Z Critical Value Calculator

This calculator finds the z critical value from the standard normal distribution for a given significance level (α). Simply enter the significance level (in decimal form) and click the “Calculate” button.

The calculator will give you z critical values for left-tailed, right-tailed, and two-tailed tests. It will also provide you with a brief explanation of how you can obtain the value from the standard normal table.

What is a Z Critical Value?

A z-critical value is a cutoff point on the standard normal distribution, which helps in determining whether the test statistic is statistically significant. In hypothesis testing, it serves as a boundary for rejecting or failing to reject the null hypothesis.

The z critical value relies on the significance level (α). For a two-tailed test, you need to divide α by 2.

Think of it as the “boundary line” for decision-making. If your test statistic falls beyond this boundary, you reject the null hypothesis. Otherwise, you fail to reject the null hypothesis.

Application of Z Critical Values in Hypotheses Testing

Want to learn how to use z-critical values in hypothesis testing, let’s go through some examples.

Example 1. Right-Tailed Test

Suppose a school wants to know if a new teaching method improves student test scores. They collected sample data, and the calculated Z statistic from the sample is 1.85. At 5% significance level, can we conclude that the new teaching method improves students’ test scores?

Solution

Step 1: State the hypotheses and identify the Type of Test

• Null hypothesis (H₀): The new teaching method does not improve scores.
• Alternative hypothesis (H₁): The new teaching method improves scores.

Since we are testing for an improvement, this is a right-tailed test.

Step 2: Choose the significance level.

From the question, we want to test the hypothesis at 5% significance level. Thus, α = 0.05

Step 3: Find the Z critical value

Using the z critical value calculator, the right-tailed critical value at α = 0.05 is 1.645.

Step 4: Compare the Z statistic to the Z critical and make the Decision

From the question, the test statistic is z = 1.85. Since the absolute value of the z test statistic (z = 1.85) is greater than the absolute z-critical value, we reject the null hypothesis, H₀.

Step 5: Make a Conclusion

At the 5% significance level, there’s sufficient evidence to conclude that the new teaching method significantly improves student test scores.

Example 2: Left-Tailed Test

Suppose a company wants to test if a new energy drink decreases reaction time in athletes. They collected sample data, and the calculated Z statistic from the sample is -1.75. At 1% significance level, can we conclude that the new energy drink reduces reaction time?

Solution

Step 1: State the hypotheses and identify the type of test

• Null hypothesis (H₀): The new energy drink does not decrease reaction time.
• Alternative hypothesis (H₁): The new energy drink decreases reaction time.

Since we are testing for a decrease, this is a left-tailed test.

Step 2: Choose the significance level

From the question, we want to test the claim at 1% level. Hence, the significance level, α = 0.01

Step 3: Find the Z critical value

Using the Z critical value calculator, the left-tailed critical value at α = 0.01 is approximately -2.3263.

Step 4: Compare the Z statistic to the Z critical and make the decision

From the question, the z statistic is -1.75. Since the absolute value of this test statistic (1.75) is less than the absolute z-critical value (2.33), we fail to reject the null hypothesis.

Step 5: Conclusion

At the 1% significance level, there is not enough evidence to conclude that the new energy drink significantly decreases reaction time in athletes.

Example 3: Two-Tailed Test

A nutritionist wants to test whether a new diet affects the average cholesterol level of patients. The sample data yield a Z statistic of 2.10. At 10% significance level, can we conclude that the diet changes cholesterol levels?

Step 1: State the hypotheses and identify the type of test

• Null hypothesis (H₀): The new diet does not change cholesterol levels.
• Alternative hypothesis (H₁): The new diet changes cholesterol levels.

Since we are testing for any change (increase or decrease), this is a two-tailed test.

Step 2: Choose the significance level

From the scenario, we want to test the claim at 10% level. Thus, the significance level, α = 0.10

Step 3: Find the Z critical value

Using the Z critical value calculator, the two-tailed critical value at α = 0.10 is approximately ±1.645.

Step 4: Compare the Z statistic to the Z critical and make the decision

From the scenario, the test statistic is z = 2.10. Since the absolute test statistic value (z = 2.10) is greater than the absolute z-critical value (1.645), we reject the null hypothesis.

Step 5: Conclusion

At the 10% significance level, there is sufficient evidence to conclude that the new diet significantly affects cholesterol levels.

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