STATISTICS IN SUMMARY
Chapter Specifics
• Statistical inference draws conclusions about a population on the basis of data from a sample. Because we don’t have data for the entire population, our conclusions are uncertain.
510
• A confidence interval estimates an unknown parameter in a way that tells us how uncertain the estimate is. The interval itself says how closely we can pin down the unknown parameter. The confidence level is a probability that says how often in many samples the method would produce an interval that does catch the parameter. We find confidence intervals starting from the sampling distribution of a statistic, which shows how the statistic varies in repeated sampling.
• The standard deviation of the sampling distribution of the sample statistic is commonly referred to as the standard error.
• We estimate a population proportion p using the sample proportion of an SRS from the population. Confidence intervals for p are based on the sampling distribution of . When the sample size n is large, this distribution is approximately Normal.
• We estimate a population mean μ using the sample mean of an SRS from the population. Confidence intervals for μ are based on the sampling distribution of . When the sample size n is large, the central limit theorem says that this distribution is approximately Normal. Although the details of the methods differ, inference about μ is quite similar to inference about a population proportion p because both are based on Normal sampling distributions.
The reason we collect data is not to learn about the individuals that we observed but to infer from the data to some wider population that the individuals represent. Chapters 1 through 6 tell us that the way we produce the data (sampling, experimental design) affects whether we have a good basis for generalizing to some wider population—in particular, whether a sample statistic provides insight into the value of the corresponding population parameter. Chapters 17 through 20 discuss probability, the formal mathematical tool that determines the nature of the inferences we make. Chapter 18 discusses sampling distributions, which tell us how statistics computed from repeated SRSs behave and hence what a statistic (in particular, a sample proportion) computed from our sample is likely to tell us about the corresponding parameter of the population (in particular, a population proportion) from which the sample was selected.
In this chapter, we discuss the basic reasoning of statistical estimation of a population parameter, with emphasis on estimating a population proportion and population mean. To an estimate of a population parameter, such as a population proportion, we attach a margin of error and a confidence level. The result is a confidence interval. The sampling distribution, first introduced in Chapter 3 and discussed more fully in Chapter 18, provides the mathematical basis for constructing confidence intervals and understanding their properties. We will provide more advice on interpreting confidence intervals in Chapter 23.
511
CASE STUDY EVALUATED Going back to the survey results presented in the Case Study at the beginning of the chapter, the report presented by the BRFSS gave confidence intervals. With 95% confidence, the true percentage of California residents who meet the daily recommendations for fruit consumption is between 17.3% to 18.1%, and the true percentage of California residents who meet the daily recommendations for vegetable consumption is between 12.6% and 13.4%. Interpret these intervals in plain language that someone who knows no statistics will understand.
Online Resources
• The Snapshots video Inference for One Proportion discusses confidence intervals for a population proportion in the context of a sample survey conducted during the 2012 presidential elections.
• The StatClips Examples video Confidence Intervals: Intervals for Proportions Example C provides an example of how to calculate a 90% confidence interval for a proportion.
• The Snapshots video Confidence Intervals discusses confidence intervals for a population mean in the context of an example involving birds killed by wind turbines.