Critical Values for a One-Sample Two-Tailed Z-Test at α 0.05

When conducting a one-sample two-tailed z-test at a significance level of α 0.05, the critical values of the z-scores are crucial for making a decision about the null hypothesis. These values are derived from the standard normal distribution and represent the z-scores that cut off the upper and lower 2.5% of the distribution, ensuring that the total area in both tails is 5%. In other words, the critical values are the points at which the test statistic must fall to reject the null hypothesis.

Understanding the Significance Level

To better understand this, let us break down the concept of a two-tailed test at the 0.05 level of significance. A significance level of 0.05 means that there is a 5% chance of incorrectly rejecting the null hypothesis if it is actually true. In a two-tailed test, this 5% is divided equally into the two tails of the distribution, making each tail 2.5%. This symmetry ensures that we are equally likely to reject the null hypothesis in either direction.

Finding the Critical Values

To find the critical values, we must consult a z-table or statistical appendix. These tables provide the z-scores corresponding to specific areas under the standard normal curve. For a two-tailed test at the 0.05 level of significance, we need to find the z-scores that capture the upper and lower 2.5% of the distribution.

Using Z-Tables

Here are the steps to find these critical values:

Divide the significance level by 2: 0.05 / 2 0.025. This is because the 5% is split equally between the upper and lower tails.

Locate the 0.025 value in the upper part of the z-table. This represents the area to the right of the critical value in the upper tail and to the left of the critical value in the lower tail.

Find the corresponding z-score for 0.025 in the body of the z-table. This z-score is typically between 1.5 and 2.0, indicating the separation between the critical region and the non-critical region.

Interpreting the Results

The critical z-scores for a two-tailed test at the 0.05 level of significance are:

Lower critical value: z -1.96

Upper critical value: z 1.96

These values signpost the boundaries of the critical region. If the calculated z-score falls below -1.96 or above 1.96, you would reject the null hypothesis. Otherwise, you would fail to reject the null hypothesis, indicating that there is not enough evidence to suggest a significant difference beyond the random variation expected under the null hypothesis.

Practical Application and Interpretation

Understanding the critical values and how to apply them in a one-sample two-tailed z-test is essential for researchers and data analysts. By knowing these critical values, you can interpret your z-scores and make informed decisions about the statistical significance of your results.

Conclusion

In summary, for a one-sample two-tailed z-test at a significance level of α 0.05, the critical values are z -1.96 and z 1.96. These values help us determine whether the observed data is statistically significant compared to the null hypothesis. By carefully calculating and comparing the test statistic to these critical values, we can draw meaningful conclusions from our data.

Keywords: one-sample z-test, critical values, significance level, z-score, two-tailed test