Module 7: Linking Probability to Statistical Inference
Module 7: Linking Probability to Statistical Inference
Statistical Inference (3 of 3)
Statistical Inference (3 of 3)
Learning OUTCOMES
- Test a hypothesis about a population proportion using a simulated sampling distribution or a normal model of the sampling distribution. State a conclusion in context.
Now we focus on the second type of inference: hypothesis testing and the logic behind it.
In hypothesis testing, we make a claim about a parameter and test it. On this page, we make a claim about a population proportion and use a sample proportion from data to test our claim. This is very similar to the thinking we did with simulations in the previous module.
Example
Test a Claim about Health Insurance Coverage
With data from the 2010 National Health Interview Survey, the Centers for Disease Control and Prevention (CDC) estimates that 22% of U.S. adults (age 18–64) did not have health insurance in 2010. Is the percentage higher this year? In a hypothesis test, we translate the research question into a claim about the population.
Claim: The percentage of U.S. adults (ages 18–64) who do not have health insurance is higher than 22% this year.
To test the claim, we assume that the percentage is 22% this year. Then we gather a random sample from the population to test the claim. Suppose 25% of a random sample of 600 U.S. adults (age 18–64) do not have health insurance this year. What can we conclude? Obviously, this sample has more than 22% uninsured adults. But does this data suggest the percentage of the U.S. adult population (age 18–24) who are uninsured is greater than 22%?
To test the claim, we begin with a population with [latex]p = 0.22[/latex] and take random samples of 600 people at a time.
- If a sample proportion of 0.25 is likely to occur when sampling from a population with [latex]p = 0.22[/latex], then this sample could have come from a population with 22% uninsured. The evidence from the sample is not strong enough to conclude that the population percentage is greater than 22%.
- If a sample proportion is unlikely when sampling from a population with [latex]p = 0.22[/latex], then the sample provides evidence that the proportion of population who are uninsured is greater than 22%.
Likely or unlikely? It depends on how much the sample proportions vary. We need to use a simulation or a mathematical model to represent the sampling distribution of sample proportions.
Simulation: We used a simulation to select 2,000 random samples of 600 people, each from a population with [latex]p = 0.22[/latex]. Judging from the simulation, a sample proportion of 0.25 is unlikely. Sample proportions of 0.25 or greater do not occur very often. In this simulation, only 90 out of the 2,000 random samples (4.5%) had proportions of 0.25 or greater.
Normal Probability Model of the Sampling Distribution: We can also apply what we know from our work with a normal model of the sampling distribution. Visually, the simulated sampling distribution looks like it has a nor