Unit 5 — Sampling Theory I

Management examples of sampling, principles of sampling, simple random sample, Excel functions for selecting a random sample. Sampling distributions for the statistic arithmetic mean — the concept of standard error.

1. Why We Sample, and How Core syllabus concept

1Statistic vs parameter — the distinction everything rests on

The deck opens with it slide 2:

Statistic — a numerical value computed from a sample.
Parameter — a numerical value associated with the population.
"Essentially, we would like to know the parameter. But in most cases it is hard to know the parameter since the population is too large. So we have to estimate the parameter by some proper statistic computed from the sample."

That paragraph is the answer to "why do we sample?" and it is also the reason for the symbol convention used everywhere below:

Quantity	Population (parameter)	Sample (statistic)	Sampling distribution
Mean	μ	x̄	μ_x̄ and σ_x̄
Standard deviation	σ	s	σ_x̄ = the standard error
Proportion	p	p̂	μ_p̂ and σ_p̂
Size	N	n	—

Your class notes fix the convention in one line: "μ ke jagah μ_x̄, σ ke jagah σ_x̄" — in sampling, μ is replaced by μ_x̄ and σ by σ_x̄ Class notes p. 2.

2Types of sampling

The syllabus names simple random sampling specifically; Levin §6.2–6.3 gives the full set pp. 271–282:

Method	How it works	Use it when
Simple random	Every element has an equal chance of selection; every sample of size n is equally likely.	The population list is available and reasonably homogeneous.
Systematic	Take every k^th element after a random start.	You have an ordered list and want speed. Risky if the list has a hidden cycle.
Stratified	Split the population into homogeneous strata, then sample randomly within each.	The population has distinct subgroups you must represent (e.g. all four year-groups).
Cluster	Split into clusters, randomly choose whole clusters, survey everyone in them.	The population is geographically spread and travel is the main cost.
Non-random (judgement / convenience / quota) — cheap and fast, but the sampling error cannot be quantified, so none of the formulas below apply. Levin §6.3 p. 279.

Excel functions the syllabus asks for: =RAND() gives a uniform random number in [0,1) — put it beside each record, sort by it, take the top n. =RANDBETWEEN(1, N) picks a random record number directly. The Data Analysis ToolPak's Sampling tool does both periodic and random sampling in one dialog.

Sources

Primary Sampling Distribution deck — statistic vs parameter slide 2

Textbook Levin §6.1–6.4 Introduction to sampling, random sampling, non-random sampling, design of experiments pp. 268–285

2. The Sampling Distribution of the Mean Core syllabus concept

1The definition — write it exactly like this

The deck slide 3 and your class notes Class notes p. 1 agree word for word:

A sampling distribution of the mean is the probability distribution of the means of all the possible samples of a given size that could be drawn out of a given population.

The crucial thing to grasp: x̄ is itself a random variable. Draw a different sample and you get a different x̄. Since it is a random variable, it has its own distribution, its own mean, and its own standard deviation — and that standard deviation is what we call the standard error.

2Worked example — building one by hand

Your class built a sampling distribution from scratch, which is the fastest way to see why it exists Class notes p. 1. Five employees — Carol 3, Debbie 3, Elizabeth 7, Frank 9, George 14 years of service. Take all possible samples of size 3.

Step 1 — How many samples are there? ⁵C₃ = 5! / (3! · 2!) = 10 samples
Step 2 — List every sample mean and how often it occurs

Sample mean x̄ 4⅓ 5 6⅓ 6⅔ 8 8⅔ 10 Total

Frequency f 1 1 2 1 2 2 1 10

That frequency table is the sampling distribution of the mean.
Step 3 — Check the population mean against the mean of the sampling distribution μ = (3 + 3 + 7 + 9 + 14) / 5 = 36/5 = 7.2 μ_x̄ = Σ f·x̄ / Σf = 72 / 10 = 7.2 ✓
They are equal — which is the first clause of the Central Limit Theorem, and no coincidence.

Sample mean x̄	4⅓	5	6⅓	6⅔	8	8⅔	10	Total
Frequency f	1	1	2	1	2	2	1	10

Sources

Primary Sampling Distribution deck — definition, worked Excel example slides 3–7

Class Five-employee sampling distribution built by hand p. 1

Textbook Levin §6.5–6.6 Introduction to sampling distributions; in more detail pp. 286–301

3. The Central Limit Theorem Core syllabus concept

1The three statements

The deck states it in three bullets slide 8:

The mean of the population is always equal to the mean of the sampling distribution of the mean. μ_x̄ = μ
As the sample size increases, the sampling distribution of the mean approaches normality, regardless of the shape of the population distribution.
That relationship between the shape of the population and the shape of the sampling distribution is what we call the Central Limit Theorem.

Your class notes put statement 2 in the most usable form Class notes p. 1: "even if the data is NOT normal, in sample as per CLT we can consider data to be normal."

2Why it matters so much

The deck's own answer slide 9: "It is perhaps the most important theorem in inferential statistics. The significance of CLT is that without the knowledge of the frequency distribution of the population, it permits us to use the sample statistics to make inferences about the population parameters."

The practical consequence, and the reason every exam question here works: because x̄ is approximately normal, you may standardise it with a z and read the ordinary normal table — even though you know nothing about the shape of the underlying population. Every Q1 (a) and Q5 (b) silently relies on this.

The conventional threshold is n ≥ 30: at that point the sampling distribution is close enough to normal for any population shape. If the population is already normal, x̄ is exactly normal for any n.

ENCentral limit theorem — sampling distribution of sample means The Organic Chemistry Tutor ENIntro to sampling distributions, CLT, standard error of x̄ chrisferb ENThe central limit theorem Simple Learning Pro

Sources

Primary Sampling Distribution deck — CLT and its significance slides 8–9

Lecture Sample Size deck — CLT plus the diagram of sample means for various n slides 20–21

Textbook Levin §6.6 Sampling distributions in more detail pp. 289–301

4. The Standard Error of the Mean Core syllabus concept · the highest-value formula in this unit

1Definition

The deck's definition slide 10:

Standard error of a statistic (mean or proportion) is the standard deviation of the sampling distribution of that statistic.

It is not the standard deviation of the data. It measures how much the sample mean bounces around from sample to sample.

Standard Error of Mean

\[ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \qquad (\mu_{\bar{x}} = \mu) \]

Finite Population Multiplier

\[ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \times \sqrt{\frac{N - n}{N - 1}} \qquad \text{(when } N \text{ is given and } n/N \ge 0.05\text{)} \]

Your class notes circle σ_x̄ = σ/√n and label it "standard error of mean" Class notes p. 2; the finite form is circled and labelled "finite population multiplier" on p. 5. Deck: slide 20.

2Standard error and sample size — the relationship the exam loves

The deck spells out the chain of consequences slide 19:

Standard error is a measure of the dispersion of the sample means around the population mean.
As it decreases, the values taken by sample means cluster more closely around the population mean.
So if SE decreases, the value of any sample mean will probably be closer to the population mean.
Putting it differently: if SE decreases, the precision with which the sample mean estimates the population mean increases.

And since σ_x̄ = σ/√n, the only lever you control is n. Quadruple the sample and you halve the standard error — the diminishing returns that Unit 6 Concept 3 makes explicit.

3Method — the four lines that answer Q1 (a) and Q5 (b)

Step 1 — Standard error

\[ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \]

Step 2 — Standardise

\[ z = \frac{\bar{x} - \mu_{\bar{x}}}{\sigma_{\bar{x}}} \]

⚠ The single most common error in this whole unit

Dividing by σ instead of σ_x̄. In the Final's Q1 (a) that would give z = (515−520)/25 = −0.2 instead of −1.2, and an answer of 0.1586 instead of 0.7698. If the question is about a sample mean x̄, the denominator is always σ/√n. Your class notes write the z formula with μ_x̄ and σ_x̄ subscripted every single time Class notes p. 4 for exactly this reason. Copy that habit.

4Worked example — the deck's bank-employees problem

A population of annual earnings of class II employees in all nationalised banks has mean $19,000 and SD $2,000. A random sample of 30 employees is drawn. What is the probability that their average annual earnings will be more than $19,750? slide 11, worked in class Class notes pp. 3–5.

Step 1 — Write down what you have μ = 19,000 σ = 2,000 n = 30 ⟹ μ_x̄ = 19,000
Step 2 — Standard error σ_x̄ = σ/√n = 2000/√30 = 2000/5.477 = 365.15
Step 3 — Draw it, then standardise
Sketch the sampling-distribution curve centred at 19,000, mark 19,750, shade the right tail.
z = (19,750 − 19,000) / 365.15 = 750 / 365.15 = 2.05
Step 4 — Read the table P(x̄ > 19,750) = P(Z > 2.05) = 0.5 − A(2.05) = 0.5 − 0.4798 = 0.0202 = 2.02%
Only about a 2% chance that a sample of 30 averages above $19,750 — the sample mean rarely strays that far from $19,000 because the standard error is only $365.

Your class notes reach 0.0202 p. 5. An earlier attempt on p. 4 reads the table as 0.48991 and gets 0.02018 — the same number to three decimals.

5The finite population multiplier — when to bother

The deck's rule slide 21:

The fraction n/N is called the sampling fraction.
When N is very large relative to n, the FPM takes a value close to 1 and changes nothing.
"As a rule of thumb, when the sampling fraction is less than 0.05, the finite population multiplier need not be used."

⚠ A discrepancy between your class notes and the deck

Your class notes write the threshold as n/N < 0.5 p. 5, p. 6. The deck says 0.05 slide 21, and so does Levin §6.7 p. 302. Almost certainly a dropped zero in the notes. Use 0.05. Practically it rarely matters: if a question hands you N at all, it usually wants the FPM used — the Final's Q4 (a) has n/N = 200/5000 = 0.04, below the threshold, and the official key applies the correction anyway Key p. 4. Safest rule: if N is given, use the FPM and say why.

6Worked example — the FPM in action

From the Re-Exam's Q4 (a) figures: N = 2,000 employees, n = 150, p̂ = 0.6 Key p. 3. Shown here for the mean-style structure; the proportion version is in Unit 6 Concept 2.

Step 1 — Compute the sampling fraction n/N = 150 / 2000 = 0.075 ⟹ 0.075 > 0.05, so apply the FPM
Step 2 — Compute the multiplier √[(N − n)/(N − 1)] = √[(2000 − 150)/(2000 − 1)] = √(1850/1999) = √0.9255 = 0.9620
Step 3 — Read what it did
It shrank the standard error by about 3.8%. That is the whole effect: sampling without replacement from a finite pool leaves less uncertainty than sampling from an infinite one, because each draw removes some of the population you were unsure about. As n → N the multiplier → 0, which is correct: measure everyone and there is no sampling error left at all.

ENSampling distribution & standard error of the mean — numericals Gourav Manjrekar ENMean and standard error of a sampling distribution of sample means Math and Stats Help ENFinite population correction factor in simple random sampling Fareeza Fayyaz

Practice G1–G6 — standard error, sampling distribution of the mean, FPM →

Sources

Primary Sampling Distribution deck — standard error, the bank example, SE vs sample size, FPM, sampling fraction slides 10–23

Class Standard error, the bank problem worked twice, the FPM rule pp. 2–6

Textbook Levin §6.7 An operational consideration in sampling: the relationship between sample size and standard error pp. 302–307

Exam Final Q1 (a) p. 1 · Q5 (b) p. 3 · Re-Exam Q1 (a) p. 1 · Q5 (b) p. 3

Unit 05 — Sampling Theory I

1. Why We Sample, and How Core syllabus concept

1Statistic vs parameter — the distinction everything rests on

2Types of sampling

2. The Sampling Distribution of the Mean Core syllabus concept

1The definition — write it exactly like this

2Worked example — building one by hand

3. The Central Limit Theorem Core syllabus concept

1The three statements

2Why it matters so much

4. The Standard Error of the Mean Core syllabus concept · the highest-value formula in this unit

1Definition

2Standard error and sample size — the relationship the exam loves

3Method — the four lines that answer Q1 (a) and Q5 (b)

4Worked example — the deck's bank-employees problem

5The finite population multiplier — when to bother

6Worked example — the FPM in action