Stating Hypotheses In Exercises 11–16, the statement represents a claim. Write its complement and state which is H0 and which is Ha.


p < 0.45

Verified step by step guidance

1. Step 1: Understand the problem. The problem involves writing the complement of the given claim and identifying which hypothesis is the null hypothesis (H0) and which is the alternative hypothesis (Ha). The given claim is 'p < 0.45', where 'p' represents the population proportion.
2. Step 2: Recall the definitions of hypotheses. The null hypothesis (H0) is a statement of no effect or no difference and typically includes equality (e.g., '=', '≥', or '≤'). The alternative hypothesis (Ha) is the claim being tested and represents a statement of inequality (e.g., '≠', '<', or '>').
3. Step 3: Write the complement of the given claim. The complement of 'p < 0.45' is 'p ≥ 0.45'. This is because the complement includes all values of 'p' that are not less than 0.45.
4. Step 4: Assign the hypotheses. The null hypothesis (H0) is the complement of the claim, so H0: p ≥ 0.45. The alternative hypothesis (Ha) is the original claim, so Ha: p < 0.45.
5. Step 5: Finalize the hypotheses. The null hypothesis (H0) is H0: p ≥ 0.45, and the alternative hypothesis (Ha) is Ha: p < 0.45. These hypotheses are now ready to be used in a hypothesis test.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating two competing hypotheses: the null hypothesis (H0), which represents a statement of no effect or no difference, and the alternative hypothesis (Ha), which represents the claim being tested. The goal is to determine whether there is enough evidence to reject H0 in favor of Ha.

Null and Alternative Hypotheses

The null hypothesis (H0) is a statement that indicates no significant difference or effect exists in the context of the study. In contrast, the alternative hypothesis (Ha) represents the claim that there is a significant difference or effect. For the given statement 'p < 0.45', H0 would typically be 'p ≥ 0.45', while Ha would be 'p < 0.45'.

Complement of a Hypothesis

The complement of a hypothesis refers to the opposite scenario of the original claim. In hypothesis testing, if the original claim is that a parameter is less than a certain value (e.g., p < 0.45), the complement would state that the parameter is greater than or equal to that value (e.g., p ≥ 0.45). Understanding the complement is crucial for correctly formulating H0 and Ha.