Formulating Hypotheses

1. The Framework of Hypothesis Testing

Hypothesis testing provides a formal procedure for deciding between two competing statements about a population, based on evidence from a sample. Think of it as the statistical equivalent of a legal trial.

In a courtroom, a defendant is presumed innocent until proven guilty. The prosecution must present compelling evidence to overturn this initial presumption. Hypothesis testing operates on a similar principle.

• The Null Hypothesis ($H_0$) is the statement of "no effect," "no difference," or the status quo. It's the initial assumption, the statistical equivalent of "presumed innocent." It is the hypothesis that we are trying to find evidence against.
• The Alternative Hypothesis ($H_a$ or $H_1$) is the claim we want to establish. It represents the "research hypothesis" or the new theory we are testing. It's the statistical equivalent of "guilty."

The entire process revolves around assessing the strength of the evidence from our sample data against the null hypothesis. We ask: "If the null hypothesis were true, how likely is it that we would observe data like ours?" If this likelihood is very low, we reject the null hypothesis in favor of the alternative.

Let's start by reading a brief introduction to this concept.

To summarize the key idea from the reading:

• Null Hypothesis ($H_0$): A statement about a population parameter assumed to be true. It always contains a statement of equality ($=,\le ,\ge$).
• Alternative Hypothesis ($H_a$): A statement that directly contradicts the null hypothesis. It never contains a statement of equality ($\ne ,>,<$).

The following image provides a concise visual summary of these two concepts.

2. Formulating Hypotheses: Types and Examples

The way we formulate our hypotheses depends on the research question. Specifically, it depends on whether we're looking for any difference or a difference in a specific direction. This leads to three types of tests.

To see how this is done in practice, let's watch a video that walks through several examples of translating a word problem into formal hypotheses.

Let's formalize the three types of tests you saw in the video. Let $\mu$ be the population parameter of interest (e.g., mean) and ${\mu }_0$ be the specific value we are testing against.

a) Two-Tailed Test (Non-Directional)

This is used when we are interested if the parameter is simply different from a certain value. We don't specify the direction of the difference.

• Research Question: "A machine is supposed to produce straws with a 4mm diameter. We want to test if the machine is out of calibration."
• Hypotheses:
  • $H_0:\mu =4$ (The machine is correctly calibrated)
  • $H_a:\mu \ne 4$ (The machine is not correctly calibrated)

This was the first example in the video. The alternative hypothesis covers both possibilities: $\mu >4$ and $\mu <4$.

b) One-Tailed Test: Upper-Tailed (Right-Tailed)

This is used when we are interested if the parameter is greater than a certain value.

• Research Question: "A researcher believes that teens sleep longer than 10 hours per day, contrary to the belief that they sleep no longer than 10 hours."
• Hypotheses:
  • $H_0:\mu \le 10$ (The average sleep time is 10 hours or less)
  • $H_a:\mu >10$ (The average sleep time is more than 10 hours)

This was the second example in the video. The alternative hypothesis is the "research" claim that the researcher wants to prove.

c) One-Tailed Test: Lower-Tailed (Left-Tailed)

This is used when we are interested if the parameter is less than a certain value.

• Research Question: "A teacher believes the school board's claim that at least 60% of students bring a phone is too high."
• Hypotheses (for a proportion $p$):
  • $H_0:p\ge 0.60$ (The proportion is 60% or more)
  • $H_a:p<0.60$ (The proportion is less than 60%)

This was the third example in the video. The teacher is trying to find evidence for their belief, which becomes the alternative hypothesis.

A Note on Formulation: In many texts and software implementations, the null hypothesis for one-tailed tests is often simplified to an equality (e.g., $H_0:\mu =10$). This is because the test is statistically performed at the boundary condition. The logic holds: if you can find evidence to reject that the mean is exactly 10 in favor of it being greater, you have implicitly found evidence that it's not less than 10. For clarity in formulation, it's best to use the inequality ($\le$ or $\ge$) as it more accurately reflects the full statement of the null.

3. Application: A/B Testing

A very common application of hypothesis testing in data science and technology is A/B testing. Imagine your EV company wants to know if changing the color of the "Book a Test Drive" button on the website from blue to green will increase clicks.

Here's how you'd formulate the hypotheses:

• Let $p_A$ be the click-through rate for the original blue button (Version A).
• Let $p_B$ be the click-through rate for the new green button (Version B).

The research question is: "Does the green button have a higher click-through rate?"

• Null Hypothesis ($H_0$): There is no improvement or the change is detrimental.
  $H_0:p_B\le p_A\text{or equivalently}{p}_B-p_A\le 0$
• Alternative Hypothesis ($H_a$): The green button is better.
  $H_a:p_B>p_A\text{or equivalently}{p}_B-p_A>0$

This is an upper-tailed test. You would then collect data by showing different users the two versions, and use a statistical test to decide whether to reject $H_0$ and conclude that the green button is indeed superior.

4. Check Your Understanding

To solidify your understanding, translate the following research questions into null and alternative hypotheses. Define the parameter you are testing in each case.

1. An R&D engineer claims that a new battery manufacturing process has reduced the average weight of a battery pack, which was previously 450 kg.
2. A quality control department wants to verify that the resistance of a specific component is exactly 50 Ohms, as specified. Any deviation, higher or lower, is a problem.
3. The marketing team launches a campaign, believing it will increase the proportion of visitors to the company website to more than the historical average of 5%.

Conclusion

In this lesson, we have taken the first crucial step into the world of hypothesis testing. You learned that this statistical framework is a formal process for making decisions from data, analogous to a courtroom trial.

Key Takeaways:

• Null Hypothesis ($H_0$): Represents the default assumption or "status quo." It always includes the equality condition ($=,\le ,\ge$).
• Alternative Hypothesis ($H_a$): Represents the research claim or the effect you're trying to prove. It is the mathematical opposite of $H_0$ ($\ne ,>,<$).
• Three Test Types: The formulation depends on the research question, leading to two-tailed ($\ne$), upper-tailed ($>$), or lower-tailed ($<$) tests.
• Burden of Proof: The goal of a hypothesis test is to see if there is enough evidence in the sample data to reject the null hypothesis in favor of the alternative.

Preview of the Next Lesson:

Now that you can formulate hypotheses, the next logical question is: "How do we decide whether to reject the null hypothesis?" In the next lesson, we will begin to answer this by exploring one of the most fundamental and powerful methods for constructing a statistical test. You will learn to derive the Likelihood Ratio Test (LRT) statistic for simple hypotheses. This will link today's topic with the concept of likelihood that you have encountered before, providing a principled way to quantify the evidence for one hypothesis over another.