12.6: Interval Estimation
- Page ID
- 65583
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Earlier we mentioned that numerical summaries computed on a sample can be used to estimate the corresponding value of the population parameter. In the case of the five exams sampled from a class, the mean grade in the sample can be used to estimate the mean grade of all the exams in the class. In that section we also learned that such an estimate was subject to error and that the typical size of this error is estimated by the standard error. One way to report an estimate is by including the associated standard error using what is known as the plus/minus notation.
For example, suppose the professor chooses five exams at random from the class, and the mean of the grades on the sampled exams is 72.7. The professor also uses the sample to compute the standard error of the mean grade of the sample, and this comes out to be 4.5. Many researchers would report this result as the mean class grade on the exam is estimated to be \(72.7\pm 4.5\). The idea with this notation is that the best guess about the mean grade on the exam for the class is 72.7, but the researcher knows there is error in this estimate. If 72.7 is the best guess, then the error could be taken to be equally as probable on each side of the estimate. This is an assumption, but it is true in many cases. That means that one could reasonably assume that the true mean class grade could be as small as \(72.7-4.5=68.2\) or as large as \(72.5+4.5=77.2\). That is, it is reasonable to assume the the true class grade is between 68.2 and 77.2. In statistics this would usually be written as an interval using the notation \([68.2,77.2]\).
Of course, sometimes the error is a bit more than the standard error, so maybe a researcher should use the interval \(72.7\pm 2\times 4.5\) to be more certain. This would give the interval \([73.7,81.7]\). The researcher should be more certain that this interval will really contain the true mean, but how much more sure are they? Maybe the researcher should use the interval \(72.7\pm 3\times 4.5\) instead.
A confidence interval takes this idea and formalizes it within a statistical framework. The key idea that statistical theory can add to this framework is the idea of controlling risk. Just as with a statistical hypothesis test, there is error involved when using an interval to estimate a parameter value. The researcher does not make any real decision here because once the interval is computed, the conclusion is that the true parameter value is contained between the lower and upper bounds of the interval. If the interval really does contain the value of the population parameter, then the researcher is correct. However, because we don't know exactly how big the error is, it can happen that the true value of the population parameter is not within the interval. In this case an error has been made.
As with statistical testing, a researcher would like to make the probability as small as possible that this type of error occurs. But again, there is a cost to making this probability too small. As the error probability becomes smaller, the interval becomes wider. At some point the interval is so wide that it does not convey any information. For the professor and their exams scores, suppose they decide to make the error probability something like 0.00000001. The corresponding interval would be \([47.1,97.5]\). This interval is so wide that the professor cannot really tell how well the students performed. Usually, the error probability will be set to 0.05 or 0.01.
These types of intervals are known as interval estimators. When a level of risk is specified along with the interval estimator, the resulting estimator is called a confidence interval. As with hypothesis testing, the error probability is specified to be a value \(\alpha\). The term confidence refers to the probability that the interval contains the true population value and is usually in terms of a percentage. For example, a 99% confidence interval refers to an interval estimator with \(1-\alpha=0.99\) so that \(\alpha=0.01\) or the risk that the interval does not contain the true parameter value is 0.01.
A \(100\times (1-\alpha)\% \) confidence interval is a range of values calculated from an observed sample that is guaranteed to contain the true parameter value with probability \(1-\alpha\).
A very important part of the definition has to do with how the probability is interpreted. Because of the way probability is interpreted in this case, the risk level refers to the probability that the confidence interval contains the true parameter value before the random sample is taken. This is because after the random sample is taken, there is nothing random left in the process. Statisticians and other researchers are very specific when they interpret a confidence interval, saying, for example, that they are 99% confident that the true parameter value is contained in the interval instead of saying that the probability that the interval contains the true population parameter is 0.99.