7.2. Null and Alternative Hypotheses#
In statistical hypothesis testing, we assess two competing statements about a population to make inferences based on sample data. These statements are the null hypothesis and the alternative hypothesis. The null hypothesis typically represents the absence of an effect or relationship, while the alternative hypothesis suggests the presence of an effect or relationship. The goal is to determine whether the sample data provide sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis.
Definition - Null Hypothesis
The null hypothesis (\(H_{0}\)) in statistics is a statement asserting that there is no significant difference or relationship between the variables being studied. It can be seen as the status quo or the default assumption. When you fail to accept the null hypothesis, it indicates that there is evidence to suggest a relationship or difference between the variables, and this often prompts the need for further investigation or action.
Definition - Alternative Hypothesis
The alternative hypothesis (\(H_{a}\)) is a claim about the population that contradicts the null hypothesis. It represents what the researcher is trying to prove or establish through their study. When there is enough evidence to reject the null in favor of the alternative hypothesis, it suggests that the research has found support for the proposed claim or relationship.
Example 7.4
A researcher is investigating whether a new teaching method improves student test scores in mathematics.
Null Hypothesis (\(H_0\)): The new teaching method has no effect on student test scores in mathematics.
Alternative Hypothesis (\(H_a\)): The new teaching method increases student test scores in mathematics.
7.2.1. Decision Making in Hypothesis Testing#
In hypothesis testing, we evaluate two competing claims: the null hypothesis (\(H_0\)) and the alternative hypothesis (\(H_a\)). These hypotheses are mutually exclusive, representing opposing statements about a population parameter. Our task is to scrutinize the sample data and make an informed decision based on the evidence.
After analyzing the sample data, we arrive at one of two possible decisions:
Reject \(H_0\): If the sample data provides compelling evidence in favor of \(H_a\), we reject \(H_0\). This suggests a significant relationship or effect exists in the population.
Fail to Reject \(H_0\): If the sample data does not offer sufficient evidence to support \(H_a\), we retain \(H_0\). This indicates a lack of evidence for a significant relationship or effect in the population.
It’s crucial to interpret these results carefully, avoiding unwarranted claims about the population based solely on sample evidence.
In formulating hypotheses, we use specific mathematical symbols to express our claims:
The null hypothesis (\(H_0\)) typically includes an equality sign (=, ≥, or ≤), representing no significant difference or effect.
The alternative hypothesis (\(H_a\)) never includes an equality sign and uses symbols that contradict \(H_0\) (≠, >, or <).
The choice of symbols depends on the research question and the nature of the hypothesis test. For instance:
To test if a value exceeds a claim, \(H_0\) might use ≤, while \(H_a\) uses >.
To test if a value is less than a claim, \(H_0\) might use ≥, while \(H_a\) uses <.
Table 7.1 provides a quick reference for selecting the appropriate symbols based on the research question and the type of comparison being made.
\(H_0\) (Null Hypothesis) |
\(H_a\) (Alternative Hypothesis) |
|---|---|
equal to ( \(=\) ) |
not equal to ( \(\neq\) ) or greater than ( \(>\) ) or less than ( \(<\) ) |
greater than or equal to ( \(\geq\) ) |
less than ( \(<\) ) |
less than or equal to ( \(\leq\) ) |
greater than ( \(>\) ) |
Remark
Some researchers may use = in \(H_0\) even when > or < is used in \(H_a\). This practice is acceptable as it doesn’t affect the decision to reject or fail to reject \(H_0\). The hypothesis test evaluates sample data evidence, and the choice of symbols doesn’t impact the validity of the statistical analysis.
Example 7.5 (Average Screen Time)
A digital wellness company is investigating adult screen time habits. They want to test if the average daily screen time for adults exceeds 3 hours.
Null Hypothesis (\(H_0\)): The average daily screen time for adults is 3 hours or less.
Alternative Hypothesis (\(H_a\)): The average daily screen time for adults is more than 3 hours.
Mathematical Representation:
where \(\mu\) represents the population mean daily screen time in hours.
Example 7.6 (Body Mass Index (BMI))
A health researcher is investigating whether the mean BMI in a specific population differs from the World Health Organization’s standard for a normal BMI of 25 kg/m².
Null Hypothesis (\(H_0\)): The population mean BMI is equal to 25 kg/m².
Alternative Hypothesis (\(H_a\)): The population mean BMI is not equal to 25 kg/m².
Mathematical Representation:
where \(\mu\) represents the population mean BMI in kg/m².
Example 7.7 (Plant Growth)
An agricultural scientist is examining whether a new organic fertilizer increases the average growth rate of tomato plants beyond the standard rate of 2 inches per week.
Null Hypothesis (\(H_0\)): The average growth rate of tomato plants with the new organic fertilizer is 2 inches per week or less.
Alternative Hypothesis (\(H_a\)): The average growth rate of tomato plants with the new organic fertilizer is more than 2 inches per week.
Mathematical Representation:
where \(\mu\) represents the population mean growth rate of tomato plants in inches per week.
Example 7.8 (Average Sleep Duration)
A team of sleep researchers is investigating the sleep patterns of teenagers to determine if their average sleep duration differs from the recommended 8 hours per night.
Null Hypothesis (\(H_0\)): The average sleep duration of teenagers is 8 hours per night.
Alternative Hypothesis (\(H_a\)): The average sleep duration of teenagers is not 8 hours per night.
Mathematical Representation:
where \(\mu\) represents the population mean sleep duration of teenagers in hours per night.
Example 7.9 (Employee Retention Rate)
A human resources department wants to evaluate if the introduction of a new employee wellness program has increased the annual employee retention rate beyond the industry standard of 85%.
Null Hypothesis (\(H_0\)): The annual employee retention rate is 85% or less after implementing the new wellness program.
Alternative Hypothesis (\(H_a\)): The annual employee retention rate is more than 85% after implementing the new wellness program.
Mathematical Representation:
where \(p\) represents the population proportion of employees retained annually after the implementation of the new wellness program.
Example 7.10 (Water Consumption)
A public health campaign aims to increase daily water consumption. Researchers want to determine if the average water consumption per person has increased beyond the previous average of 2 liters per day.
Null Hypothesis (\(H_0\)): The average water consumption per person is 2 liters per day or less.
Alternative Hypothesis (\(H_a\)): The average water consumption per person is more than 2 liters per day.
Mathematical Representation:
where \(\mu\) represents the population mean water consumption in liters per person per day.
Note
Please refer to Table 4.1 for a detailed list of mathematical symbols and their meanings. This table will assist you in recognizing and understanding the various signs used throughout this document.