Finding Z Critical Value

A z-critical value is a cutoff point on a standard normal distribution (z-distribution) used in hypothesis testing to determine whether the results are statistically significant or not. The value is compared with the z-test statistic to determine whether to reject the null hypothesis or not. The decision is to reject the null hypothesis if the absolute value of the z-statistic is greater than the absolute z-critical value. Otherwise, you fail to reject the null hypothesis.

Given its importance in hypothesis testing, every student needs to be able to find it manually from z tables, using an online calculator, or using technologies such as Excel. Therefore, in this guide, we’ll show you how you can easily find the z-critical value given either the significance level (α) or the confidence level.

How to Find Z Critical Value: Step-by-Step

Are you struggling with finding the z critical value for hypothesis testing or confidence interval problems? Here are the four steps you should follow:

• Step 1: Identify the Significance Level (α) or confidence level
• Step 2: Determine the type of test. The test can be two-sided, right-sided, or left-sided
• Step 3: Calculate the area.
  • For a two-sided test, the area is α/2
  • For a right-sided test, the area is 1-α
  • For a left-sided test, the area is α
• Step 4: Look up the z-critical value from the appropriate z table.

Looking for a quick and accurate strategy to find z critical value without using tables? Try the free online z critical value calculator or check out our complete guide on how to find the z critical value using Excel.

Finding Z Critical Value in Hypothesis Testing: Example Problems

In hypothesis testing, finding the z-critical value requires you to know the significance level (α) and whether the test is right-sided, left-sided, or two-sided. To help you learn how to find the correct z critical value based on the type of test, check out the following examples:

Example 1: Right-Sided Test Scenario

A beverage company claims that the average volume of its soda bottles is at least 500 ml. A quality inspector wants to test whether the mean volume is greater than 500 ml using a significance level of α = 0.05. What is the correct z critical value for this hypothesis test?

Step 1: Identify the Significance Level (α)

From the problem, the significance level is α = 0.05

Step 2: Determine the Type of Test

The alternative hypothesis involves “greater than.” In particular, we want to test the hypothesis that the mean volume is greater than 500 ml (i.e, μ > 500 ml). Thus, this is a right-sided test.

Step 3: Calculate the Area

For a right-tailed test, the area is 1-α. Thus, area = 1- 0.05

= 0.95

Step 4: Look up the Z Critical Value from a Positive Z-Table

You’ll need to look up 0.95 in the positive z table. The corresponding z value is 1.65, as shown below:

Therefore, the appropriate z critical value for the test at α = 0.05 is 1.65

Example 2: Left-Sided Test Scenario

A hospital administrator believes that the average patient waiting time in the emergency room is less than 30 minutes. To test this claim, a hypothesis test is conducted at a significance level of 0.01. What is the z critical value appropriate for this test?

Step 1: Identify the Significance Level (α)

Based on the question, we need to test the hypothesis at a 0.01 significance level. Thus, α=0.01

Step 2: Determine the Type of Test

The alternative hypothesis involves “less than.” That is, we want to test whether the average waiting time is less than 30 minutes (μ < 30). Thus, the test is a left-sided test.

Step 3: Calculate the Area

For a left-sided test, the area is α. Thus, area = α

= 0.01

Step 4: Look Up the Z Critical Value from a Negative Z Table

From the negative z-table, the z-score corresponding to a cumulative area of 0.01 is -2.33, as shown below:

Thus, the appropriate z-critical value for the test is −2.33

Example 3: Two-Sided Test Scenario

A standardized test is designed to have a mean score of 100. An education researcher wants to test whether the current population mean score is different from 100. The researcher wants to test this claim at 5% level of significance. Find the appropriate z critical values for the test.

Step 1: Identify the Significance Level (α)

Based on the scenario, the significance level is 5%. Thus, α= 0.05

Step 2: Determine the Type of Test

The alternative hypothesis is “not equal to.” In other words, we want to test the alternative hypothesis μ ≠ 100. This is a non-directional test and hence a two-sided test.

Step 3: Calculate the Area

For a two-sided test, you need to divide the area by 2. That is, area = α/2.

Therefore, the area for the two-sided test becomes 0.05/2

= 0.025.

Step 4: Look up the Z Critical Value from either a Positive or a Negative Z Table

If you want to use the positive z-table, you need to use 1-α/2 as the area.

That is, 1-α/2 = 1-0.025

= 0.975

Now, looking up the value from a positive z-table, the corresponding z value for an area of 0.975 is 1.96, as shown below:

Since the standard normal table is symmetrical, the two-sided z-critical value becomes ±1.96.

Alternatively, you can use a negative z-table to get the same value. In this case, you’ll need to look up the corresponding z score of α/2.

Thus, looking up an area of 0.025 from a negative z-table gives -1.96, as shown below:

Since the standard normal table is symmetrical, the appropriate two-sided z critical value at alpha = 0.05 is ±1.96.

Finding Z Critical Value for Confidence Intervals: Example Problems

When constructing a confidence interval for a population mean (using the z distribution), the z critical value depends solely on the confidence level. Therefore, to learn how to find the appropriate z critical values for any confidence interval, consider the following examples.

Example 1. 95% Confidence Interval

A peanut manufacturer wants to estimate the mean weight of its jars of peanuts. A sample of 60 jars is collected, and the sample mean weight is 3.2 pounds. The manufacturer wants a 95% confidence interval for the true mean weight of all its peanut jars. What is the z-critical value for constructing the interval?

Step 1: Identify the Significance Level

From the scenario, we want to construct the 95% confidence interval. Thus, the confidence level is 95%. To find the significance level, we use the formula:

Significance level, α = 1-confidence level

Thus, α = 1-0.95

= 0.05

Step 2: Divide α by 2

Since confidence intervals are always two-sided, you need to divide alpha by 2.

Therefore, α/2 = 0.05/2

= 0.025

Step 3: Look Up the Z Critical Value from Either a Positive or Negative Z-Table

If using a positive z-table, find the cumulative area to the left as follows:

area = 1-0.025

= 0.975

Looking up the value from the table gives 1.96. Thus, the appropriate z critical value for the 95% confidence interval is ±1.96.

Example 2: 90% Confidence Interval

An economist wants to estimate the average weekly household expenditure on groceries in a city. A random sample is taken, and a 90% confidence interval is to be constructed for the population mean. Find the z critical value for the interval.

Step 1: Identify the Significance Level

Since the confidence level is 90%, the appropriate significance level is 1-0.90

= 0.10

Step 2: Divide α by 2

α​/2 = 0.10/2

= 0.05

$$\frac{\alpha}{2} = \frac{0.10}{2} = 0.05$$2α​=20.10​=0.05

Step 3: Look up the Z Critical Value either from the Positive or the Negative Z-table

Using the positive table, the area becomes: 1-0.05

= 0.95

Looking up 0.95 from the positive table gives z = +1.645. But since the z table is symmetrical, the z critical value for 90% confidence interval is ±1.645

Commonly Used Z Critical Values for Confidence Levels

The table below shows the most common z critical values for constructing confidence intervals at different confidence levels.

Frequently Asked Questions