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Sampling & Statistical Inference

Paper-II Statistics (Post-specific) · Statistics Syllabus

Useful for: SSC CGL

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Overview

Inference uses sample data to estimate population values and to test hypotheses with known error probabilities.

Key points

Sampling: simple random, stratified, systematic, cluster, multi-stage.
Standard error of the mean = σ/√n.
95% confidence interval for the mean (large sample) = x̄ ± 1.96 σ/√n.
Type I error: rejecting a true null (probability α); Type II: accepting a false null (probability β).
z-test for large samples or known σ; t-test for small samples with unknown σ; χ² for frequencies and independence.

Explanation

Power of a test = 1 − β.
Larger samples give smaller standard errors and narrower intervals.
Degrees of freedom for a one-sample t-test = n − 1.
A p-value below α leads to rejecting the null hypothesis.

Step by step

Identify the measure or test asked and its formula.
Organise the data (table, totals) before computing.
Interpret the result (sign, size, significance).

Examples

σ = 10, n = 100 → standard error = 1.
x̄ = 50, σ = 10, n = 100 → 95% CI = 50 ± 1.96 = 48.04 to 51.96.
Probability of a Type I error equals the significance level α.

Traps and advanced points

Trap: Using σ instead of σ/√n.
Trap: Confusing Type I and Type II errors.
Trap: Using a z-test for a small sample with unknown σ.

Quick revision

SE = σ/√n.
x̄ ± 1.96 SE.
α = Type I.

Practice MCQs

In simple random sampling, every unit of the population has
1. a chance only if it is large
2. no chance of selection
3. an equal chance of selection
4. a fixed position
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Answer: (c) an equal chance of selection. Lottery and random-number methods are used.
Rejecting a true null hypothesis is called a
1. Type I error
2. Type II error
3. sampling frame
4. standard error
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Answer: (a) Type I error. Its probability is the level of significance α.
The standard error of the sample mean is
1. σ × n
2. σ/√n
3. σ/n²
4. √σ/n
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Answer: (b) σ/√n. It falls as the sample size grows.
Dividing the population into homogeneous groups and sampling from each is
1. convenience sampling
2. cluster sampling
3. stratified sampling
4. quota sampling
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Answer: (c) stratified sampling. Strata are internally similar.
Which test is used to compare the means of two small samples?
1. sign of the mean
2. chi-square test of independence
3. F-test for variances only
4. t-test
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Answer: (d) t-test. The t-test suits small samples with unknown σ.
Selecting every 10th name from a list after a random start is
1. systematic sampling
2. stratified sampling
3. cluster sampling
4. judgement sampling
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Answer: (a) systematic sampling. The interval k is fixed.