The question in the image asks for the relationship between confidence intervals and hypothesis testing. Specifically, it mentions testing the null hypothesis $H_0: \mu = \mu_0$ versus the alternative hypothesis $H_1: \mu \neq \mu_0$. The key point is to connect the outcome of a confidence interval with hypothesis testing.

If a 95% confidence interval for $\mu$ does not contain $\mu_0$, we reject $H_0$ at the 5% significance level. This is because, at a 95% confidence level, if $\mu_0$ is not within the interval, it suggests that $\mu_0$ is unlikely to be the true value of the population mean $\mu$, thus rejecting the null hypothesis. If $\mu_0$ were inside the confidence interval, we would not reject $H_0$.

So, the correct answer in this case is to reject $H_0$.

Would you like more details on how confidence intervals and hypothesis testing are related?

Here are 5 related questions to think about:

1. What happens if $\mu_0$ lies within the confidence interval?
2. How does the significance level $\alpha$ affect the confidence interval?
3. What is the difference between a two-tailed test and a one-tailed test in this context?
4. Can you reject $H_0$ using a confidence interval for other types of tests (e.g., proportions)?
5. How does increasing the sample size affect the width of a confidence interval?

Tip: Always align the confidence level with the significance level when comparing confidence intervals with hypothesis tests. A 95% confidence interval corresponds to a 5% significance level.