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One-Tailed Vs. Two-Tailed Tests

Introduction If you’ve ever analyzed data using built-in t-test functions, such as those in R or SciPy, here’s a question for you: have you ever adjusted the default setting for the alternative hypothesis? If your answer is no—or if you’re not even sure what this means—then this blog post is for you! The alternative hypothesis parameter, commonly referred to as “one-tailed” versus “two-tailed” in statistics, defines the expected direction of the difference between control and treatment groups. In a two-tailed test, we assess whether there is any difference in mean values between the groups, without specifying a direction. A one-tailed test, on the other hand, posits a specific direction—whether the control group’s mean is either less than or greater than that of the treatment group. Choosing between one- and two-tailed hypotheses might seem like a minor detail, but it affects every stage of A/B testing: from test planning to Data Analysis and results interpretation. This article builds a theoretical foundation on why the hypothesis direction matters and explores the pros and cons of each approach. One-tailed vs. two-tailed hypothesis testing: Understanding the difference To understand the importance of choosing between one-tailed and two-tailed hypotheses, let’s briefly review the basics of the t-test, the commonly used method in A/B testing. Like other Hypothesis Testing methods, the t-test begins with a conservative assumption: there is no difference between the two groups (the null hypothesis). Only if we find strong evidence against this assumption can we reject the null hypothesis and conclude that the treatment has had an effect. But what qualifies as “strong evidence”? To that end, a rejection region is determined under the null hypothesis and all results that fall within this region are deemed so unlikely that we take them as evidence against the feasibility of the null hypothesis. The size of this rejection region is based on a predetermined probability, known as alpha (α), which represents the likelihood of incorrectly rejecting the null hypothesis.  What does this have to do with the direction of the alternative hypothesis? Quite a bit, actually. While the alpha level determines the size of the rejection region, the alternative hypothesis dictates its placement. In a one-tailed test, where we hypothesize a specific direction of difference, the rejection region is situated in only one tail of the distribution. For a hypothesized positive effect (e..g., that the treatment group mean is higher than the control group mean), the rejection region would lie in the right tail, creating a right-tailed test. Conversely, if we hypothesize a negative effect (e.g., that the treatment group mean is less than the control group mean), the rejection region would be placed in the left tail, resulting in a left-tailed test. In contrast, a two-tailed test allows for the detection of a difference in either direction, so the rejection region is split between both tails of the distribution. This accommodates the possibility of observing extreme values in either direction, whether the effect is positive or negative. To build intuition, let’s visualize how the rejection regions appear under the different hypotheses. Recall that according to the null hypothesis, the difference between the two groups should center around zero. Thanks to the central limit theorem, we also know this distribution approximates a normal distribution. Consequently, the rejection areas corresponding to the different alternative hypothesis look like that: Why does it make a difference? The choice of direction for the alternative hypothesis impacts the entire A/B testing process, starting with the planning phase—specifically, in determining the sample size. Sample size is calculated based on the desired power of the test, which is the probability of detecting a true difference between the two groups when one exists. To compute power, we examine the area under the alternative hypothesis that corresponds to the rejection region (since power reflects the ability to reject the null hypothesis when the alternative hypothesis is true). Since the direction of the hypothesis affects the size of this rejection region, power is generally lower for a two-tailed hypothesis. This is due to the rejection region being divided across both tails, making it more challenging to detect an effect in any one direction. The following graph illustrates the comparison between the two types of hypotheses. Note that the purple area is larger for the one-tailed hypothesis, compared to the two-tailed hypothesis: In practice, to maintain the desired power level, we compensate for the reduced power of a two-tailed hypothesis by increasing the sample size (Increasing sample size raises power, though the mechanics of this can be a topic for a separate article). Thus, the choice between one- and two-tailed hypotheses directly influences the required sample size for your test.  Beyond the planning phase, the choice of alternative hypothesis directly impacts the analysis and interpretation of results. There are cases where a test may reach significance with a one-tailed approach but not with a two-tailed one, and vice versa. Reviewing the previous graph can help illustrate this: for example, a result in the left tail might be significant under a two-tailed hypothesis but not under a right one-tailed hypothesis. Conversely, certain results might fall within the rejection region of a right one-tailed test but lie outside the rejection area in a two-tailed test. How to decide between a one-tailed and two-tailed hypothesis Let’s start with the bottom line: there’s no absolute right or wrong choice here. Both approaches are valid, and the primary consideration should be your specific business needs. To help you decide which option best suits your company, we’ll outline the key pros and cons of each. At first glance, a one-tailed alternative may appear to be the clear choice, as it often aligns better with business objectives. In industry applications, the focus is typically on improving specific metrics rather than exploring a treatment’s impact in both directions. This is especially relevant in A/B testing, where the goal is often to optimize conversion rates or enhance revenue. If the treatment doesn’t lead to a significant improvement the examined change won’t be implemented. Beyond this conceptual advantage, we have already mentioned one key benefit of a one-tailed hypothesis: it requires a smaller sample size. Thus, choosing a one-tailed alternative can save both time and resources. To illustrate this advantage, the following graphs show the required sample sizes for one- and two-tailed hypotheses with different power levels (alpha is set at 5%). In this context, the decision between one- and two-tailed hypotheses becomes particularly important in sequential testing—a method that allows for ongoing data analysis without inflating the alpha level. Here, selecting a one-tailed test can significantly reduce the duration of the test, enabling faster decision-making, which is especially valuable in dynamic business environments where prompt responses are essential. However, don’t be too quick to dismiss the two-tailed hypothesis! It has its own advantages. In some business contexts, the ability to detect “negative significant results” is a major benefit. As one client once shared, he preferred negative significant results over inconclusive ones because they offer valuable learning opportunities. Even if the outcome wasn’t as expected, he could conclude that the treatment had a negative effect and gain insights into the product. Another benefit of two-tailed tests is their straightforward interpretation using confidence intervals (CIs). In two-tailed tests, a CI that doesn’t include zero directly indicates significance, making it easier for practitioners to interpret results at a glance. This clarity is particularly appealing since CIs are widely used in A/B testing platforms. Conversely, with one-tailed tests, a significant result might still include zero in the CI, potentially leading to confusion or mistrust in the findings. Although one-sided confidence intervals can be employed with one-tailed tests, this practice is less common. Conclusions By adjusting a single parameter, you can significantly impact your A/B testing: specifically, the sample size you need to collect and the interpretation of the results. When deciding between one- and two-tailed hypotheses, consider factors such as the available sample size, the advantages of detecting negative effects, and the convenience of aligning confidence intervals (CIs) with hypothesis testing. Ultimately, this decision should be made thoughtfully, taking into account what best fits your business needs. (Note: all the images in this post were created by the author)

Introduction

If you’ve ever analyzed data using built-in t-test functions, such as those in R or SciPy, here’s a question for you: have you ever adjusted the default setting for the alternative hypothesis? If your answer is no—or if you’re not even sure what this means—then this blog post is for you!

The alternative hypothesis parameter, commonly referred to as “one-tailed” versus “two-tailed” in statistics, defines the expected direction of the difference between control and treatment groups. In a two-tailed test, we assess whether there is any difference in mean values between the groups, without specifying a direction. A one-tailed test, on the other hand, posits a specific direction—whether the control group’s mean is either less than or greater than that of the treatment group.

Choosing between one- and two-tailed hypotheses might seem like a minor detail, but it affects every stage of A/B testing: from test planning to Data Analysis and results interpretation. This article builds a theoretical foundation on why the hypothesis direction matters and explores the pros and cons of each approach.

One-tailed vs. two-tailed hypothesis testing: Understanding the difference

To understand the importance of choosing between one-tailed and two-tailed hypotheses, let’s briefly review the basics of the t-test, the commonly used method in A/B testing. Like other Hypothesis Testing methods, the t-test begins with a conservative assumption: there is no difference between the two groups (the null hypothesis). Only if we find strong evidence against this assumption can we reject the null hypothesis and conclude that the treatment has had an effect.

But what qualifies as “strong evidence”? To that end, a rejection region is determined under the null hypothesis and all results that fall within this region are deemed so unlikely that we take them as evidence against the feasibility of the null hypothesis. The size of this rejection region is based on a predetermined probability, known as alpha (α), which represents the likelihood of incorrectly rejecting the null hypothesis. 

What does this have to do with the direction of the alternative hypothesis? Quite a bit, actually. While the alpha level determines the size of the rejection region, the alternative hypothesis dictates its placement. In a one-tailed test, where we hypothesize a specific direction of difference, the rejection region is situated in only one tail of the distribution. For a hypothesized positive effect (e..g., that the treatment group mean is higher than the control group mean), the rejection region would lie in the right tail, creating a right-tailed test. Conversely, if we hypothesize a negative effect (e.g., that the treatment group mean is less than the control group mean), the rejection region would be placed in the left tail, resulting in a left-tailed test.

In contrast, a two-tailed test allows for the detection of a difference in either direction, so the rejection region is split between both tails of the distribution. This accommodates the possibility of observing extreme values in either direction, whether the effect is positive or negative.

To build intuition, let’s visualize how the rejection regions appear under the different hypotheses. Recall that according to the null hypothesis, the difference between the two groups should center around zero. Thanks to the central limit theorem, we also know this distribution approximates a normal distribution. Consequently, the rejection areas corresponding to the different alternative hypothesis look like that:

Why does it make a difference?

The choice of direction for the alternative hypothesis impacts the entire A/B testing process, starting with the planning phase—specifically, in determining the sample size. Sample size is calculated based on the desired power of the test, which is the probability of detecting a true difference between the two groups when one exists. To compute power, we examine the area under the alternative hypothesis that corresponds to the rejection region (since power reflects the ability to reject the null hypothesis when the alternative hypothesis is true).

Since the direction of the hypothesis affects the size of this rejection region, power is generally lower for a two-tailed hypothesis. This is due to the rejection region being divided across both tails, making it more challenging to detect an effect in any one direction. The following graph illustrates the comparison between the two types of hypotheses. Note that the purple area is larger for the one-tailed hypothesis, compared to the two-tailed hypothesis:

In practice, to maintain the desired power level, we compensate for the reduced power of a two-tailed hypothesis by increasing the sample size (Increasing sample size raises power, though the mechanics of this can be a topic for a separate article). Thus, the choice between one- and two-tailed hypotheses directly influences the required sample size for your test. 

Beyond the planning phase, the choice of alternative hypothesis directly impacts the analysis and interpretation of results. There are cases where a test may reach significance with a one-tailed approach but not with a two-tailed one, and vice versa. Reviewing the previous graph can help illustrate this: for example, a result in the left tail might be significant under a two-tailed hypothesis but not under a right one-tailed hypothesis. Conversely, certain results might fall within the rejection region of a right one-tailed test but lie outside the rejection area in a two-tailed test.

How to decide between a one-tailed and two-tailed hypothesis

Let’s start with the bottom line: there’s no absolute right or wrong choice here. Both approaches are valid, and the primary consideration should be your specific business needs. To help you decide which option best suits your company, we’ll outline the key pros and cons of each.

At first glance, a one-tailed alternative may appear to be the clear choice, as it often aligns better with business objectives. In industry applications, the focus is typically on improving specific metrics rather than exploring a treatment’s impact in both directions. This is especially relevant in A/B testing, where the goal is often to optimize conversion rates or enhance revenue. If the treatment doesn’t lead to a significant improvement the examined change won’t be implemented.

Beyond this conceptual advantage, we have already mentioned one key benefit of a one-tailed hypothesis: it requires a smaller sample size. Thus, choosing a one-tailed alternative can save both time and resources. To illustrate this advantage, the following graphs show the required sample sizes for one- and two-tailed hypotheses with different power levels (alpha is set at 5%).

In this context, the decision between one- and two-tailed hypotheses becomes particularly important in sequential testing—a method that allows for ongoing data analysis without inflating the alpha level. Here, selecting a one-tailed test can significantly reduce the duration of the test, enabling faster decision-making, which is especially valuable in dynamic business environments where prompt responses are essential.

However, don’t be too quick to dismiss the two-tailed hypothesis! It has its own advantages. In some business contexts, the ability to detect “negative significant results” is a major benefit. As one client once shared, he preferred negative significant results over inconclusive ones because they offer valuable learning opportunities. Even if the outcome wasn’t as expected, he could conclude that the treatment had a negative effect and gain insights into the product.

Another benefit of two-tailed tests is their straightforward interpretation using confidence intervals (CIs). In two-tailed tests, a CI that doesn’t include zero directly indicates significance, making it easier for practitioners to interpret results at a glance. This clarity is particularly appealing since CIs are widely used in A/B testing platforms. Conversely, with one-tailed tests, a significant result might still include zero in the CI, potentially leading to confusion or mistrust in the findings. Although one-sided confidence intervals can be employed with one-tailed tests, this practice is less common.

Conclusions

By adjusting a single parameter, you can significantly impact your A/B testing: specifically, the sample size you need to collect and the interpretation of the results. When deciding between one- and two-tailed hypotheses, consider factors such as the available sample size, the advantages of detecting negative effects, and the convenience of aligning confidence intervals (CIs) with hypothesis testing. Ultimately, this decision should be made thoughtfully, taking into account what best fits your business needs.

(Note: all the images in this post were created by the author)

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Colorado Eyes the AI Data Center Boom with Bold Incentive Push

Even as states work on legislation to limit data center development, it is clear that some locations are looking to get a bigger piece of the huge data center spending that the AI wave has created. It appears that politicians in Colorado took a look around and thought to themselves “Why is all that data center building going to Texas and Arizona? What’s wrong with the Rocky Mountain State?” Taking a page from the proven playbook that has gotten data centers built all over the country, Colorado is trying to jump on the financial incentives for data center development bandwagon. SB 24-085: A Statewide Strategy to Attract Data Center Investment Looking to significantly boost its appeal as a data center hub, Colorado is now considering Senate Bill 24-085, currently making its way through the state legislature. Sponsored by Senators Priola and Buckner and Representatives Parenti and Weinberg, this legislation promises substantial economic incentives in the form of state sales and use tax rebates for new data centers established within the state from fiscal year 2026 through 2033. Colorado hopes to position itself strategically to compete with neighboring states in attracting lucrative tech investments and high-skilled jobs. According to DataCenterMap.com, there are currently 53 data centers in the state, almost all located in the Denver area, but they are predominantly smaller facilities. In today’s era of massive AI-driven hyperscale expansion, Colorado is rarely mentioned in the same breath as major AI data center markets.  Some local communities have passed their own incentive packages, but SB 24-085 aims to offer a unified, statewide framework that can also help mitigate growing NIMBY (Not In My Backyard) sentiment around new developments. The Details: How SB 24-085 Works The bill, titled “Concerning a rebate of the state sales and use tax paid on new digital infrastructure

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Microsoft will invest $80B in AI data centers in fiscal 2025

And Microsoft isn’t the only one that is ramping up its investments into AI-enabled data centers. Rival cloud service providers are all investing in either upgrading or opening new data centers to capture a larger chunk of business from developers and users of large language models (LLMs).  In a report published in October 2024, Bloomberg Intelligence estimated that demand for generative AI would push Microsoft, AWS, Google, Oracle, Meta, and Apple would between them devote $200 billion to capex in 2025, up from $110 billion in 2023. Microsoft is one of the biggest spenders, followed closely by Google and AWS, Bloomberg Intelligence said. Its estimate of Microsoft’s capital spending on AI, at $62.4 billion for calendar 2025, is lower than Smith’s claim that the company will invest $80 billion in the fiscal year to June 30, 2025. Both figures, though, are way higher than Microsoft’s 2020 capital expenditure of “just” $17.6 billion. The majority of the increased spending is tied to cloud services and the expansion of AI infrastructure needed to provide compute capacity for OpenAI workloads. Separately, last October Amazon CEO Andy Jassy said his company planned total capex spend of $75 billion in 2024 and even more in 2025, with much of it going to AWS, its cloud computing division.

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John Deere unveils more autonomous farm machines to address skill labor shortage

Join our daily and weekly newsletters for the latest updates and exclusive content on industry-leading AI coverage. Learn More Self-driving tractors might be the path to self-driving cars. John Deere has revealed a new line of autonomous machines and tech across agriculture, construction and commercial landscaping. The Moline, Illinois-based John Deere has been in business for 187 years, yet it’s been a regular as a non-tech company showing off technology at the big tech trade show in Las Vegas and is back at CES 2025 with more autonomous tractors and other vehicles. This is not something we usually cover, but John Deere has a lot of data that is interesting in the big picture of tech. The message from the company is that there aren’t enough skilled farm laborers to do the work that its customers need. It’s been a challenge for most of the last two decades, said Jahmy Hindman, CTO at John Deere, in a briefing. Much of the tech will come this fall and after that. He noted that the average farmer in the U.S. is over 58 and works 12 to 18 hours a day to grow food for us. And he said the American Farm Bureau Federation estimates there are roughly 2.4 million farm jobs that need to be filled annually; and the agricultural work force continues to shrink. (This is my hint to the anti-immigration crowd). John Deere’s autonomous 9RX Tractor. Farmers can oversee it using an app. While each of these industries experiences their own set of challenges, a commonality across all is skilled labor availability. In construction, about 80% percent of contractors struggle to find skilled labor. And in commercial landscaping, 86% of landscaping business owners can’t find labor to fill open positions, he said. “They have to figure out how to do

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2025 playbook for enterprise AI success, from agents to evals

Join our daily and weekly newsletters for the latest updates and exclusive content on industry-leading AI coverage. Learn More 2025 is poised to be a pivotal year for enterprise AI. The past year has seen rapid innovation, and this year will see the same. This has made it more critical than ever to revisit your AI strategy to stay competitive and create value for your customers. From scaling AI agents to optimizing costs, here are the five critical areas enterprises should prioritize for their AI strategy this year. 1. Agents: the next generation of automation AI agents are no longer theoretical. In 2025, they’re indispensable tools for enterprises looking to streamline operations and enhance customer interactions. Unlike traditional software, agents powered by large language models (LLMs) can make nuanced decisions, navigate complex multi-step tasks, and integrate seamlessly with tools and APIs. At the start of 2024, agents were not ready for prime time, making frustrating mistakes like hallucinating URLs. They started getting better as frontier large language models themselves improved. “Let me put it this way,” said Sam Witteveen, cofounder of Red Dragon, a company that develops agents for companies, and that recently reviewed the 48 agents it built last year. “Interestingly, the ones that we built at the start of the year, a lot of those worked way better at the end of the year just because the models got better.” Witteveen shared this in the video podcast we filmed to discuss these five big trends in detail. Models are getting better and hallucinating less, and they’re also being trained to do agentic tasks. Another feature that the model providers are researching is a way to use the LLM as a judge, and as models get cheaper (something we’ll cover below), companies can use three or more models to

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OpenAI’s red teaming innovations define new essentials for security leaders in the AI era

Join our daily and weekly newsletters for the latest updates and exclusive content on industry-leading AI coverage. Learn More OpenAI has taken a more aggressive approach to red teaming than its AI competitors, demonstrating its security teams’ advanced capabilities in two areas: multi-step reinforcement and external red teaming. OpenAI recently released two papers that set a new competitive standard for improving the quality, reliability and safety of AI models in these two techniques and more. The first paper, “OpenAI’s Approach to External Red Teaming for AI Models and Systems,” reports that specialized teams outside the company have proven effective in uncovering vulnerabilities that might otherwise have made it into a released model because in-house testing techniques may have missed them. In the second paper, “Diverse and Effective Red Teaming with Auto-Generated Rewards and Multi-Step Reinforcement Learning,” OpenAI introduces an automated framework that relies on iterative reinforcement learning to generate a broad spectrum of novel, wide-ranging attacks. Going all-in on red teaming pays practical, competitive dividends It’s encouraging to see competitive intensity in red teaming growing among AI companies. When Anthropic released its AI red team guidelines in June of last year, it joined AI providers including Google, Microsoft, Nvidia, OpenAI, and even the U.S.’s National Institute of Standards and Technology (NIST), which all had released red teaming frameworks. Investing heavily in red teaming yields tangible benefits for security leaders in any organization. OpenAI’s paper on external red teaming provides a detailed analysis of how the company strives to create specialized external teams that include cybersecurity and subject matter experts. The goal is to see if knowledgeable external teams can defeat models’ security perimeters and find gaps in their security, biases and controls that prompt-based testing couldn’t find. What makes OpenAI’s recent papers noteworthy is how well they define using human-in-the-middle

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