1.Previous review
Following the previous template "2.5 Type I and Type II errors in Hypothesis Testing," where we introduced what a Type I Error (α) and a Type II Error (β) are and provided corresponding examples, let's further review what was mentioned in "2.3 Probabilities and Significant Level_Inferential Statistics (2)"
2. Two tailed or one tailed test？
3.1 Two tailed test
3.2 Left tailed test
3.3 Right tailed test
In "2.3 Probabilities and Significant Level_Inferential Statistics (2)," we only introduced the method of determining whether to use a left-tailed or right-tailed test by posing corresponding questions. Today, let's combine what we have learned about the principle of setting rejection regions and the corresponding Type I Error vs. Type II Error from "2.5 Type I and Type II errors in Hypothesis Testing" to further understand:
One-tailed or two-tailed test?
Right-tailed or left-tailed test?
This will allow us to deeply understand why to set a left-tailed or right-tailed test in the context of visualizing the size or probability of the Type II Error or ß error area.
Throughout this process, we will use the Bayeslab online AI analysis tool. With the help of AI Agent's visualization capabilities, we can obtain Python analytical code tailored by the AI Agent with just simple prompts. This approach enables us to efficiently perform Python data visualization and routine business analysis. The outcome includes not only a set of visualized images but also Python source code, data analysis files, and even assists individuals with no statistical or Python coding background to complete professional data analysis.
Note:
For choosing the appropriate formula when the data distribution is normal for different situations involving samples and populations, please see the summary in the image below:
OK, let’s get started.
2.How to identify one tailed or two tailed test?
Let’s continue using the IQ test example we have previously discussed.
Now, a sample of 30 teenagers from a certain region has been taken to conduct an IQ test. The data is shown in IQ_Score.csv, and the calculated mean of this sample is 106.64.
The question is whether we can consider the average IQ of teenagers in this region to be 108? (Given a significance level α=0.05)
The above questions clearly ask whether there is a significant difference between μ and μ₀.
If there is a difference, it can be either greater or smaller, meaning there are two rejection regions, correct?
Now, with μ₀ = 108 as the center symmetry axis of the normal distribution curve, this obviously indicates a two-tailed test, correct?
Now, we calculate its significance based on the confidence interval of μ = 106.64 ± 95%. This results in two normal distribution curves: one centered at μ = 106.64 and the other at μ₀ = 108. The overlap of these two curves will display the corresponding Type I Error vs Type II Error areas.
Type I Error: the blue shaded area below
Type II Error: the red shaded area below
This is a two-tailed test: There are rejection regions on both sides, indicating significant deviations both below “μ₀ = 108” and above μ₀ = 108, each with a probability of α/2.
3.How to tell if it is a right-tailed or left-tailed test?
However, if the rejection region is on one side, it becomes a one-tailed test. One-tailed tests are further divided into right-tailed and left-tailed tests, and both use the same Z distribution or t distribution formulas.
Z = (x̄ — μ₀) / (σ / √n)
t = (x̄ — μ₀) / (s / √n)
In this case, the assumed rejection regions are opposite (opposite signs).
Left-tailed test:
H₀: μ ≥ μ₀
H₁: μ < μ₀
Z ≤ -Zₐ or t ≤ -tₐ
Right-tailed test:
H₀: μ ≤ μ₀
H₁: μ > μ₀
Z ≥ Zₐ or t ≥ tₐ
3.1 left tailed test example
If the question is whether μ < μ₀, i.e., whether μ is significantly lower than μ₀, for the IQ test example, it would be phrased as “Is this test score significantly lower than 108?” In this case:：
H₀: μ ≥ μ₀
H₁: μ < μ₀ meaning Z ≤ -Zₐ
This indicates that the rejection region is on the left side, i.e., Z ≤ -Zₐ. When the rejection region is on the left, we call it a left-tailed test.
( The above visualization was generated by Bayeslab Prompts )
3.2 Right tailed test example
If we now set it as a right-tailed test, with the rejection region on the right side, making Z ≥ -Zₐ, what would the visualization look like?
Specifically, we use the fine-tuning button in Bayeslab and add a prompt to achieve this:
If the rejection region is set on the right side, we can clearly see that the “Type II Error,” represented by the blue area, has increased, correct?
This means that if the question is “Is this test score significantly lower than 108?” and we choose to set the rejection region on the left side, the resulting Type II Error (ß) will be smaller compared to setting the rejection region on the right side.
In other words, with the significance level or Type I Error (α) fixed at 0.05, we can further reduce the probability of a Type II Error by correctly setting the direction of the rejection region as a left-tailed or right-tailed test.
Let’s review the definitions of Type I and Type II errors:
Type I Error (α): The probability of rejecting the null hypothesis when it is actually true.
Type II Error (ß): The probability of failing to reject the null hypothesis when the alternative hypothesis is actually true.
● Type I error (α):
Rejecting a correct null hypothesis and accepting an incorrect alternative hypothesis, also known as an α error.
● Type II error (β):
Accepting an incorrect null hypothesis and rejecting a correct alternative hypothesis, also known as a β error.
4. Comparison of different Hypothesis
By visualizing the effects of the following three different hypotheses and comparing the sizes of the Type II Error (ß) areas for each hypothesis, the differences will be more intuitive.
Similarly, if the question is “Is this test score significantly higher than 108?” then we should place the rejection region on the right side, meaning Z ≥ Zₐ.
In other words, placing the rejection region on the right side creates a right-tailed test, where the resulting Type II Error area is significantly smaller than in a left-tailed test.
The key takeaway today is that by choosing the appropriate direction — left-tailed or right-tailed — when performing a one-tailed test, you can effectively reduce the likelihood of Type I and Type II errors.
5. Formula of left tailed test and right tailed test
the summary table is as follows:
Stay tuned, subscribe to Bayeslab , and let everyone master the wisdom of statistics at a low cost.
About Bayeslab
Bayeslab: Website
The AI First Data Workbench
X: @BayeslabAI
Documents: https://bayeslab.gitbook.io/docs
Blogs:https://bayeslab.ai/blog
