Sample vs Population

A population is the entire group we want to study (e.g., all Year 12 students in NSW). A sample is a subset of the population that we actually observe and measure.

Population parameters (fixed but usually unknown): mean μ, standard deviation σ, proportion p.

Sample statistics (vary from sample to sample): sample mean $x̄$ , sample standard deviation s, sample proportion $&hat;p$ .

We use sample statistics as estimates of population parameters. The key question in statistics is: how reliable are these estimates?

Example: To estimate the average height of all Australian 17-year-olds (μ), we measure a random sample of 100 students and calculate $x̄$ = 171.2 cm.

Different random samples would give different values of $x̄$ — this variation is called sampling variability.

The Central Limit Theorem (CLT)

The Central Limit Theorem is one of the most important results in statistics. It states that for sufficiently large sample sizes, the distribution of the sample mean is approximately normal, regardless of the shape of the population distribution.

X̄ ~ N(μ, σ 2 n)

The sample mean has mean μ and standard deviation σ/√n (called the standard error).

As the sample size $n$ increases, the sampling distribution becomes more tightly clustered around the true population mean μ. A general guideline is that n ≥ 30 is sufficient for the CLT to apply.

Key implication: If σ = 15 and n = 25, the standard error is σ/√n = 15/5 = 3.

If n = 100, the standard error is 15/10 = 1.5. Larger samples give more precise estimates.

Sampling Variability

Sampling variability refers to the fact that different random samples from the same population will produce different statistics. The sampling distribution describes the pattern of these varying statistics across all possible samples.

Standard error of the mean: SE = σ/√n measures how much sample means typically vary from the population mean.

Standard error of a proportion: SE = √(p(1−p)/n) measures how much sample proportions typically vary.

Understanding sampling variability allows us to construct confidence intervals and perform hypothesis tests, quantifying the uncertainty in our estimates.

Example: A population proportion is p = 0.6. For a sample of n = 200:

SE = √(0.6 × 0.4 / 200) = √(0.0012) ≈ 0.0346. About 95% of sample proportions will fall within 0.6 ± 2(0.0346), i.e., between 0.531 and 0.669.

Key Vocabulary

Sampling Distribution

The probability distribution of a statistic obtained from all possible samples of a given size from a population.

Central Limit Theorem

States that the sampling distribution of the sample mean is approximately normal for large n, regardless of population shape.

Standard Error

The standard deviation of a sampling distribution. For the sample mean: SE = σ/√n.

Parameter vs Statistic

A parameter describes a population (fixed); a statistic describes a sample (varies).

Worked Examples

1

A population has μ = 80 and σ = 12. Find the mean and standard error of the sampling distribution of $X̄$ for n = 36.

Step 1: Mean of the sampling distribution = μ = 80.

Step 2: Standard error = σ/√n = 12/√36 = 12/6 = 2.

Answer: $X̄$ ~ N(80, 2²). The mean is 80 and the standard error is 2.

2

Using Example 1, find P( $X̄$ > 83).

Step 1: Standardise: z = (83 − 80)/2 = 1.5.

Step 2: P(Z > 1.5) = 1 − P(Z ≤ 1.5) = 1 − 0.9332.

Answer: P( $X̄$ > 83) ≈ 0.0668 or about 6.7%.

3

A poll finds 55% of 400 voters support a candidate. Find the standard error of the sample proportion.

Step 1: $&hat;p$ = 0.55, n = 400.

Step 2: SE = √( $&hat;p$ (1 − $&hat;p$ )/n) = √(0.55 × 0.45 / 400) = √(0.000619).

Answer: SE ≈ 0.0249 or about 2.5%.

Knowledge Check

Select the correct answer for each question. Click "Check Answer" to see if you are right.

Question 1

If σ = 20 and n = 100, what is the standard error of the sample mean?

0.2 200 20 2

Question 2

According to the Central Limit Theorem, what shape does the sampling distribution of the sample mean approximate for large n?

Normal (bell-shaped) Uniform Skewed right Same shape as the population

Question 3

Which of the following reduces sampling variability?

Decreasing the sample size Increasing the sample size Using a biased sampling method Increasing the population size

Question 4

A population has μ = 50 and σ = 10. For n = 25, what is P(47 < $X̄$ < 53)?

0.6827 0.9332 0.8664 0.2345

Question 5

Which of the following is a population parameter (not a sample statistic)?

x̄

= 25.3 s = 4.7

&hat;p

= 0.45 μ = 30

Sampling Distributions

Sample vs Population

The Central Limit Theorem (CLT)

Sampling Variability

Key Vocabulary

Sampling Distribution

Central Limit Theorem

Standard Error

Parameter vs Statistic

Worked Examples

A population has μ = 80 and σ = 12. Find the mean and standard error of the sampling distribution of $X̄$ for n = 36.

Using Example 1, find P( $X̄$ > 83).

A poll finds 55% of 400 voters support a candidate. Find the standard error of the sample proportion.

Knowledge Check

Key Concepts Summary

Sampling Distributions

Sample vs Population

The Central Limit Theorem (CLT)

Sampling Variability

Key Vocabulary

Sampling Distribution

Central Limit Theorem

Standard Error

Parameter vs Statistic

Worked Examples

A population has μ = 80 and σ = 12. Find the mean and standard error of the sampling distribution of X̄ for n = 36.

Using Example 1, find P(X̄ > 83).

A poll finds 55% of 400 voters support a candidate. Find the standard error of the sample proportion.

Knowledge Check

Key Concepts Summary

A population has μ = 80 and σ = 12. Find the mean and standard error of the sampling distribution of $X̄$ for n = 36.

Using Example 1, find P( $X̄$ > 83).