To calculate a confidence interval (two-sided), you need to follow these steps:

1. Let's say the sample size is 100.
2. Find the mean value of your sample. Assume it's 3.
3. Determine the standard deviation of the sample. Let's say it's 0.5.
4. Choose the confidence level. The most common confidence level is 95%.
5. In the statistical table find the Z(0.95)-score, i.e., the 97.5th quantile of N(0,1) – in our case, it's 1.959.
6. Compute the standard error as σ/√n = 0.5/√100 = 0.05.
7. Multiply this value by the z-score to obtain the margin of error: 0.05 × 1.959 = 0.098.
8. Add and subtract the margin of error from the mean value to obtain the confidence interval. In our case, the confidence interval is between 2.902 and 3.098.

To find a Z critical value for a given confidence level α:

1. Check if you perform a one- or two-tailed test.
2. For a one-tailed test:
   • Left-tailed: critical value is the α-th quantile of the standard normal distribution N(0,1).
   • Right-tailed: critical value is the (1-α)-th quantile.
3. Two-tailed test: critical value equals ±(1-α/2)-th quantile of N(0,1).
4. No quantile tables? Use CDF tables! (The quantile function is the inverse of the CDF.)
5. Verify your answer with an online critical value calculator.

The Z critical value for a 95% confidence interval is:

• 1.96 for a two-tailed test;
• 1.64 for a right-tailed test; and
• -1.64 for a left-tailed test.