To test the hypothesis that wives in unstable marriages have a neuroticism rating greater than 19.43, we need to establish our null and alternative hypotheses, as well as determine the type of tail for the test. Here is a step-by-step solution:
### Step 1: Establish the Null Hypothesis (H0)
The null hypothesis is a statement of no effect or no difference. It represents a baseline or a standard to be tested against. In this situation, we will assume that the mean neuroticism rating for wives in unstable marriages is the same as the mean rating for wives in stable marriages.
Null Hypothesis (H0): The mean neuroticism rating for wives in unstable marriages is equal to 19.43.
This can be mathematically represented as:
[tex]\[ H_0: \mu = 19.43 \][/tex]
### Step 2: Establish the Alternative Hypothesis (H1)
The alternative hypothesis is a statement that indicates the presence of an effect or a difference. In this case, the alternative hypothesis will be that the mean neuroticism rating for wives in unstable marriages is greater than the mean rating for wives in stable marriages.
Alternative Hypothesis (H1): The mean neuroticism rating for wives in unstable marriages is greater than 19.43.
This can be mathematically represented as:
[tex]\[ H_1: \mu > 19.43 \][/tex]
### Step 3: Determine the Type of Tail
The type of tail in hypothesis testing refers to the direction of the effect that is being tested. There are three types of tests:
1. One-tailed (right-tailed): Tests if the sample mean is greater than the population mean.
2. One-tailed (left-tailed): Tests if the sample mean is less than the population mean.
3. Two-tailed: Tests if the sample mean is significantly different (either greater or less) from the population mean.
In this case, since we are testing if the mean neuroticism rating for wives in unstable marriages is greater than the mean rating for wives in stable marriages, we will use:
Type of Tail: One-tailed (right-tailed)
### Summary of Hypotheses and Test Type
- Null Hypothesis (H0): The mean neuroticism rating for wives in unstable marriages is 19.43.
- Alternative Hypothesis (H1): The mean neuroticism rating for wives in unstable marriages is greater than 19.43.
- Type of Tail: One-tailed (right-tailed)
This setup allows us to proceed with the statistical test to determine if there is sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis.
