Statistical Inference (3 of 3)

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.

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.

Is the Asthma Rate Higher for Children Living in Poverty?

With data from the 2009 National Health Interview Survey, the Centers for Disease Control and Prevention estimated that 9.4% of U.S. children had asthma. Is the percentage higher for children living in poverty?

Suppose we select a random sample of 50 poor children and find that 9 of the 50 have asthma.

The conditions are not met for use of a normal model, so we ran a simulation with [latex]p = 0.094[/latex].

Comment

Are you wondering how extreme the results have to be before we conclude that the result is unusual? Well, it is a judgment call. No single cutoff point determines that a result is unusual. But in practice, we often agree on a cutoff point before we collect the data. In Inference for One Proportion, we discuss this idea further. However, if you are worried about this issue when taking a Checkpoint for this module, you can consider a result to be unusual if the probability is less than 5%.

Let’s Summarize

Introduction to Statistical Inference

The two types of inference procedures in this course are confidence intervals and hypothesis tests. The goal of a confidence interval is to estimate a parameter value. The goal of a hypothesis test is to test a claim about a parameter. Both types of inference are based on the sampling distribution of sample statistics. For both, we report probabilities that state what would happen if we used the inference method many times.

Confidence Intervals

The purpose of a confidence interval is to estimate a population parameter on the basis of a sample statistic. Sample statistics vary, so there is always error in our estimate, but we will never know how much. We therefore use the standard error, which is the average error in our sample estimates, to create a margin of error. The margin of error is related to our confidence that the interval contains the population parameter.

We investigated the 95% confidence interval for a population proportion in depth. When a normal model is a good fit for the sampling distribution, the 95% confidence interval has a margin of error equal to 2 standard errors.

[latex]\text{sample statistic} \pm \text{margin of error}[/latex]

[latex]\text{sample proportion} \pm 2(\text{standard error})[/latex]

[latex]\hat{p} \pm 2 \sqrt{\frac{p(1 - p)}{n}}[/latex]

 

We say we are 95% confident that the calculated interval contains the population proportion, meaning that 95% of the time, these intervals will actually contain the population proportion, and we will be right. Five percent of the time, we will be wrong. We can never tell if a confidence interval does or does not contain the population proportion we are trying to estimate.

Hypothesis Tests

The purpose of a hypothesis test is to use sample data to test a claim about a population parameter. We investigated testing a claim about a population proportion informally.

We make a claim about a population proportion. From the claim, we state an assumption about the value of the population proportion. Could the data have come from this population? Or is the sample proportion too far off? It depends on how much random samples from this population vary. We construct a simulation or a normal model to represent the sampling distribution that occurs when sampling from a population with this assumed value. We make a judgment about whether the data is likely or unlikely to occur in the sampling distribution. If the data supports our claim and is unlikely, then we doubt our assumption about the population proportion.

For example, if last year 20% of the U.S adult population smoked, we might claim that the percentage of smokers in the United States this year is greater. So in the simulation we set [latex]p = 0.20[/latex] and see if the data causes us to question this claim.

• If a sample proportion is likely to occur in the sampling distribution, then this sample result could have come from a population with the assumed value. In this situation, the data do not lead us to doubt our assumption about the value of the parameter. We therefore conclude that the evidence from the sample is not strong enough to support our original claim

• In our example, a sample proportion that is likely to occur means we do not question the assumption that we made when we set [latex]p = 0.20[/latex]. We cannot conclude that the percentage of smokers in the United States is greater than 20% this year.

• If a sample proportion supports our claim and is unlikely to occur in the sampling distribution, then it is unlikely that this sample result came from a population with the assumed value. In this situation, the data lead us to doubt our assumption about the value of the parameter. We conclude that the evidence from the sample is strong enough to support the claim.

• In our example, a sample proportion that is unusually large means that the data makes us doubt the assumption we made when we set [latex]p = 0.20[/latex]. We therefore is probably greater than 20% this year.

Likely or unlikely? It depends on how much the sample proportions vary. If the normal model is a good fit for the sampling distribution, we can calculate a z-score and use a simulation to associate a probability with our “likely” or “unlikely” statement. Recall what we learned in “Distribution of Sample Proportions” to calculate the z-score.

[latex]\text{standard error} = \sqrt{\frac{p(1 - p)}{n}}[/latex]

[latex]Z = \frac{\text{statistic} - \text{parameter}}{\text{standard error}} = \frac{\hat{p} - p}{\text{standard error}}[/latex]

 

We can also write this as one formula:

[latex]Z = \frac{\hat{p} - p}{\sqrt{\frac{p(1 - p)}{n}}}[/latex]