Chapter 8: Hypothesis Testing – Structured Study Notes

Chapter 8: Hypothesis Testing

Introduction to Hypothesis Testing

Hypothesis testing is a fundamental procedure in inferential statistics, used to make decisions or inferences about population parameters based on sample data. The process involves formulating two competing hypotheses and using sample evidence to determine which is more plausible.

• Inferential statistics involves drawing conclusions about populations from samples.

• Two main activities:

  1. Estimating a population parameter (see Chapter 7).

  2. Testing a hypothesis or claim about a population parameter (focus of this chapter).

• A hypothesis is a claim or statement about a population parameter.

Types of Hypotheses

• Null Hypothesis (H0): The default or status quo claim, assumed true unless evidence suggests otherwise.

• Alternative Hypothesis (H1 or Ha): The claim we seek evidence for; accepted if sample evidence strongly contradicts H0.

• We test H0 against H1 using sample data.

Example: A medical researcher claims the mean body temperature of healthy adults is not equal to 98.6°F. Here, H0: μ = 98.6, H1: μ ≠ 98.6.

Errors in Hypothesis Testing

There are two types of errors that can occur in hypothesis testing:

• Type I Error (α): Rejecting H0 when it is actually true (false positive).

• Type II Error (β): Failing to reject H0 when H1 is actually true (false negative).

The significance level (α) is the probability of making a Type I error, commonly set at 0.05 or 0.01.

Test Statistics

A test statistic is a value calculated from sample data, used to decide whether to reject H0. The choice of test statistic depends on the parameter being tested and the sample size.

• Test statistic for mean (σ known): $z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}$

• Test statistic for mean (σ unknown): $t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$

• Test statistic for proportion: $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$

Steps in Hypothesis Testing

1. State the parameter of interest (e.g., μ, p).

2. Formulate the null and alternative hypotheses.

3. Select the significance level (α).

4. Determine the appropriate test statistic (z, t, etc.).

5. Define the rejection region(s) based on α and the test type (one-tailed or two-tailed).

6. Calculate the test statistic from the sample data.

7. Make a decision:

   • If the test statistic falls in the rejection region, reject H0.

   • If not, fail to reject H0.

8. Interpret the result in the context of the problem.

P-Value Approach

The P-value is the probability, assuming H0 is true, of obtaining a result as extreme as or more extreme than the observed sample result. If the P-value is less than α, we reject H0.

Critical Values and Rejection Regions

Critical values define the boundaries of the rejection region(s) for a given significance level. For the z-test, common critical values are:

Test Type	α	Critical Value(s)
Lower-tailed	0.05	-1.645
Upper-tailed	0.05	1.645
Two-tailed	0.05	±1.96
Lower-tailed	0.01	-2.33
Upper-tailed	0.01	2.33
Two-tailed	0.01	±2.58

Examples of Hypothesis Testing

Example 1: Metal Lathe Quality Control

• Problem: Inspectors want to determine if a machine is producing bearings with a mean diameter different from 5 inches.

• Hypotheses: H0: μ = 5, H1: μ ≠ 5

• Sample Data: n = 100, x̄ = 4.85, σ = 0.08, α = 0.01

• Test Statistic: $z = \frac{4.85 - 5}{0.08/\sqrt{100}} = -18.75$

• Conclusion: Since the test statistic is far in the rejection region, we reject H0.

Example 2: Testing a Filling Machine

• Problem: A machine is supposed to fill cans with a mean of 12 ounces. A sample of 100 cans yields x̄ = 12.1, s = 0.71, α = 0.01.

• Hypotheses: H0: μ = 12, H1: μ ≠ 12

• Test Statistic: $t = \frac{12.1 - 12}{0.71/\sqrt{100}} = 1.41$

• Critical Value: For α = 0.01, two-tailed, df = 99, tcrit ≈ ±2.626

• Conclusion: Since 1.41 does not exceed the critical value, we fail to reject H0.

Note: If σ is unknown, use the t-distribution.

Summary Table: Steps in Hypothesis Testing

Step	Description
1	State parameter and hypotheses
2	Select significance level (α)
3	Choose test statistic
4	Define rejection region
5	Compute test statistic from data
6	Make decision (reject or fail to reject H0)
7	Interpret result in context

Additional Info:

• For small samples (n < 30) and unknown σ, use the t-distribution.

• For large samples (n ≥ 30), the Central Limit Theorem allows use of the z-test even if the population is not normal.

• Always check assumptions before performing a hypothesis test.