Hypothesis Testing: The Five Steps Approach
Hypothesis Testing: The Five Steps Approach
Overview of Hypothesis Testing
Core Purpose: The fundamental goal of hypothesis testing is to evaluate the null hypothesis (H_0).
Supporting the Research Hypothesis: If we decide to reject the null hypothesis, it directly provides support for our actual research hypothesis (H_1).
Why Test the Null?
We know or presume the value of the null hypothesis in the population.
This knowledge allows us to easily construct a sampling distribution for the null hypothesis.
Process: We compare our sample value (statistic) to the value specified by the null hypothesis.
Decision Basis: Our decision to reject the null hypothesis is based on the likelihood that our observed sample values could have come from the population under the conditions assumed by the null hypothesis (i.e., if the null were true).
The Five Steps in Hypothesis Testing
Develop Research Hypothesis (H_1): Formulate a specific, testable prediction (e.g., "Men self-disclose more frequently than women").
Measure Variables and Obtain Sample Data: Define how variables will be measured and then collect relevant data from a sample (N).
Compare Sample Data to the Sampling Distribution Under the Null Hypothesis: Determine what the distribution of sample statistics would look like if the null hypothesis were true in the population.
Select a Criterion for Rejecting the Null: Establish a threshold (e.g., a significance level \alpha) that will help in deciding whether the sample data is too unlikely under the null hypothesis to be attributed to chance.
Use Criterion to Decide: Apply the selected criterion to ascertain if the observed sample value differs sufficiently from the null hypothesis value. The alternative is that the observed difference is merely due to sampling error.
The Significance of the Null Hypothesis
Population Value Specification: The null hypothesis always specifies a particular value in the population. This value is most often zero, indicating no effect, no difference, or no association, but it can sometimes be another specific value.
Goal of Rejection: Our primary aim in research is typically to reject the null hypothesis, as this rejection provides empirical support for our research hypothesis.
Uncertainty of Research Hypothesis Values: We often lack precise knowledge about the exact values associated with our research hypothesis. For instance:
We may not know the exact magnitude of difference between treatment and control groups.
We might not know the precise strength of an association between two variables.
We may not know the exact percentage of voter preference.
What We Do Know We Don't Want: A key insight is that we usually know what value we DON'T want the outcome to be, which is often zero (i.e., no difference or no relationship).
Examples of Null Hypotheses:
Political Polls: For a poll with two candidates, the null hypothesis can be that there is no difference in support, meaning each candidate has 50\% preference. (P1 = P2 = 0.5 or P1 - P2 = 0).
Continuous Variables: The null hypothesis can state that a sample mean is not different from some other relevant population value. If two values are identical, their difference is zero.
Example: Testing if a Coin is Fair
This example illustrates the hypothesis testing steps in a concrete scenario.
Scenario Setup
Initial Assumption (H_0): The coin is fair unless proven otherwise.
Question: What evidence would compel you to believe the coin is not fair?
Thought Experiment: If you flip a coin 10 times, how many heads (or tails) would you need to observe to conclude it's unfair?
Simulation Note: Even with a truly fair coin, extreme outcomes (e.g., 10 heads or 10 tails) are possible, though not likely.
Threshold: This highlights the need to establish a threshold for rejecting the null hypothesis.
Probability Requirement: We must determine the probabilities associated with obtaining a given result (e.g., 7 heads) assuming the coin is fair.
Applying the Five Steps to the Coin Example
Research Hypothesis (H_1): The coin is NOT fair.
Develop Design to Test Null Hypothesis (H_0): The coin IS fair.
We choose to flip the coin 10 times (N=10).
Let's say we conduct the experiment and observe 7 heads and 3 tails.
Resist the Urge: It's crucial to acknowledge that 7 heads is possible with a fair coin.
Determine Sampling Distribution Under Null Hypothesis: We need to know the likelihood of each possible outcome if the coin were fair.
For 10 flips, the probability of getting r heads when the coin is fair (\pi = 0.5) for each flip can be calculated. The probabilities are:
P(0 \text{ heads}) = 0.0010
P(1 \text{ head}) = 0.0098
P(2 \text{ heads}) = 0.0439
P(3 \text{ heads}) = 0.1172
P(4 \text{ heads}) = 0.2051
P(5 \text{ heads}) = 0.2461
P(6 \text{ heads}) = 0.2051
P(7 \text{ heads}) = 0.1172
P(8 \text{ heads}) = 0.0439
P(9 \text{ heads}) = 0.0098
P(10 \text{ heads}) = 0.0010
Observation: 4, 5,, and 6 heads together account for 67\% of the possible outcomes when the coin is fair.
Visualization: Plotting these probabilities yields a bell-shaped distribution, resembling a normal curve.
Select a Criterion for Rejecting the Null Hypothesis: This involves deciding how much risk we are willing to take that the coin is fair, given the probability of