To understand the relationship between confidence intervals and hypothesis testing, let's delve into both concepts.
### Confidence Intervals:
A confidence interval gives an estimated range of values which is likely to include an unknown population parameter. This range is calculated from a given set of sample data. For example, a 99% confidence interval means that we can be 99% certain that the interval contains the true population parameter.
### Hypothesis Testing:
Hypothesis testing is a statistical method used to make decisions about the population parameters based on sample data.
- Null Hypothesis (H₀): This is the hypothesis that there is no effect or no difference, and it is what we seek to test against.
- Alternative Hypothesis (H₁): This is the hypothesis that there is an effect or a difference.
In the context of this question, the hypotheses are:
- \( H_0: \mu = \mu_0 \)
- \( H_1: \mu \neq \mu_0 \)
### Relationship:
When we calculate a 99% confidence interval for the population mean (μ), this interval provides us with a range of values which we believe, with 99% confidence, includes the true mean. This interval can also be used in hypothesis testing.
1. 99% Confidence Interval and Hypothesis Testing:
If a 99% confidence interval for μ does not contain μ₀, we have significant evidence to reject the null hypothesis \( H_0 \) at the 1% significance level.
2. 99% Confidence Interval and Decision Making:
- If μ₀ is within the 99% confidence interval, we do not reject the null hypothesis \( H_0 \).
- If μ₀ is not within the 99% confidence interval, we reject the null hypothesis \( H_0 \) at the 1% level of significance.
### Specific Answer to the Question:
When testing [tex]\( H_0: \mu = \mu_0 \)[/tex] versus [tex]\( H_1: \mu \neq \mu_0 \)[/tex], if a 99% confidence interval does not contain [tex]\( \mu_0 \)[/tex], we reject [tex]\( H_0 \)[/tex] at the 1% level.
