Last modified: February 05, 2025

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Null Hypotheses and Alternative Hypotheses

Statistical hypothesis testing is a fundamental method used in research to make inferences about populations based on sample data. Understanding the concepts of null and alternative hypotheses, as well as how to calculate and interpret p-values, is crucial for conducting robust and meaningful analyses. This section delves into these concepts, providing a comprehensive overview to enhance your statistical toolkit.

After reading this material you should be able to tell:

• What is a null hypothesis and how it relates to the p-value
• How to calculate the p-value
• How to interpret the p-value

What is Null Hypothesis?

In statistical hypothesis testing, we construct two opposing hypotheses to make inferences about a population parameter based on sample data:

1. The null hypothesis ($H_0$) represents the assumption that there is no effect, no difference, or no change in the population. It serves as the default assumption in hypothesis testing. For example, if we are testing a new drug, the null hypothesis might state that the drug has no effect on patients compared to a placebo.
2. The alternative hypothesis ($H_1$ or $H_a$) challenges the null hypothesis by suggesting that there is an effect, a difference, or a change in the population. Continuing with the previous example, the alternative hypothesis would assert that the drug does have an effect on patients.

Confused now? That's understandable. Choosing a null hypothesis is often guided by heuristics, and there is no strictly defined method for selecting it. By examining numerous examples, you will begin to develop an intuition for what researchers typically mean by the null hypothesis and how they choose it. This practical exposure is important for truly mastering the concept.

The relationship between the null and alternative hypotheses is foundational to the scientific method, as it allows researchers to test assumptions and determine whether an observed effect is genuine or simply due to random chance. Null hypothesis significance testing (NHST) is used across various academic disciplines, including medicine, psychology, neuroscience, genetics, economics, linguistics, and more.

Now, pay close attention! If that wasn't confusing enough, there's more to the story. Another concept is closely related, and together they form the foundation of NHST. As we already learned the primary goal of NHST is to use statistical evidence from sample data to decide whether to reject the null hypothesis in favor of the alternative hypothesis. This decision is based on the p-value, which quantifies the probability of obtaining results at least as extreme as those observed, assuming that the null hypothesis is true.

The plot illustrates the distribution of a test statistic under the assumption that the null hypothesis $H_0$ is true. Here's a detailed explanation of the components:

• The green curve (null distribution) represents the probability density function (pdf) of the test statistic under the null hypothesis ($H_0$). This distribution is determined by the specific test being used (e.g., Z-distribution for large samples, t-distribution for smaller samples) and is often centered around a mean value that reflects the null hypothesis (e.g., zero difference).
• The vertical dashed red line marks the observed value of the test statistic from the sample data, such as $z=2.0$. This observed statistic is calculated from the sample data and is used to determine how likely such an observation is under the null hypothesis.
• The orange shaded area (tail area) shows the probability of observing a test statistic as extreme as, or more extreme than, the observed value under $H_0$. This area represents the p-value. In a one-tailed test, the p-value is the probability of observing a test statistic greater than or equal to the observed value (or less than or equal to, depending on the direction of the test). In a two-tailed test, the p-value includes both tails, representing the probability of observing a test statistic as extreme as the observed value in either direction (both greater than and less than).

The Language of Hypothesis Testing

• If the evidence (based on the p-value) is strong enough, we reject $H_0$, suggesting that there is statistically significant evidence in favor of the alternative hypothesis.
• When the evidence is insufficient, we fail to reject $H_0$. This does not mean we accept $H_0$ as true, but rather that we do not have enough evidence to conclude it is false.
• It's important to avoid "accepting" $H_0$, as statistical tests are meant to assess whether there is sufficient evidence against $H_0$, not to prove that $H_0$ is true.
• Type I error occurs when we incorrectly reject $H_0$ even though it is true (a false positive).
• Type II error happens when we fail to reject $H_0$ even though $H_1$ is true (a false negative).

Understanding P-Values Through an Analogy

Imagine you're a detective tasked with determining whether a suspect is guilty of a crime. In this scenario, the null hypothesis ($H_0$) is that the suspect is innocent, while the alternative hypothesis ($H_a$) is that the suspect is guilty.

As you gather evidence from the crime scene—such as fingerprints, DNA samples, and eyewitness accounts—you assess how likely it is to find this evidence if the suspect were truly innocent. The p-value represents the probability of obtaining evidence as strong as what you have (or stronger) under the assumption that the suspect is innocent.

In a one-tailed test, you might only consider evidence that points towards guilt, calculating the probability of finding such incriminating evidence. This means you're looking at one direction of the distribution of possible evidence. Conversely, in a two-tailed test, you also account for evidence that could exonerate the suspect, considering both directions—whether the evidence is strongly incriminating or strongly exonerating.

If the p-value is low, say below a 5% significance level, it suggests that the observed evidence is unlikely to have occurred by chance if the suspect were innocent. This would lead you to reject the null hypothesis and conclude that there is sufficient evidence to support the suspect's guilt. On the other hand, a high p-value would indicate that the evidence is not strong enough to rule out innocence, and you would fail to reject the null hypothesis.

Interpretation of the P-value

The p-value is the probability of obtaining a test statistic at least as extreme as the one observed, assuming that the null hypothesis $H_0$ is true.

• The p-value does not represent the probability that $H_0$ is true or false. It simply measures how consistent the data are with the null hypothesis.
• A large p-value means the data are consistent with $H_0$, but this does not prove that $H_0$ is true.
• A small p-value (when $p<\alpha$) provides strong evidence against $H_0$, leading to the rejection of the null hypothesis.
• A large p-value (when $p\ge \alpha$) indicates insufficient evidence to reject $H_0$, so we fail to reject it.

Example: Coin Toss Experiment

• The null hypothesis ($H_0$) states that the coin is fair, meaning the probability of heads is $P\left(\text{heads}\right)=0.5$.
• The alternative hypothesis ($H_1$) suggests that the coin is not fair, so the probability of heads is $P\left(\text{heads}\right)\ne 0.5$.
• The experiment involved a total of 10 tosses ($n=10$).
• The observed outcome was 4 heads and 6 tails.

Statistical Test

We use the binomial test to calculate the p-value for the observed outcome.

I. Compute the probability of observing exactly 4 heads under $H_0$:

$$P(X=4)=(\frac{10}{4})\left(0.5{)}^4\right(0.5{)}^6=\frac{210\times 0.0625\times 0.015625}{1}=0.2051$$

II. Compute the probabilities for more extreme outcomes (0 to 4 heads and 6 to 10 heads) since it's a two-tailed test.

III. Total P-value:

$$\text{p-value}=2\times \left(P\right(X\le 4\left)\right)=2\times \left(\sum _{k=0}^{4}P\right(X=k\left)\right)$$

IV. Sum of Probabilities:

$$ \begin{align} P(X = 0) &= 0.00098 \\ P(X = 1) &= 0.00977 \\ P(X = 2) &= 0.04395 \\ P(X = 3) &= 0.11719 \\ P(X = 4) &= 0.20508 \\ \sum_{k=0}^{4} P(X = k) &= 0.37697 \end{align} $$

V. Compute P-value:

$$\text{p-value}=2\times 0.37697=0.75394$$

Decision

• The significance level ($\alpha$) is typically set at 0.05.
• In this case, the comparison shows that the p-value is $0.75394$, which is greater than $0.05$.
• As a result, we are failing to reject $H_0$, meaning we cannot conclude that the coin is unfair based on this data.
• However, this does not mean we are accepting $H_0$. The result does not prove the coin is fair; it simply indicates that the observed data are consistent with a fair coin.

Additional Considerations

• Statistical power is the probability of correctly rejecting $H_0$ when $H_1$ is true.
• The sample size influences power, with larger samples leading to increased power.
• A one-tailed test checks for an effect in one specific direction, such as $P\left(\text{heads}\right)>0.5$.
• A two-tailed test examines for an effect in either direction, for example, $P\left(\text{heads}\right)\ne 0.5$.
• Confidence intervals provide an estimated range of values that are likely to include the population parameter.
• In relation to hypothesis testing, if the confidence interval does not include the null hypothesis value, $H_0$ can be rejected.