Wiki Page, Math 13 - Perez Sandbox : Module 18 - Wrap Up "Introduction to Statistical Inference"-2

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2026-01-17T21:15:25-08:00

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.

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 p=0.20 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 p=0.20 . 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 p=0.20 . We therefore conclude that the percentage of smokers in the U.S. 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 an applet to associate a probability with our “likely” or “unlikely” statement. Recall what we learned in "Distribution of Sample Proportions" to calculate the z-score.

We can also write this as one formula:

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