Hypothesis Testing with One Sample – Study Notes

Chapter 7: Hypothesis Testing with One Sample

7.1 Introduction to Hypothesis Testing

Hypothesis testing is a fundamental statistical method used to make inferences about population parameters based on sample data. It provides a structured process for evaluating claims or assertions about a population.

• Hypothesis Test: A process that uses sample statistics to test a claim about the value of a population parameter.

• Statistical Hypothesis: A statement about a population parameter, such as the mean or proportion.

• Two hypotheses are always stated: one representing the claim and the other its complement. When one is false, the other must be true.

Example: If a manufacturer claims a car has a mean gas mileage of 50 mpg, a hypothesis test can be used to evaluate this claim using sample data.

7.2 Stating Null and Alternative Hypotheses

Every hypothesis test involves two competing hypotheses:

• Null Hypothesis (H0): A statement of equality (e.g., μ = 50, p = 0.61). It is the hypothesis assumed to be true at the start of the test.

• Alternative Hypothesis (Ha): The complement of the null hypothesis, containing a statement of strict inequality (e.g., μ ≠ 50, p < 0.61, μ > 18).

To write hypotheses:

1. Translate the verbal claim into a mathematical statement.

2. Write its complement as the other hypothesis.

3. Always assume the null hypothesis is true for the purpose of the test.

Examples:

• If a school claims 61% of students participate in extracurricular activities: H0: p = 0.61; Ha: p ≠ 0.61.

• If a dealership claims oil changes take less than 15 minutes: H0: μ ≥ 15; Ha: μ < 15.

• If a company claims furnace life is more than 18 years: H0: μ ≤ 18; Ha: μ > 18.

7.3 Types of Errors in Hypothesis Testing

Because decisions are based on samples, there is always a risk of making an incorrect decision. Two types of errors can occur:

• Type I Error (α): Rejecting the null hypothesis when it is actually true.

• Type II Error (β): Failing to reject the null hypothesis when it is actually false.

Example: If the USDA limit for salmonella in turkey is 13.5%, a Type I error would mean falsely concluding contamination exceeds the limit, while a Type II error would mean failing to detect contamination above the limit. Type II errors can be more serious if they result in health risks.

7.4 Level of Significance

The level of significance (denoted by α) is the maximum probability of making a Type I error that the researcher is willing to accept. Common values are 0.05, 0.01, and 0.10.

• Setting a small α reduces the chance of a Type I error.

• The probability of a Type II error is denoted by β.

7.5 Statistical Tests and Test Statistics

After stating the hypotheses and significance level, a random sample is taken and a test statistic is calculated. The test statistic is compared to the parameter in the null hypothesis and converted to a standardized value (such as z or t) for decision-making.

• Test Statistic: The statistic calculated from sample data used to decide whether to reject H0.

• Standardized Test Statistic: The test statistic converted to a standard scale (e.g., z, t).

7.6 P-values

The P-value is the probability, assuming H0 is true, of obtaining a sample statistic as extreme or more extreme than the observed value. It is used to decide whether to reject H0:

• If P-value ≤ α, reject H0.

• If P-value > α, fail to reject H0.

7.7 Nature of the Test: One-Tailed and Two-Tailed Tests

The form of the alternative hypothesis determines the type of test:

• Left-tailed Test: Ha uses < (e.g., μ < μ0).

• Right-tailed Test: Ha uses > (e.g., μ > μ0).

• Two-tailed Test: Ha uses ≠ (e.g., μ ≠ μ0).

Examples:

• Claim: Proportion is 61%. Ha: p ≠ 0.61 (two-tailed).

• Claim: Mean oil change time is less than 15 minutes. Ha: μ < 15 (left-tailed).

• Claim: Mean furnace life is more than 18 years. Ha: μ > 18 (right-tailed).

7.8 Making and Interpreting Decisions

After calculating the P-value, compare it to α to make a decision:

• If P-value ≤ α, reject H0.

• If P-value > α, fail to reject H0.

Interpretation depends on which hypothesis represents the claim:

• If the claim is H0 and you reject H0, there is enough evidence to reject the claim.

• If the claim is Ha and you reject H0, there is enough evidence to support the claim.

7.9 Steps for Hypothesis Testing

1. State the claim mathematically and verbally. Identify H0 and Ha.

2. Specify the level of significance (α).

3. Determine the appropriate standardized sampling distribution (z, t, etc.).

4. Calculate the test statistic and its standardized value.

5. Find the P-value.

6. Make a decision based on the P-value and α.

7. Write a statement interpreting the decision in the context of the original claim.

7.10 Strategies for Hypothesis Testing

• If you want to support a claim, word it as the alternative hypothesis (Ha).

• If you want to reject a claim, word it as the null hypothesis (H0).

• You cannot use a hypothesis test to support a claim if it is the null hypothesis.

Example: If a research team wants to support the claim that mean recovery time is less than 96 hours, Ha: μ < 96. If the opposing team wants to reject this claim, they make H0: μ ≤ 96.

Key Formulas

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

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

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

Summary Table: Types of Errors

Decision	H0 True	H0 False
Reject H0	Type I Error (α)	Correct Decision
Fail to Reject H0	Correct Decision	Type II Error (β)

Additional info: This summary covers the foundational concepts, terminology, and procedures for hypothesis testing with one sample, as outlined in a typical introductory statistics course.