Unit-IV: Sampling Theory and Large Sample Tests

Standard Error of Sample Mean

In statistics, when we draw multiple random samples of size n from a population, the sample means (x̄) will vary from sample to sample. The standard deviation of the sampling distribution of a statistic is known as its standard error (SE).

The standard error of the sample mean measures the statistical accuracy of an estimate and quantifies the amount of error expected in our estimate of a population mean.

For a population with standard deviation σ, the standard error of the sample mean (SE(x̄)) depends on whether sampling is done with or without replacement from a finite or infinite population.

• Infinite Population or Sampling with Replacement: SE(x̄) = σ / √n
• Finite Population (with replacement correction using Finite Population Correction factor): SE(x̄) = (σ / √n) × √((N - n) / (N - 1)), where N is the population size and n is the sample size.

Key Concepts and Formulas

• Population Mean: μ
• Sample Mean: x̄
• Population Standard Deviation: σ
• Sample Size: n

Large Sample Tests

Large sample theory is based on the Central Limit Theorem, which states that regardless of the underlying distribution of the population, the sampling distribution of the sample mean will approximate a normal distribution as the sample size becomes large (typically n ≥ 30).

When n ≥ 30, the test statistic follows a standard normal distribution, Z ~ N(0, 1).

General steps for conducting a large sample hypothesis test:

1. Set up the null hypothesis (H₀) and alternative hypothesis (H₁).
2. Choose the level of significance (α), commonly 5% (0.05) or 1% (0.01).
3. Calculate the appropriate test statistic (Z).
4. Determine the critical value(s) from the standard normal distribution table.
5. Compare the calculated Z with the critical table value and state the statistical conclusion.

Critical Values Table for Z-Test

Test for Single Mean

We use this test when we want to determine if the mean of a sample significantly differs from a hypothesized population mean (μ₀).

Hypothesis Formulation

• Two-tailed test: H₀: μ = μ₀ vs H₁: μ ≠ μ₀
• Right-tailed test: H₀: μ ≤ μ₀ vs H₁: μ > μ₀
• Left-tailed test: H₀: μ ≥ μ₀ vs H₁: μ < μ₀

Test Statistic Formula

Z = (x̄ - μ₀) / (σ / √n)

If the population standard deviation σ is unknown and the sample size is large (n ≥ 30), we can substitute σ with the sample standard deviation s.

Practical Example

A factory claims that the average lifespan of its lightbulbs is 1000 hours with a standard deviation of 50 hours. A random sample of 100 bulbs is tested, yielding a sample mean of 985 hours. Test the claim at the 5% level of significance.

• H₀: μ = 1000
• H₁: μ ≠ 1000
• x̄ = 985, μ₀ = 1000, σ = 50, n = 100
• Z = (985 - 1000) / (50 / √100) = -15 / (50 / 10) = -15 / 5 = -3.0
• Since |-3.0| ≥ 1.96, we reject H₀. The sample mean significantly differs from the claimed population mean.

Test for Difference of Means

This test is used to compare the means of two independent populations to determine if the difference between them is statistically significant.

Hypothesis Formulation

• Two-tailed test: H₀: μ₁ = μ₂ (or μ₁ - μ₂ = 0) vs H₁: μ₁ ≠ μ₂
• Right-tailed test: H₀: μ₁ ≤ μ₂ vs H₁: μ₁ > μ₂
• Left-tailed test: H₀: μ₁ ≥ μ₂ vs H₁: μ₁ < μ₂

Test Statistic Formula

Z = (x̄₁ - x̄₂) / √( (σ₁² / n₁) + (σ₂² / n₂) )

Where:

• x̄₁, x̄₂ = Means of sample 1 and sample 2
• n₁, n₂ = Sizes of sample 1 and sample 2
• σ₁, σ₂ = Standard deviations of population 1 and population 2

Exam Note: If population standard deviations are unknown, substitute them with sample standard deviations s₁ and s₂ provided the sample sizes are large (n₁ ≥ 30 and n₂ ≥ 30).

Related Problems and Special Cases

If it is assumed that both samples come from populations with equal variances (σ₁ = σ₂ = σ), a pooled standard deviation (s_p) can be calculated:

s_p = √[ ((n₁ - 1)s₁² + (n₂ - 1)s₂²) / (n₁ + n₂ - 2) ]

The test statistic for pooled variance is:

Z = (x̄₁ - x̄₂) / (s_p × √( (1 / n₁) + (1 / n₂) ))

Common Mistakes to Avoid

• Confusing population standard deviation (σ) with sample standard deviation (s) when sample size rules apply.
• Forgetting to take the square root of the variance terms in the denominator of the difference of means formula.
• Misinterpreting critical values for one-tailed versus two-tailed hypothesis tests.