Calculate The Following Cutoffs on A Standard Normal Distribution

What is a standard normal distribution?

The standard normal distribution, often referred to as the z-distribution, is a specific case of the normal distribution where the mean (μ) is 0 and the standard deviation (σ) is 1. This distribution is widely used in statistics because it allows for easy comparison of different normal distributions through z-scores.

The probability density function for the standard normal distribution is:

Where z represents the number of standard deviations from the mean. The total area under the curve is 1, representing 100% probability.

How to calculate cutoffs

Cutoffs on a standard normal distribution are typically expressed as z-scores. To find a cutoff value:

1. Determine the desired probability level (e.g., 0.05 for 5% significance)
2. Look up the corresponding z-score in standard normal distribution tables or use statistical software
3. For two-tailed tests, divide the alpha level by 2 and find the corresponding z-score

For example, to find the z-score for a one-tailed test with α = 0.05, you would look for the z-score that leaves 5% of the area in the tail.

Common cutoff values

Here are some commonly used cutoff values for the standard normal distribution:

These values are commonly used in hypothesis testing, quality control, and other statistical applications.

Interpretation of results

When you calculate a cutoff value, it represents the z-score that corresponds to a specific probability level. For example:

• A z-score of 1.96 corresponds to a probability of 0.05 in the tail (5%)
• This means there's a 5% chance of observing a value as extreme as this or more under the null hypothesis
• In hypothesis testing, you would reject the null hypothesis if your calculated z-score exceeds this cutoff value

Cutoff values help determine whether results are statistically significant. Lower cutoff values (closer to 0) indicate more significant results.