Fundamentals of Hypothesis Testing: One-Sample Tests

Chapter 9: Fundamentals of Hypothesis Testing – One-Sample Tests

Objectives of Hypothesis Testing

This chapter introduces the foundational concepts and procedures for hypothesis testing in business statistics, focusing on one-sample tests for means and proportions.

• Understand the principles of hypothesis testing

• Apply hypothesis testing to means and proportions

• Identify and evaluate assumptions of hypothesis tests

• Recognize pitfalls and ethical issues in hypothesis testing

• Learn strategies to avoid common pitfalls

What is a Hypothesis?

Definition and Examples

• Hypothesis: A claim or assertion about a population parameter (e.g., mean or proportion).

• Population Mean Example: The mean monthly cell phone bill in a city is $\mu = 42$.

• Population Proportion Example: The proportion of adults in a city with cell phones is $\pi = 0.88$.

The Null and Alternative Hypotheses

The Null Hypothesis ($H_0$)

• States the claim or assertion to be tested (e.g., $H_0: \mu = 30$).

• Always refers to a population parameter, not a sample statistic.

• Assumed true unless evidence suggests otherwise ("innocent until proven guilty").

• Contains "=", "≤", or "≥" signs.

The Alternative Hypothesis ($H_1$ or $H_a$)

• The opposite of the null hypothesis (e.g., $H_1: \mu \neq 30$).

• Challenges the status quo and is what the researcher typically seeks to support.

• Never contains "=", "≤", or "≥" signs.

The Hypothesis Testing Process

Step-by-Step Procedure

1. State the claim: Formulate $H_0$ and $H_1$ (e.g., $H_0: \mu = 50$, $H_1: \mu \neq 50$).

2. Sample the population: Collect data and compute the sample mean ($\bar{X}$).

3. Compare sample results to the claim: If the sample mean is significantly different from the hypothesized mean, consider rejecting $H_0$.

4. Decision: Use probability and critical values to determine if the observed result is unlikely under $H_0$.

The Test Statistic and Critical Values

Key Concepts

• If the sample mean is close to the hypothesized mean, do not reject $H_0$.

• If the sample mean is far from the hypothesized mean, reject $H_0$.

• Critical Value: The cutoff point(s) that define the rejection region(s) for $H_0$.

Risks in Decision Making: Type I and Type II Errors

Definitions

• Type I Error ($\alpha$): Rejecting a true null hypothesis (false alarm). Probability is $\alpha$ (level of significance).

• Type II Error ($\beta$): Failing to reject a false null hypothesis (missed opportunity). Probability is $\beta$.

Possible Outcomes Table

Decision	$H_0$ True	$H_0$ False
Do Not Reject $H_0$	Correct Decision Confidence = $1-\alpha$	Type II Error $P$(Type II Error) = $\beta$
Reject $H_0$	Type I Error $P$(Type I Error) = $\alpha$	Correct Decision Power = $1-\beta$

Additional info:

• The confidence coefficient ($1-\alpha$) is the probability of not rejecting $H_0$ when it is true.

• The power of a test ($1-\beta$) is the probability of correctly rejecting $H_0$ when it is false.

Factors Affecting Type II Error ($\beta$)

• $\beta$ increases as the difference between the hypothesized parameter and its true value decreases.

• $\beta$ decreases as $\alpha$ increases.

• $\beta$ decreases as sample size ($n$) increases.

• $\beta$ decreases as population standard deviation ($\sigma$) decreases.

Level of Significance and the Rejection Region

• The level of significance ($\alpha$) determines the size of the rejection region(s).

• For a two-tail test, the rejection regions are in both tails of the sampling distribution.

Hypothesis Tests for the Mean

Choosing the Appropriate Test

• $\sigma$ Known: Use the Z test.

• $\sigma$ Unknown: Use the t test.

Z Test of Hypothesis for the Mean ($\sigma$ Known)

• Test statistic formula:

$Z_{STAT} = \frac{\bar{X} - \mu}{\sigma / \sqrt{n}}$

• Compare $Z_{STAT}$ to critical Z values for the chosen $\alpha$.

Critical Value Approach to Testing

1. State $H_0$ and $H_1$.

2. Choose $\alpha$ and $n$.

3. Determine the test statistic and sampling distribution.

4. Find the critical values for the rejection region(s).

5. Collect data and compute the test statistic.

6. Make a decision: If the test statistic falls in the rejection region, reject $H_0$; otherwise, do not reject $H_0$.

Example: Z Test for the Mean

• Claim: The mean diameter of a manufactured bolt is 30mm ($\sigma = 0.8$).

• Hypotheses: $H_0: \mu = 30$, $H_1: \mu \neq 30$ (two-tail test).

• Sample: $n = 100$, $\bar{X} = 29.84$.

• Critical values: $\pm 1.96$ for $\alpha = 0.05$.

• Test statistic:

$Z_{STAT} = \frac{29.84 - 30}{0.8 / \sqrt{100}} = -2.0$

• Since $-2.0 < -1.96$, reject $H_0$.

p-Value Approach to Testing

Definition and Interpretation

• p-value: Probability of obtaining a test statistic as extreme or more extreme than the observed value, assuming $H_0$ is true.

• If p-value $< \alpha$, reject $H_0$; otherwise, do not reject $H_0$.

The 5-Step p-Value Approach

1. State $H_0$ and $H_1$.

2. Choose $\alpha$ and $n$.

3. Determine the test statistic and sampling distribution.

4. Collect data, compute the test statistic and p-value.

5. Make a decision: If p-value $< \alpha$, reject $H_0$.

Summary Table: Critical Value vs. p-Value Approaches

Step	Critical Value Approach	p-Value Approach
1	State $H_0$ and $H_1$	State $H_0$ and $H_1$
2	Choose $\alpha$ and $n$	Choose $\alpha$ and $n$
3	Determine test statistic and sampling distribution	Determine test statistic and sampling distribution
4	Find critical values	Compute p-value
5	Compare test statistic to critical values	Compare p-value to $\alpha$

Additional info:

• Both approaches lead to the same conclusion regarding $H_0$.

• p-value approach is more commonly used with statistical software.