Last modified: March 11, 2025

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Hypothesis Testing

Hypothesis testing is a tool in statistics that drives much of scientific research. It lets us draw conclusions about entire populations based on the information we collect from samples. You'll find it applied in many areas—from evaluating how well a new drug works in clinical trials to unraveling the mysteries of customer behavior in business analytics.

A hypothesis is a statement that might be true.

Inputs and Outputs of Hypothesis Testing

Inputs:

1. The null hypothesis ($H_0$) represents the default assumption or the status quo, indicating no effect or no difference. Hypothesis testing aims to challenge this statement.
2. The alternative hypothesis ($H_1$ or $H_a$) is the claim that the test seeks to support, indicating the presence of an effect or difference.
3. The significance level ($\alpha$) is a pre-determined threshold, usually set at 0.05, which defines the risk of rejecting the null hypothesis when it is actually true (Type I error).
4. Sample data refers to the collected data from observations, experiments, or surveys, providing the basis for calculating the test statistic and the p-value.

Output:

The p-value is the probability of observing data as extreme as, or more extreme than, the sample data, assuming the null hypothesis is true. A small p-value (typically ≤ $\alpha$) provides strong evidence against the null hypothesis.

Overview of Hypothesis Testing Steps

Hypothesis testing is a structured process involving several steps:

1. The process begins by formulating hypotheses, where the null and alternative hypotheses are defined based on the research question.
2. Next, a significance level ($\alpha$) is chosen, often set at 0.05, but it can be adjusted depending on the study's requirements or field norms.
3. Data collection is conducted systematically to ensure the data is representative and free from bias.
4. After collecting the data, the test statistic is calculated using an appropriate formula to convert the sample data into a value suitable for hypothesis testing.
5. The p-value is then determined, representing the probability of obtaining the observed or more extreme test statistic under the null hypothesis.
6. In decision making, $H_0$ is rejected if the p-value is less than $\alpha$; otherwise, you fail to reject $H_0$.
7. Finally, interpreting results involves understanding the decision in the context of the research question. Failing to reject $H_0$ does not prove it is true, only that there isn’t strong evidence against it.

Example: Marble Bags

Imagine two bags: Bag A with a mix of 5 white and 5 black marbles, and Bag B with only black marbles.

Bag A            Bag B
  _____            _____
 / • •  \         / O O  \
|  • •  |        |  O O  |    O = White Marble
|  O O  |        |  O O  |    • = Black Marble
|  O O  |        |  O O  |    
|  • O  |        |  O O  |
 \_____/          \_____/

Suspecting you have Bag B, you decide to test this hypothesis:

• The null hypothesis ($H_0$) states that the bag in question is Bag A.
• The alternative hypothesis ($H_a$) asserts that the bag in question is Bag B.

Drawing n marbles and finding them all black leads to calculating p-values to test these hypotheses. For Bag A, the chance of drawing a black marble is 0.5. Hence, drawing n black marbles consecutively from Bag A has a probability of $(0.5)^n$.

• For n=2, p-value = $(0.5)^2 = 0.25$
• For n=3, p-value = $(0.5)^3 = 0.125$
• For n=5, p-value = $(0.5)^5 = 0.03125$

A smaller p-value indicates stronger evidence against $H_0$. As n increases, the likelihood that you have Bag B (only black marbles) increases.

Null and Alternative Hypotheses for a Mean

When testing the population mean, hypothesis testing considers three possibilities, each with distinct null and alternative hypotheses:

Types of Tests

I. Left-Tailed Test

• The null hypothesis ($H_0$) states that the population mean $\mu$ is equal to a specific value $\mu_0$ ($\mu = \mu_0$).
• The alternative hypothesis ($H_a$) claims that $\mu$ is less than $\mu_0$ ($\mu < \mu_0$).

II. Right-Tailed Test

• The null hypothesis ($H_0$) suggests that $\mu$ is equal to $\mu_0$ ($\mu = \mu_0$).
• The alternative hypothesis ($H_a$) asserts that $\mu$ is greater than $\mu_0$ ($\mu > \mu_0$).

III. Two-Tailed Test

• The null hypothesis ($H_0$) states that $\mu$ is equal to $\mu_0$ ($\mu = \mu_0$).
• The alternative hypothesis ($H_a$) proposes that $\mu$ is not equal to $\mu_0$ ($\mu \neq \mu_0$).

The null hypothesis always assumes that the population mean $\mu$ equals a predetermined value $\mu_0$. The alternative hypothesis presents a contrary statement: the population mean $\mu$ is less than, greater than, or not equal to $\mu_0$.

Important Note: Left-tailed and right-tailed tests are typically used when the effect is expected to occur in only one direction or when only one-directional effects are relevant. In most research scenarios, a two-tailed test is preferred unless there's strong justification for a one-tailed test.

Examples

I. Testing the Effectiveness of a New Diet (Two-Tailed Test)

• The null hypothesis ($H_0$) is that the average daily energy expenditure $\mu$ equals the standard diet average $\mu_0$.
• The alternative hypothesis ($H_a$) is that the average daily energy expenditure $\mu$ differs from the standard diet average $\mu_0$ ($\mu \neq \mu_0$).

II. Evaluating Customer Service Efficiency (Left-Tailed Test)

• The null hypothesis ($H_0$) states that the average resolution time $\mu$ equals the industry standard of 10 minutes.
• The alternative hypothesis ($H_a$) is that the average resolution time $\mu$ is less than 10 minutes ($\mu < 10$).

III. Assessing the Impact of a New Teaching Method (Right-Tailed Test)

• The null hypothesis ($H_0$) suggests that the average test score $\mu$ equals the district average of 75%.
• The alternative hypothesis ($H_a$) asserts that the average test score $\mu$ exceeds the district average of 75% ($\mu > 75$).

The P-value

Once the data is collected and the sample statistic computed, the researcher computes the P-value.

The P-value is the probability of obtaining a measurement at least as extreme as the one we measured, under the assumption that the null hypothesis is true.

• In a left-tailed test, the shaded area represents the p-value, which is the area under the curve to the left of the sample statistic.
• In a right-tailed test, the shaded area represents the p-value, which is the area under the curve to the right of the sample statistic.

By at least as extreme, we mean a value at least as far to the left or right of the measured value.

Choosing the Right Statistical Test

Selecting a suitable statistical test is critical in hypothesis testing, and several factors determine the appropriate choice:

Factors to Consider

1. The type of data (categorical, ordinal, interval, or ratio) significantly influences the choice of statistical test, as different types of data require different methods.
2. The number of variables under analysis determines the test type, with specific tests designed for univariate (one variable), bivariate (two variables), and multivariate (more than two variables) analyses.
3. The distribution of the data is crucial, as parametric tests assume a normal distribution. If the data does not meet this assumption, a non-parametric test is more appropriate.
4. The study design, such as comparing groups or measuring changes over time within a group, also plays a role in selecting the correct statistical test.

Examples of Statistical Tests

• A t-test is ideal for comparing the means of two groups with interval or ratio data that is normally distributed. For non-normally distributed data or ordinal data, consider a non-parametric alternative like the Mann-Whitney U test.
• To compare means across more than two groups, an Analysis of Variance (ANOVA) is appropriate.
• For categorical data, a chi-square test is often used to examine differences in proportions.
• Left-tailed and right-tailed tests are specific to the direction of the hypothesis. A left-tailed test is used when the research hypothesis suggests a decrease or lower value, while a right-tailed test is for an increase or higher value.
• Two-tailed tests are applied when the research question does not specify a direction of effect. These tests are more conservative and broadly applicable.
• Parametric tests assume underlying statistical distributions and typically require interval or ratio data. In contrast, non-parametric tests do not assume a specific distribution and are often used with ordinal data or when the assumptions of parametric tests are not met.

The following table summarizes some common statistical tests and their applications:

Example: Hypothesis Test for the Mean

An agronomist suggests that a new fertilizer increases the average yield of a particular crop to more than 2 tons per hectare. To test this claim, a study is conducted where the new fertilizer is applied to randomly selected plots. The yield of 25 plots is measured, resulting in a mean yield of 2.1 tons per hectare and a standard deviation of 0.3 tons per hectare. Is the new fertilizer effective at increasing the average yield at a significance level of $\alpha = 0.05$?

Hypothesis Setup:

• Null Hypothesis ($H_0$): $\mu = 2$ tons per hectare (The fertilizer does not increase yield)
• Alternative Hypothesis ($H_a$): $\mu > 2$ tons per hectare (The fertilizer increases yield)

Test Statistic:

For the test statistic, we use the one-sample z-test since the sample size is greater than 30:

$$z = \frac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}$$

where:

• $\bar{x}$ is the sample mean,
• $\mu_0$ is the hypothesized population mean,
• $\sigma$ is the population standard deviation (approximated here by the sample standard deviation),
• $n$ is the sample size.

Plugging in the values:

$$z = \frac{2.1 - 2}{0.3/\sqrt{25}}$$

$$z = \frac{0.1}{0.06}$$

$$z \approx 1.667$$

We look up the critical z-value for a right-tailed test at $\alpha = 0.05$, which is approximately 1.645. Since our calculated z-value of 1.667 is greater than 1.645, we reject the null hypothesis.

There is sufficient evidence at the $\alpha = 0.05$ significance level to support the claim that the new fertilizer increases the average yield of the crop to more than 2 tons per hectare.