend
Y = a + bX by least squares
Topic 1
Sampling distributions and the Central Limit Theorem
- Sampling distribution: the probability distribution of a statistic (mean, proportion) over all possible samples of a given size.
Standard error of mean
σ ÷ √n
Standard error of proportion
√[p(1 − p) ÷ n]
Finite population correction
Multiply SE by √[(N − n) ÷ (N − 1)]
- Central Limit Theorem: for large samples (n ≥ 30), the sampling distribution of the mean is approximately normal with mean μ and standard error σ/√n, whatever the shape of the population.
Topic 2
Estimation and confidence intervals
- Point estimate: a single value (sample mean x̄ for μ). Interval estimate: a range likely to contain the parameter.
- Properties of a good estimator: unbiasedness, consistency, efficiency, sufficiency.
CI for mean (σ known or large n)
x̄ ± Z × σ ÷ √n
CI for mean (small n)
x̄ ± t × s ÷ √n
CI for proportion
p ± Z × √[p(1 − p) ÷ n]
Sample size for mean
n = (Z σ ÷ E)²
Sample size for proportion
n = Z² p(1 − p) ÷ E²
Example
Sample of 100 customers: mean spend ₹800, σ = ₹200. 95% CI = 800 ± 1.96 × 20 = ₹760.8 to ₹839.2. To estimate a proportion within ±5% at 95% confidence with p = 0.5: n = 1.96² × 0.25 ÷ 0.0025 ≈ 385.
Topic 3
Index numbers
An index number measures relative change in a variable (prices, quantities) over time relative to a base period (= 100).
Laspeyres
Σp1q0 ÷ Σp0q0 × 100 (base-year weights)
Paasche
Σp1q1 ÷ Σp0q1 × 100 (current-year weights)
Fisher's ideal
√(Laspeyres × Paasche)
Dorbish–Bowley
(L + P) ÷ 2
Marshall–Edgeworth
Σp1(q0 + q1) ÷ Σp0(q0 + q1) × 100
- Tests of consistency: time reversal test (P01 × P10 = 1), factor reversal test (P01 × Q01 = value ratio) — Fisher's index satisfies both (hence "ideal"); unit test, circular test.
- Base shifting: new index = old index ÷ index of new base year × 100.
- Splicing: linking an old series and a new series with different bases into a continuous series.
- Deflating: real value = money value ÷ price index × 100 (real wages, real GDP).
Example
Money wages rose from ₹20,000 to ₹30,000 while CPI rose from 100 to 140. Real wage = 30,000 ÷ 140 × 100 = ₹21,429 — real income rose only about 7%.
- Indian indices: CPI (Combined, base 2012 — being revised to 2024), WPI (base 2011–12), IIP, Sensex, Nifty.
Topic 4
Time series analysis
Secular trend (T)
Long-term movement
Seasonal variation (S)
Regular within-year patterns
Cyclical variation (C)
Business cycles over years
Irregular variation (I)
Random, unpredictable events
- Models: additive Y = T + S + C + I; multiplicative Y = T × S × C × I.
- Measuring trend: freehand, semi-averages, moving averages, least squares.
Trend equation
Yc = a + bX
With X coded so ΣX = 0
a = ΣY ÷ n; b = ΣXY ÷ ΣX²
Example
Sales (₹ lakh) 2021–2025: 10, 12, 15, 16, 22; X = −2, −1, 0, 1, 2. ΣY = 75 → a = 15; ΣXY = −20 − 12 + 0 + 16 + 44 = 28; ΣX² = 10 → b = 2.8. Trend for 2026 (X = 3) = 15 + 8.4 = ₹23.4 lakh.
Key terms
- Standard error
- Standard deviation of a sampling distribution
- Central Limit Theorem
- Sample means are approximately normal for large samples
- Confidence interval
- Range likely to contain a population parameter
- Fisher's ideal index
- Geometric mean of Laspeyres and Paasche indices
- Secular trend
- Long-term movement in a time series
Quick revision
- SE of mean σ/√n; CLT for n ≥ 30.
- CI: x̄ ± Zσ/√n; sample size n = (Zσ/E)².
- Index numbers: Laspeyres, Paasche, Fisher; tests — time and factor reversal.
- Base shifting, splicing, deflating.
- Time series: T, S, C, I; least squares Yc = a + bX.
Important exam questions
Practice questions written to the PTU exam pattern for this unit's syllabus: short answers (Section A style) and long answers (Sections B and C style).
Short-answer questions
- Q1.What is a sampling distribution?
- Q2.State the Central Limit Theorem.
- Q3.Distinguish point and interval estimation.
- Q4.Why is Fisher's index called ideal?
- Q5.What is deflating of an index?
- Q6.Name the components of a time series.
Long-answer questions
- Q1.Explain sampling distributions and the Central Limit Theorem.
- Q2.Explain interval estimation and determination of sample size.
- Q3.Explain the construction of index numbers and the tests of consistency.
- Q4.Explain the components of a time series and the least-squares method of trend.
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