To find the critical value when $\sigma = 0.01$, we first need to know more context—specifically, whether you are referring to:

1. A z-distribution (standard normal distribution): In this case, $\sigma$ refers to the significance level (often called alpha, $\alpha$) of a hypothesis test.
2. A t-distribution: In this case, it would depend on the degrees of freedom.
3. Chi-square, F-distribution, or others.

Assuming you mean a z-distribution (common in hypothesis tests like a one-sample z-test), the critical value for $\sigma = 0.01$ corresponds to a confidence level of $99\%$ (since $1 - 0.01 = 0.99$).

For a two-tailed test:

• We are interested in the critical values that leave 0.01/2 = 0.005 in each tail.
• Using a z-distribution table, the critical values are approximately $\pm 2.576$.

Thus, the critical values for a two-tailed z-test at $\sigma = 0.01$ are:

$\text{Critical values} = -2.576, +2.576$

For a one-tailed test:

• If you're performing a one-tailed test, the critical value for $\sigma = 0.01$ corresponds to $z = 2.326$ for the upper tail and $z = -2.326$ for the lower tail.

Would you like to dive into more details on z-scores or expand on a different distribution?

Here are 5 relative questions you might consider:

1. How do critical values change with different significance levels ($\sigma$)?
2. How do critical values differ between one-tailed and two-tailed tests?
3. What is the role of degrees of freedom in determining critical values for the t-distribution?
4. How do you calculate p-values using critical values?
5. What is the relationship between confidence intervals and critical values?

Tip: In hypothesis testing, always double-check if you are dealing with a one-tailed or two-tailed test, as this significantly impacts your critical values.