Wiki Page, Math 13 - Perez Sandbox : Module 18 - Introduction to Statistical Inference (3a of 3)

Wiki Page
2026-01-17T21:15:24-08:00

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 p=0.22 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 p=0.22, 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 p=0.22, 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 p=0.22. 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 normal shape. The formal conditions for use of a normal model require that the expected count of successes and failures are at least 10. Here we expect 22% (132 people) of the 600 people to be uninsured. We expect 78% (468 people) of the 600 to be insured. Since the sampling distribution meets the conditions for use of a normal model, we can calculate the z-score and find the probability that a random sample has a proportion of 0.25 or greater. The z-score is 1.76, and our applet gives a probability of 0.039.

Both approaches suggest that a sample proportion of 0.25 is unlikely. We don’t expect to see 25% or more uninsured in random samples of 600 very often. We estimate the chances as 4.5% with the simulation and only about 3.9% with the normal model. This is so unusual that we conclude the data from this year did not come from a population with only 22% uninsured. Our data provides strong evidence that more than 22% in the population are uninsured.

EXAMPLE Test the claim about health insurance again with different data

Now we retest the same claim using different data.

Claim: The percentage of U.S. adults (ages 18–64) who do not have health insurance is higher than 22% this year.

Suppose 26% of a random sample of 50 U.S. adults (age 18–64) do not have health insurance this year. What can we conclude?

Simulation: We again used a simulation to select 2,000 random samples from a population with p=0.22. This time there are 50 people in each sample. Judging from the simulation, a sample proportion of 0.26 is not unlikely. Sample proportions of 0.26 or greater occur frequently. In this simulation, 608 (30.4%) of the 2,000 random samples had proportions of 0.26 or greater.

Normal Probability Model of the Sampling Distribution: Visually, the simulated sampling distribution looks like it has a normal shape. The formal conditions for use of a normal model require that the expected count of successes and failures is at least 10. Here we expect 22% (11 people) of the 50 people to be uninsured. We expect 78% (39 people) of the 50 to be insured. Since the sampling distribution meets the conditions for use of a normal model, we can calculate the z-score and find the probability that a random sample has a proportion of 0.26 or greater. The z-score is 0.68, and our applet gives a probability of 0.25.

Both approaches suggest that a sample proportion of 0.26 is not unlikely. It falls within a typical range of sample proportions that we expect to see from random samples of 50 people. The z-score is 0.68, meaning the sample proportion is less than 1 standard error from 0.22. We also see the probability that a random sample has 26% or more uninsured is high: about 30% according to the simulation and about 25% according to the normal model. This probability suggests the data from this year could have come from a population with only 22% uninsured. Even though 26% are uninsured in our sample, our data does not provide strong evidence that more than 22% of the population are uninsured this year.

Comment

How can a sample proportion of 0.25 be unusual in the first example but a sample proportion of 0.26 not be unusual in the second example? These two examples highlight an important point. We have to judge a sample result by looking at it in relation to other samples of the same size. In the first example, the samples are large (600 adults in each sample), so the sample proportions do not vary much. In this sampling distribution, a sample result of 0.25 or greater is unlikely to occur. In the second example, the samples are smaller (only 50 adults in each sample), so the sample proportions vary more. In this sampling distribution, a sample result of 0.26 or greater is likely to occur.

Mark this page as done, then click the "Next" or > button to continue.

Content by the Open Learning Initiative and licensed under CC BY.