Module 8: Inference for One Proportion
Hypothesis Testing (4 of 5)
Hypothesis Testing (4 of 5)
Learning OUTCOMES
- Recognize the logic behind a hypothesis test and how it relates to the P-value.
Hypothesis testing appears in all upcoming modules. The process and the logic of the hypothesis test will always be the same, but the details will differ somewhat.
Every hypothesis test will use a P-value to make a decision about the population(s). The P-value is the connection between probability and decision-making in inference. Now we discuss the P-value in more depth and relate it to our work in Linking Probability to Statistical Inference. Later we use both simulations and statistical software to find the P-value.
To develop a better understanding of the P-value, we need to return to the idea of a sampling distribution and a normal probability model. These are ideas from Linking Probability to Statistical Inference.
Example
What Is a P-value?
Let’s return to the familiar example of the 2008 presidential election. In that election, newspapers reported that Obama received 40% of the white male vote. We wonder if a smaller percentage of white males will support Obama in the 2012 election. We define the following hypotheses and conduct a hypothesis test.
- H0: The proportion of white males voting for Obama in 2012 is 0.40.
- Ha: The proportion of white males voting for Obama in 2012 is less than 0.40.
We select a random sample of 200 white male voters and find that 35% plan to vote for Obama in 2012. Clearly 35% is less than 40%. But is the difference statistically significant or due to chance? If the population proportion is 0.40, we expect to see sample proportions vary from this. But will sample proportions as small as or smaller than 0.35 occur very often? What’s the probability?
The probability (P-value) is about 0.078.
The P-value is the chance that a random sample of 200 white males will have, at most, 35% supporting Obama if 40% of this population supports Obama. This is quite a mouthful. We find that visualizing the sampling distribution helps us understand the P-value. Here is a diagram that may be helpful in interpreting the P-value.
In general, the P-value is the probability that sample results are as extreme as or more extreme than the result observed in the data if the null hypothesis is true. The phrase “as extreme as or more extreme than” means further from the center of the sampling distribution in the direction of the alternative hypothesis.
Note: You may recall the concept of a conditional probability from Relationships in Categorical Data with Intro to Probability. The P-value is a conditional probability. The condition is “the null hypothesis is true.”
Note: We can also look at the P-value in terms of error in the sample proportion. If 40% of this population support Obama, then our sample with 35% supporting Obama has a 5% error. From this perspective, the P-value is the chance that sample proportions supporting the alternative hypot