Unit 3 Cheat Sheet: Sampling & Estimation

This cheat sheet provides a practical mapping framework to translate raw exam questions into correct parameters, select the appropriate formula, and compute exact confidence intervals.

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1. Quick-Glance Decision Rules

• Proportions ($p$): Always use the $Z$-distribution ($SE = \sqrt{\frac{\hat{p}\hat{q}}{n}}$).
• Means ($\mu$):
  • If $n \ge 30$ or $\sigma$ is known $\implies$ $Z$-distribution ($SE = \frac{\sigma}{\sqrt{n}}$).
  • If $n < 30$ and $\sigma$ is unknown $\implies$ $t$-distribution with $df = n - 1$ ($SE = \frac{s}{\sqrt{n}}$).
• Finite Population Correction (FPC): If total population $N$ is given and $\frac{n}{N} > 0.05$ ($5\%$) $\implies$ Multiply $SE$ by $\sqrt{\frac{N - n}{N - 1}}$.

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2. Problem-Clue Mapping Matrix

Use this table to map keywords in exam questions directly to mathematical symbols:

Question Keyword / Clue	Mathematical Meaning	Extracted Symbol	Action Required
"Sample of 36 employees..."	Sample size	$n = 36$	If $n \ge 30$, use $Z$-distribution.
"Sample of 15 students..."	Small sample size	$n = 15$	If $\sigma$ unknown, use $t$-distribution ($df = 14$).
"Population standard deviation of 6..."	Known population variance	$\sigma = 6$	Use $Z = \frac{\bar{x} - \mu}{\sigma/\sqrt{n}}$.
"Sample standard deviation of 10..."	Sample standard deviation only	$s = 10$	Use $t = \frac{\bar{x} - \mu}{s/\sqrt{n}}$ if $n < 30$.
"180 reported satisfied out of 300..."	Sample proportion successes	$x = 180, \, n = 300$	Compute $\hat{p} = \frac{180}{300} = 0.60, \, \hat{q} = 0.40$.
"Assuming a total of 1000 customers..."	Finite population size	$N = 1000$	Check $\frac{n}{N} = \frac{300}{1000} = 0.30 > 0.05 \implies$ Apply FPC.
"At a 95% confidence level..."	Confidence level ($1 - \alpha$)	$1 - \alpha = 0.95$	Lookup $Z_{\alpha/2} = 1.960$ or $t_{0.025, df}$.

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3. The "Golden" Critical Values Table

Memorize these two-tailed critical values for instant exam calculations:

Confidence Level ($1 - \alpha$)	Alpha ($\alpha$)	Tail Area ($\alpha/2$)	Normal Critical Value ($Z_{\text{crit}}$)
90%	$0.10$	$0.05$	$\pm 1.645$
95%	$0.05$	$0.025$	$\pm 1.960$
99%	$0.01$	$0.005$	$\pm 2.576$

(Note: For $t$-values, always locate the column $\alpha/2$ matching your row $df = n - 1$. Refer to the Statistical Tables Guide or view the Statistical Tables PDF.)

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4. Worked Problem Archetypes

Every Unit 3 problem maps to one of the following three templates.

Archetype 1: Population Mean with Known $\sigma$ or Large Sample ($Z$-Interval)

Exam Scenario: A random sample of $36$ employees spends an average of $22\text{ hours}$ on training with a population standard deviation of $6\text{ hours}$. Construct a $95\%$ confidence interval for the true mean training time.

How to Map the Problem

1. Extract Parameters:
   • $\bar{x} = 22\text{ hours}$
   • $\sigma = 6\text{ hours}$ (population standard deviation is known)
   • $n = 36$ ($n \ge 30 \implies$ Normal $Z$-distribution applies)
   • $95\%\text{ confidence} \implies Z_{0.025} = \mathbf{1.960}$
2. Calculate Standard Error ($SE$): $SE = \frac{\sigma}{\sqrt{n}} = \frac{6}{\sqrt{36}} = \frac{6}{6} = \mathbf{1.0\text{ hour}}$
3. Calculate Margin of Error ($ME$): $ME = Z_{\alpha/2} \times SE = 1.960 \times 1.0 = \mathbf{1.960\text{ hours}}$
4. Construct Confidence Interval: $\text{CI} = \bar{x} \pm ME = 22 \pm 1.960 = \mathbf{[20.040, \, 23.960]\text{ hours}}$

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Archetype 2: Population Mean with Small Sample & Unknown $\sigma$ ($t$-Interval)

Exam Scenario: A researcher tests $15$ students using a new study method. The sample mean test score is $78$ with a sample standard deviation of $10$. Construct a $95\%$ confidence interval for the true average score.

How to Map the Problem

1. Extract Parameters:
   • $\bar{x} = 78$
   • $s = 10$ ($\sigma$ is unknown)
   • $n = 15$ ($n < 30 \implies$ Small sample, requires $t$-distribution)
   • Degrees of Freedom: $df = n - 1 = 15 - 1 = \mathbf{14}$
2. Lookup $t$-Critical Value:
   • For $95\%$ confidence ($\alpha/2 = 0.025$) with $df = 14$: $t_{0.025, 14} = \mathbf{2.145}$
3. Calculate Standard Error ($SE$): $SE = \frac{s}{\sqrt{n}} = \frac{10}{\sqrt{15}} \approx \frac{10}{3.873} \approx \mathbf{2.582}$
4. Calculate Margin of Error ($ME$): $ME = t_c \times SE = 2.145 \times 2.582 \approx \mathbf{5.538}$
5. Construct Confidence Interval: $\text{CI} = 78 \pm 5.538 = \mathbf{[72.46, \, 83.54]}$

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Archetype 3: Proportion with Finite Population Correction ($p$ + FPC)

Exam Scenario: Out of $1,000$ registered retail customers, a random sample of $300$ is surveyed. $180$ report being satisfied. Construct a $95\%$ confidence interval for the true proportion of satisfied customers.

How to Map the Problem

1. Extract Parameters:
   • $N = 1000$ (Total population)
   • $n = 300$ (Sample size)
   • $x = 180$ (Success count)
   • $\hat{p} = \frac{180}{300} = \mathbf{0.60}, \quad \hat{q} = 1 - 0.60 = \mathbf{0.40}$
   • $95\%\text{ confidence} \implies Z_{0.025} = \mathbf{1.960}$
2. Check the FPC Condition: $\frac{n}{N} = \frac{300}{1000} = 0.30\text{ (}30\%\text{)} > 0.05 \implies \text{\textbf{Must apply FPC!}}$
3. Calculate FPC Multiplier: $\text{FPC} = \sqrt{\frac{N - n}{N - 1}} = \sqrt{\frac{1000 - 300}{1000 - 1}} = \sqrt{\frac{700}{999}} \approx \mathbf{0.837079}$
4. Calculate Unadjusted and Adjusted Standard Error: $SE = \sqrt{\frac{\hat{p}\hat{q}}{n}} = \sqrt{\frac{0.60 \times 0.40}{300}} = \sqrt{0.0008} \approx \mathbf{0.028284}$ $SE_{\text{adj}} = SE \times \text{FPC} = 0.028284 \times 0.837079 \approx \mathbf{0.023676}$
5. Calculate Margin of Error & Interval: $ME = 1.960 \times 0.023676 \approx \mathbf{0.0464}$ $\text{CI} = 0.60 \pm 0.0464 = \mathbf{[0.5536, \, 0.6464]}\quad (\mathbf{[55.36\%, \, 64.64\%]})$

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5. Universal Execution Checklist for Exams

When solving any estimation problem, write down these 4 numbered steps on your exam script to ensure full method marks:

1. Step 1: Identify Parameters: List $n, \bar{x}\text{ (or } \hat{p}), \sigma\text{ (or } s), N\text{ (if given)}$.
2. Step 2: Justify Distribution Choice: State "Since $n \ge 30$, we use the $Z$-distribution" OR "Since $n < 30$ and $\sigma$ is unknown, we use the $t$-distribution with $df = n - 1$".
3. Step 3: State & Compute $SE$ and $ME$: Show the standard error formula and multiply by the critical value.
4. Step 4: State Final Interval with Units: $\text{Lower Limit} = \text{Estimate} - ME$, $\text{Upper Limit} = \text{Estimate} + ME$.

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6. Exam Traps & Pro-Tips

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Common Traps to Avoid

1. Standard Deviation vs. Standard Error: Never use $\sigma$ or $s$ directly as your margin of error. You must always divide by $\sqrt{n}$ to obtain the Standard Error ($SE$) before multiplying by the critical value.
2. Forgetting FPC: If a question explicitly specifies a total population size (e.g., "Assuming a total of 1000 customers..."), it is an intentional signal to test whether you calculate and apply the FPC factor.
3. Hypothesis Testing via Confidence Interval (Duality): When an exam question asks whether a claimed value $\mu_0$ is supported by the sample data:
   • If $\mu_0$ falls inside the $(1 - \alpha)$ interval $\implies$ Fail to reject $H_0$ (Claim is plausible).
   • If $\mu_0$ falls outside the $(1 - \alpha)$ interval $\implies$ Reject $H_0$ (Claim is significantly different).