Learning Objectives

By the end of this chapter, you will be able to:

  • Follow the systematic steps in hypothesis testing
  • Find critical values for different test types
  • Identify rejection regions for one-tailed and two-tailed tests
  • Make decisions using both critical value and p-value approaches
  • Apply hypothesis testing to practical problems

The Six Steps of Hypothesis Testing

flowchart TD
    A["Step 1: State H₀ and H₁"]
    B["Step 2: Choose α (significance level)"]
    C["Step 3: Calculate test statistic"]
    D["Step 4: Find critical value or p-value"]
    E["Step 5: Make decision"]
    F["Step 6: State conclusion"]

    A --> B --> C --> D --> E --> F

Step 1: State the Hypotheses

Rules for Formulating Hypotheses

  1. H₀ always contains equality (=, ≤, or ≥)
  2. H₁ is the complement (≠, >, or <)
  3. The claim you want to test is usually in H₁

Identifying the Type of Test

H₁ Contains Test Type Rejection Region
≠ Two-tailed Both tails
> Right-tailed Right tail only
< Left-tailed Left tail only

Step 2: Choose Significance Level (α)

Common choices:

  • α = 0.10 (less strict, exploratory research)
  • α = 0.05 (standard, most common)
  • α = 0.01 (strict, medical/safety research)

Step 3: Calculate Test Statistic

For Population Mean (σ known)

\[z = \frac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}\]

For Population Mean (σ unknown)

\[t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}\]

For Population Proportion

\[z = \frac{\hat{p} - p_0}{\sqrt{p_0(1-p_0)/n}}\]

Step 4: Find Critical Values

Critical Values for z-Tests

α One-Tailed Two-Tailed
0.10 ±1.28 ±1.645
0.05 ±1.645 ±1.96
0.01 ±2.33 ±2.576

Visualizing Critical Regions

flowchart TD
    subgraph "Two-Tailed Test (H₁: μ ≠ μ₀)"
        A["Reject | Do Not Reject | Reject"]
        B["α/2        |                      | α/2"]
        C["-z*                              +z*"]
    end

    subgraph "Right-Tailed Test (H₁: μ > μ₀)"
        D["Do Not Reject | Reject"]
        E["                        | α"]
        F["                     +z*"]
    end

    subgraph "Left-Tailed Test (H₁: μ < μ₀)"
        G["Reject | Do Not Reject"]
        H["α      |"]
        I["-z*"]
    end

Step 5: Make Decision

Critical Value Method

Two-tailed:

  • Reject H₀ if |z| > z*

Right-tailed:

  • Reject H₀ if z > z*

Left-tailed:

  • Reject H₀ if z < -z*

p-Value Method

  • Reject H₀ if p-value < α
  • Fail to reject H₀ if p-value ≥ α

Step 6: State Conclusion

Write the conclusion in context of the problem:

If Reject H₀: “At the α level of significance, there is sufficient evidence to conclude that [H₁ in words].”

If Fail to Reject H₀: “At the α level of significance, there is insufficient evidence to conclude that [H₁ in words].”

Complete Example 1: Two-Tailed Test

Problem: A government claims average processing time is 30 minutes. A sample of 64 cases shows mean time of 32 minutes with known σ = 8 minutes. Test at α = 0.05.

Solution:

Step 1: State hypotheses

  • $H_0: \mu = 30$
  • $H_1: \mu \neq 30$ (two-tailed)

Step 2: Significance level

  • α = 0.05

Step 3: Calculate test statistic \(z = \frac{\bar{x} - \mu_0}{\sigma/\sqrt{n}} = \frac{32 - 30}{8/\sqrt{64}} = \frac{2}{1} = 2.0\)

Step 4: Find critical value

  • Two-tailed at α = 0.05: z* = ±1.96

Step 5: Decision

  • |z| = 2.0 > 1.96
  • Reject H₀

Step 6: Conclusion “At the 0.05 level of significance, there is sufficient evidence to conclude that the average processing time is different from 30 minutes.”

Complete Example 2: Right-Tailed Test

Problem: A manager claims average productivity has increased from 100 units. A sample of 36 workers shows mean of 105 units with s = 18. Test at α = 0.05.

Solution:

Step 1: State hypotheses

  • $H_0: \mu = 100$
  • $H_1: \mu > 100$ (right-tailed)

Step 2: Significance level

  • α = 0.05

Step 3: Calculate test statistic Since σ unknown and n ≥ 30, we can use z: \(z = \frac{105 - 100}{18/\sqrt{36}} = \frac{5}{3} = 1.67\)

Step 4: Find critical value

  • Right-tailed at α = 0.05: z* = 1.645

Step 5: Decision

  • z = 1.67 > 1.645
  • Reject H₀

Step 6: Conclusion “At the 0.05 level of significance, there is sufficient evidence to conclude that average productivity has increased from 100 units.”

Complete Example 3: Left-Tailed Test

Problem: A hospital claims average wait time is less than 20 minutes. A sample of 49 patients shows mean wait of 18.5 minutes with s = 7 minutes. Test at α = 0.05.

Solution:

Step 1: State hypotheses

  • $H_0: \mu = 20$
  • $H_1: \mu < 20$ (left-tailed)

Step 2: Significance level

  • α = 0.05

Step 3: Calculate test statistic \(z = \frac{18.5 - 20}{7/\sqrt{49}} = \frac{-1.5}{1} = -1.5\)

Step 4: Find critical value

  • Left-tailed at α = 0.05: z* = -1.645

Step 5: Decision

  • z = -1.5 > -1.645 (not in rejection region)
  • Fail to Reject H₀

Step 6: Conclusion “At the 0.05 level of significance, there is insufficient evidence to conclude that average wait time is less than 20 minutes.”

Using p-Values

Calculating p-Values

Two-tailed test: \(p\text{-value} = 2 \times P(Z > \lvert z \rvert)\)

Right-tailed test: \(p\text{-value} = P(Z > z)\)

Left-tailed test: \(p\text{-value} = P(Z < z)\)

Example 4: p-Value Approach

Using Example 1 data (z = 2.0, two-tailed):

\[p\text{-value} = 2 \times P(Z > 2.0) = 2 \times (1 - 0.9772) = 2 \times 0.0228 = 0.0456\]

Since p-value = 0.0456 < α = 0.05, Reject H₀.

Comparison of Methods

Method Process Advantage
Critical Value Compare test stat to critical value Visual, intuitive
p-Value Compare p-value to α More informative, exact probability

Both methods always give the same decision!

Quick Reference: Critical Values

Z-Distribution

α Left-tail Right-tail Two-tailed
0.10 -1.28 1.28 ±1.645
0.05 -1.645 1.645 ±1.96
0.01 -2.33 2.33 ±2.576

t-Distribution (Selected)

df α = 0.05 (one) α = 0.05 (two)
10 1.812 2.228
20 1.725 2.086
30 1.697 2.042
60 1.671 2.000
∞ 1.645 1.960

Common Mistakes to Avoid

1. Wrong Alternative Hypothesis

❌ H₁: μ = 35 (contains equality) ✅ H₁: μ > 35 (no equality)

2. Using Wrong Critical Value

❌ Using two-tailed critical value for one-tailed test ✅ Match critical value to test type

3. Wrong Conclusion Statement

❌ “Accept H₀” ✅ “Fail to reject H₀”

4. Not Stating in Context

❌ “Reject H₀” ✅ “There is sufficient evidence that processing time has changed.”

Decision Flow Chart

flowchart TD
    A[Calculate test statistic z]
    B{What type of test?}
    C[Two-tailed:<br/>Is \|z\| > z*?]
    D[Right-tailed:<br/>Is z > z*?]
    E[Left-tailed:<br/>Is z < -z*?]
    F[Reject H₀]
    G[Fail to Reject H₀]

    A --> B
    B -->|"H₁: μ ≠"| C
    B -->|"H₁: μ >"| D
    B -->|"H₁: μ <"| E

    C -->|Yes| F
    C -->|No| G
    D -->|Yes| F
    D -->|No| G
    E -->|Yes| F
    E -->|No| G

Practice Problems

Problem 1

A company claims average delivery time is 5 days. A sample of 100 deliveries shows mean of 5.4 days with σ = 2 days. Test at α = 0.05 whether delivery time has increased.

Problem 2

Test if average score differs from 70 given:

  • Sample: n = 49, $\bar{x}$ = 68, s = 14
  • Use α = 0.01

Problem 3

For z = 2.15 in a right-tailed test: (a) Find the p-value (b) What is the decision at α = 0.05? (c) What is the decision at α = 0.01?

Problem 4

A hospital wants to test if average stay is less than 4 days. Sample: n = 36, $\bar{x}$ = 3.6, s = 1.2. Test at α = 0.05.

Problem 5

Explain why we use α = 0.05 in the left tail for a left-tailed test, but α/2 = 0.025 in each tail for a two-tailed test.

Summary

Component Key Point
H₀ Always contains equality
Type of Test Determined by H₁
Critical Region Where we reject H₀
Decision Compare test stat to critical value OR p-value to α
Conclusion State in context of problem

Next Topic

In the next chapter, we will apply these concepts to Large Sample Tests for Single Mean - z-tests when σ is known or n ≥ 30.