FAD1015 Reader's Note — Sampling, Estimation & Hypothesis Testing

A behind-the-scenes look at Parts G and H of the Comprehensive Drill. This note builds both intuition (what is actually happening) and mechanics (how to compute without thinking).

The Big Picture: What Is Statistical Inference?

You have a population (all UM students). You want to know the average height $\mu$. You cannot measure everyone. So you take a sample, compute $\bar{x}$, and use that to guess $\mu$.

That is statistical inference: using what you know (the sample) to say something about what you do not know (the population).

There are two flavours:

Flavour	Question	Example
Estimation (Part G)	"What is $\mu$?"	Estimate average height of UM students
Hypothesis Testing (Part H)	"Is $\mu$ equal to some specific value?"	Is average height = 165 cm?

They are connected. A confidence interval tells you which values of $\mu$ are plausible. A hypothesis test tells you whether a specific $\mu_0$ is among those plausible values. Same coin, two sides.

Part G — Sampling & Estimation (Problems 41–46)

The Engine: Sampling Distribution of $\bar{X}$

You take one sample of size $n$ from the population. You get one $\bar{x}$.

Now imagine doing this over and over — take 1000 different samples of size $n$, compute $\bar{x}$ for each. The 1000 $\bar{x}$ values form a distribution. That is the sampling distribution of the sample mean.

Key fact: This distribution is normal (or approximately normal) with:

Mean = $\mu$ (the true population mean — same as the original population)
Standard deviation = $\displaystyle \frac{\sigma}{\sqrt{n}}$ — this is called the Standard Error

$$ \bar{X} \sim N!\left(\mu,\ \frac{\sigma}{\sqrt{n}}\right) $$

Intuition for Standard Error

Why does $\sigma/\sqrt{n}$ shrink as $n$ gets bigger?

Think of averaging more people. If you ask 5 people their height, the average could be all over the place. If you ask 1000 people, the average is much more stable — outliers cancel out. So the spread of possible $\bar{x}$ values gets smaller. That is the $\sqrt{n}$ in the denominator — it pulls the distribution tighter.

Sample size $n$	Standard Error	Behaviour
1	$\sigma$	Same spread as population
4	$\sigma/2$	Half the spread
100	$\sigma/10$	One-tenth the spread

The Central Limit Theorem (CLT) — Why You Can Sleep at Night

The CLT says: even if the population is not normal, the sampling distribution of $\bar{x}$ becomes approximately normal when $n$ is large enough.

$n \ge 30$: good enough for most populations
$n \ge 5$: good enough if the population is fairly symmetric
If the population is already normal: any $n$ works

This is why normal distribution methods work even when your raw data looks nothing like a bell curve — the averages still behave nicely.

Confidence Intervals — The Fishing Net

A point estimate ($\bar{x} = 165$ cm) is a single number. It is almost certainly wrong — the true $\mu$ is not exactly 165.000... So instead, give a range:

$$ \bar{x} \pm \text{Margin of Error} $$

This is a Confidence Interval.

Intuition

Imagine casting a net to catch a fish ($\mu$). A wider net is more likely to catch it. A 99% net is wider than a 95% net. A larger sample gives a narrower net (because the standard error is smaller).

$$ \text{Width} \propto \frac{\text{critical value} \times \sigma}{\sqrt{n}} $$

The Big Trap: What "95% Confident" Means

Correct: "If we repeated this sampling process many times, 95% of the constructed intervals would contain the true $\mu$."

Wrong: "There is a 95% probability that $\mu$ lies in this specific interval."

$\mu$ is a fixed number, not a random variable. Either it is in the interval or it is not. The 95% is about the procedure — how reliable it is over many repetitions.

The z vs t Decision Tree (CRITICAL)

This is the #1 place to lose marks. Memorise this flowchart:

Is σ known?
├── YES → Use z (normal distribution)
│         CI = x̄ ± z·σ/√n
│
└── NO → What is n?
         ├── n ≥ 30 → Use z with s (CLT applies)
         │            CI = x̄ ± z·s/√n
         │
         └── n < 30 → Use t (Student's t)
                       df = n - 1
                       CI = x̄ ± t·s/√n

The z Values to Memorise

Confidence Level	$\alpha$	$z_{\alpha/2}$
90%	0.10	1.645
95%	0.05	1.96
99%	0.01	2.576

The t Value: Where to Find It

$t_{\alpha/2,, n-1}$ — read from the t-table:

Rows = degrees of freedom ($n-1$)
Columns = $\alpha/2$ (for a two-tailed CI)

Example: 95% CI, $n = 25$, $\sigma$ unknown

$df = 24$, $\alpha/2 = 0.025$
$t_{0.025,,24} = 2.064$

Rule of thumb: $t$ is always larger than $z$ for the same confidence level. The $t$-distribution has fatter tails because using $s$ instead of $\sigma$ adds uncertainty. As $n$ grows, $t \to z$.

Sample Size Formula (Problem 45–46)

If you want a CI with a specific margin of error $E$:

$$ n = \left(\frac{z_{\alpha/2} \cdot \sigma}{E}\right)^2 $$

Always round UP to the nearest whole number. If the formula gives 61.5, the answer is $n = 62$. A sample of 61 would give a margin of error larger than desired.

Trade-off: Higher confidence (99% vs 95%) → larger $z$ → larger $n$. More precision (smaller $E$) → larger $n$.

Part H — Hypothesis Testing (Problems 47–54)

The Courtroom Analogy

Element	Courtroom	Hypothesis Testing
Defendant	Innocent until proven guilty	$H_0$ assumed true until evidence says otherwise
Verdict	Guilty / Not guilty	Reject $H_0$ / Fail to reject $H_0$
Evidence	Witnesses, DNA	Test statistic, p-value
Threshold	"Beyond reasonable doubt"	$\alpha$ (significance level)

Step 1: State $H_0$ and $H_1$

$H_0$ (Null hypothesis): Always contains equality ($=$, $\le$, or $\ge$). It is the status quo — what you assume is true.

$H_1$ or $H_a$ (Alternative hypothesis): What you want to find evidence for. It determines the tail direction.

Problem says...	$H_0$	$H_1$	Type
"Test if mean differs from 100"	$\mu = 100$	$\mu \neq 100$	Two-tailed
"Test if mean is less than 8"	$\mu \ge 8$ (often written $\mu = 8$)	$\mu < 8$	Left-tailed
"Test if mean is greater than 95"	$\mu \le 95$ (often written $\mu = 95$)	$\mu > 95$	Right-tailed

Shortcut: $H_1$ always points in the direction of the claim. $H_0$ is the opposite.

Step 2: Choose $\alpha$

Standard values: 0.10, 0.05, 0.01. Usually given in the problem.

Step 3: Calculate the Test Statistic

Identical formula to the z/t for CIs:

$$ z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}} \qquad \text{or} \qquad t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} $$

The same decision tree for z vs t applies:

Condition	Test
$\sigma$ known	z-test
$\sigma$ unknown, $n \ge 30$	z-test with $s$
$\sigma$ unknown, $n < 30$	t-test, $df = n-1$

Step 4–5: Make a Decision (Three Ways, Same Answer)

All three methods always agree. Pick whichever is easiest.

Method 1: Critical Value (Traditional)

Compare your test statistic to a critical value from the table:

Test Type	Reject $H_0$ if...	Example ($\alpha = 0.05$)
Two-tailed ($\neq$)	$\vert\text{stat}\vert > z_{\alpha/2}$	$z > 1.96$ or $z < -1.96$
Right-tailed ($>$)	$\text{stat} > z_{\alpha}$	$z > 1.645$
Left-tailed ($<$)	$\text{stat} < -z_{\alpha}$	$z < -1.645$

Method 2: P-value

What is a p-value? The probability of observing your sample result (or something more extreme) if $H_0$ were true.

Small p-value → the sample result is unlikely under $H_0$ → evidence against $H_0$
Large p-value → the sample result is plausible under $H_0$ → not enough evidence

Decision rule:

$p \le \alpha$ → Reject $H_0$ (statistically significant)
$p > \alpha$ → Fail to reject $H_0$

The p-value tells you "how weird" your result is. A p-value of 0.0215 means: "If $H_0$ is true, there is only a 2.15% chance of seeing data this extreme." That is weird enough to reject at $\alpha = 0.05$, but not at $\alpha = 0.01$.

Method 3: Confidence Interval

Construct a $(1-\alpha)$ CI for $\mu$:

$\mu_0$ inside the CI → Fail to reject $H_0$
$\mu_0$ outside the CI → Reject $H_0$

This works because the CI shows the plausible values of $\mu$. If $\mu_0$ is not plausible (outside the CI), reject it.

Reading R Output (Problems 53–54)

When you see R output like:

One Sample t-test
data:  sample
t = 2.456, df = 24, p-value = 0.0215
alternative hypothesis: true mean is not equal to 50
95 percent confidence interval:
  50.23 54.77
sample estimates:
mean of x
  52.5

Read it:

Field	What it tells you	Action
`t = 2.456`	Test statistic	Compare to critical value if using traditional method
`df = 24`	$n - 1$, so $n = 25$
`p-value = 0.0215`	Probability under $H_0$	If $p < \alpha$, reject $H_0$
`alternative hypothesis: true mean is not equal to 50`	This is a two-tailed test ($H_1: \mu \neq 50$)
`95 percent CI: (50.23, 54.77)`	Does NOT contain $\mu_0 = 50$	Reject $H_0$ at $\alpha = 0.05$
`mean of x: 52.5`	Sample mean $\bar{x}$

Mechanical Reference Tables

Choosing z vs t

Situation	Use	Formula
$\sigma$ known	z	$\bar{x} \pm z_{\alpha/2}\cdot\sigma/\sqrt{n}$
$\sigma$ unknown, $n \ge 30$	z with $s$	$\bar{x} \pm z_{\alpha/2}\cdot s/\sqrt{n}$
$\sigma$ unknown, $n < 30$	t	$\bar{x} \pm t_{\alpha/2,,n-1}\cdot s/\sqrt{n}$

Critical z Values

$\alpha$ (two-tailed)	$z_{\alpha/2}$
0.10 (90% CI)	1.645
0.05 (95% CI)	1.96
0.01 (99% CI)	2.576

One-Tailed vs Two-Tailed Critical Values

For one-tailed tests at $\alpha$:

90% → $z = 1.282$
95% → $z = 1.645$
99% → $z = 2.326$

For two-tailed tests at $\alpha$:

90% → $z = 1.645$
95% → $z = 1.96$
99% → $z = 2.576$

Writing Hypotheses Cheat Sheet

Keywords in problem	$H_1$	Tail
"different from", "changed", "not equal"	$\neq$	Two
"greater than", "increased", "more than", "above"	$>$	Right
"less than", "decreased", "fewer than", "below"	$<$	Left

P-value Decision

if p <= alpha:
    reject H0
else:
    fail to reject H0

Common Traps

Rounding sample size: Always round up. $n = 61.1$ → $n = 62$.
z vs t: $\sigma$ unknown + $n < 30$ = t. Everything else = z.
$H_0$ always has equality. Even if the problem says "test if mean is less than 100", $H_0$ is $\mu \ge 100$ (or $\mu = 100$ in many textbooks).
CI and hypothesis testing agree. If you reject $H_0$ with a test, the corresponding CI will exclude $\mu_0$.
P-value is not the probability $H_0$ is true. It is the probability of the data given $H_0$ is true. Subtle but important.
$n-1$ is the degrees of freedom. In R output, $df = n-1$.

FAD1015 Reader's Note — Sampling, Estimation & Hypothesis Testing (Parts G & H)