Unit 4 Cheat Sheet: Hypothesis Testing

This cheat sheet provides a structured framework to map keywords, select the correct test, and evaluate hypotheses using the 3-method format.

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Cycle Test Scope Note

For your upcoming cycle test, Proportion Tests and Paired Sample $t$-Tests are excluded.

This cheat sheet is explicitly divided:

• PART A: Core Means Testing (What you need to study now for the cycle test).
• PART B: Proportions & Paired Tests (Kept for final semester exam reference).

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PART A: CYCLE TEST FOCUS (Means & Independent Samples)

1. Keyword-to-Tail Mapping

The first step in any hypothesis test is translating the English claim into mathematical symbols to determine the direction of the test:

Question Keyword	Alternative Hypothesis ($H_1$)	Test Direction	Rejection Region
"Is different from," "has changed," "is not equal"	$\mu \neq \mu_0$	Two-Tailed	Split across both tails ($\alpha/2$)
"Is greater than," "has increased," "lasts longer"	$\mu > \mu_0$	Right-Tailed	Upper tail only ($\alpha$)
"Is less than," "has decreased," "is underweight"	$\mu < \mu_0$	Left-Tailed	Lower tail only ($\alpha$)

(Note: The Null Hypothesis $H_0$ ALWAYS contains the equality condition: $=, \le, \text{ or } \ge$.)

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2. Quick-Glance Test Selection (Means)

• One-Sample Test (Comparing a single sample to a benchmark $\mu_0$):

  • If $n \ge 30$ or $\sigma$ is known $\implies$ $Z$-Test $SE = \frac{\sigma}{\sqrt{n}} \implies Z_{\text{cal}} = \frac{\bar{x} - \mu_0}{SE}$
  • If $n < 30$ and $\sigma$ is unknown $\implies$ $t$-Test ($df = n - 1$) $SE = \frac{s}{\sqrt{n}} \implies t_{\text{cal}} = \frac{\bar{x} - \mu_0}{SE}$

• Two-Sample Independent Test (Comparing two completely separate groups):

  • If $n_1, n_2 \ge 30 \implies$ Two-Sample $Z$-Test $SE_{\text{diff}} = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$
  • If $n_1, n_2 < 30 \implies$ Two-Sample $t$-Test with Pooled Variance ($df = n_1 + n_2 - 2$) $s_p^2 = \frac{\sum (x_{1i} - \bar{x}_1)^2 + \sum (x_{2i} - \bar{x}_2)^2}{n_1 + n_2 - 2}$ $SE_{\text{diff}} = \sqrt{s_p^2 \left( \frac{1}{n_1} + \frac{1}{n_2} \right)}$
  • Test Statistic: $Z_{\text{cal}} \text{ or } t_{\text{cal}} = \frac{(\bar{x}_1 - \bar{x}_2) - 0}{SE_{\text{diff}}}$

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3. The 3-Method Evaluation Framework

Regardless of whether you use a $Z$ or $t$ test, the decision logic is always identical across these three methods:

Method 1: Critical Region Technique

1. Look up the critical value ($Z_{\text{crit}}$ or $t_{\text{crit}}$) from the table using $\alpha$ and $df$ (see the Statistical Tables Guide or Statistical Tables PDF).
2. Decision Rule: Reject $H_0$ if the calculated statistic falls past the critical boundary into the tail(s):
   • Left-Tailed: Reject if $Z_{\text{cal}} < -Z_{\text{crit}}$
   • Right-Tailed: Reject if $Z_{\text{cal}} > +Z_{\text{crit}}$
   • Two-Tailed: Reject if $\lvert Z_{\text{cal}} \rvert > Z_{\text{crit}}$

Method 2: P-Value Technique

1. Use the table to find the probability tail area beyond your calculated test statistic. (Multiply by 2 if it is a two-tailed test).
2. Decision Rule: Reject $H_0$ if $p\text{-value} \le \alpha$.

Method 3: Confidence Interval Technique

1. Construct the $(1 - \alpha)$ confidence interval: $\bar{x} \pm (\text{Critical Value} \times SE)$.
2. Decision Rule: Reject $H_0$ if the claimed parameter ($\mu_0$, or $0$ for two-sample differences) falls outside the calculated interval.

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4. Worked Archetypes (Cycle Test Focus)

Archetype 1: One-Sample $Z$-Test (Two-Tailed)

Scenario: Claim: average sugar content is $15\text{g}$ with $\sigma = 3\text{g}$. A sample of $n = 50$ cans has mean $\bar{x} = 14\text{g}$. Test at $\alpha = 0.05$.

1. Hypotheses: $H_0: \mu = 15$ vs $H_1: \mu \neq 15$ ($\alpha/2 = 0.025$).
2. Standard Error & Test Statistic: $SE = \frac{3}{\sqrt{50}} \approx 0.4243 \implies Z_{\text{cal}} = \frac{14 - 15}{0.4243} \approx \mathbf{-2.36}$
3. Critical Region Method: $Z_{\text{crit}} = \pm 1.96$. Since $-2.36 < -1.96 \implies$ Reject $H_0$.
4. P-Value Method: $p = 2 \times P(Z < -2.36) = 2(0.0091) = 0.0182 \le 0.05 \implies$ Reject $H_0$.
5. CI Method: $\text{CI}_{95\%} = 14 \pm 1.96(0.4243) = [13.17, 14.83]$. Since $15 \notin [13.17, 14.83] \implies$ Reject $H_0$.

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Archetype 2: One-Sample $t$-Test (One-Tailed)

Scenario: Manufacturer claims battery life is at least $20\text{ hours}$. Sample of $n = 30$ has $\bar{x} = 18\text{ hours}$ and $s = 4\text{ hours}$. Test at $\alpha = 0.05$.

1. Hypotheses: $H_0: \mu \ge 20$ vs $H_1: \mu < 20$ (Left-tailed, $df = 30 - 1 = 29$).
2. Standard Error & Test Statistic: $SE = \frac{4}{\sqrt{30}} \approx 0.7303 \implies t_{\text{cal}} = \frac{18 - 20}{0.7303} \approx \mathbf{-2.74}$
3. Critical Region Method: Left-tailed $t_{\text{crit}} = -1.699$. Since $-2.74 < -1.699 \implies$ Reject $H_0$.
4. P-Value Method: $p = P(t_{29} < -2.74) = 0.0052 \le 0.05 \implies$ Reject $H_0$.
5. CI Method: $\text{Upper Bound} = 18 + 1.699(0.7303) = 19.24\text{ hours}$. Since claimed $20 > 19.24 \implies$ Reject $H_0$.

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Archetype 3: Two-Sample Independent $Z$-Test

Scenario: Battery A ($n_1 = 50, \bar{x}_1 = 280\text{h}, s_1 = 20\text{h}$) vs Battery B ($n_2 = 40, \bar{x}_2 = 260\text{h}, s_2 = 25\text{h}$). Test if Battery A lasts longer at $\alpha = 0.05$.

1. Hypotheses: $H_0: \mu_A \le \mu_B$ vs $H_1: \mu_A > \mu_B$ (Right-tailed, $\alpha = 0.05$).
2. Standard Error & Test Statistic: $SE_{\text{diff}} = \sqrt{\frac{20^2}{50} + \frac{25^2}{40}} = \sqrt{8 + 15.625} \approx 4.8606$ $Z_{\text{cal}} = \frac{280 - 260}{4.8606} \approx \mathbf{+4.11}$
3. Decision: $Z_{\text{crit}} = +1.645$. Since $+4.11 > +1.645 \implies$ Reject $H_0$. Battery A lasts significantly longer.

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PART B: EXCLUDED FROM CYCLE TEST (Proportions & Paired Tests)

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Excluded from Upcoming Cycle Test

The following tests are explicitly excluded from your current cycle test. They are documented here solely for your comprehensive semester exam reference.

1. Proportion Tests (Categorical Data)

Always uses the $Z$-Distribution.

• One-Sample Proportion ($p$ vs $p_0$): $SE_0 = \sqrt{\frac{p_0 q_0}{n}} \implies Z_{\text{cal}} = \frac{\hat{p} - p_0}{SE_0}$

• Two-Sample Proportion ($p_1$ vs $p_2$):

  • First, calculate the pooled proportion ($\bar{p}$): $\bar{p} = \frac{x_1 + x_2}{n_1 + n_2}, \quad \bar{q} = 1 - \bar{p}$
  • Then calculate standard error and $Z_{\text{cal}}$: $SE_{\text{pool}} = \sqrt{\bar{p}\bar{q} \left( \frac{1}{n_1} + \frac{1}{n_2} \right)} \implies Z_{\text{cal}} = \frac{\hat{p}_1 - \hat{p}_2}{SE_{\text{pool}}}$

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2. Paired Sample $t$-Test

Used when two measurements are taken on the same subjects (e.g., Before and After a program):

1. Reduce to One Sample: Calculate differences for each pair: $d = \text{After} - \text{Before} \quad (\text{or } \text{Before} - \text{After})$
2. Calculate Metrics of Differences:
   • Mean of differences: $\bar{d} = \frac{\sum d}{n}$
   • Standard deviation of differences: $s_d = \sqrt{\frac{\sum (d - \bar{d})^2}{n - 1}}$
3. Execute as a One-Sample $t$-Test: $SE_{\bar{d}} = \frac{s_d}{\sqrt{n}}$ $t_{\text{cal}} = \frac{\bar{d} - 0}{SE_{\bar{d}}} \quad (\text{with } df = n - 1)$