18 6.3 Introduction to Confidence Intervals
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We use inferential statistics to make generalizations about an unknown population. The simplest way of doing this is to use the sample data help us to make a point estimate of a population parameter. We realize that due to sampling variability the point estimate is most likely not the exact value of the population parameter, but should be close to it. After calculating point estimates, we can build off of them to construct interval estimates, called confidence intervals.
Confidence Intervals
A confidence interval is another type of estimate but, instead of being just one number, it is an interval of numbers. It provides a range of reasonable values in which we expect the population parameter to fall. Essentially the idea is that since a point estimate may not be perfect due to variability, we will build an interval based on a point estimate to hopefully capture the parameter of interest in the interval. There is no guarantee that a given confidence interval does capture the parameter, but there is a predictable probability of success. It is important to keep in mind that the confidence interval itself is a random variable, while the population parameter is fixed.
If you worked in the marketing department of an entertainment company, you might be interested in the mean number of songs a consumer downloads a month from iTunes. If so, you could conduct a survey and calculate the sample mean, \(\overline{x}\). You would use \(\overline{x}\) to estimate the population mean. The sample mean, \(\overline{x}\), is the point estimate for the population mean, μ.
Suppose, for the iTunes example, we do not know the population mean μ, but we do know that the population standard deviation is σ = 1 and our sample size is 100. Then, by the central limit theorem, the standard deviation for the sample mean is
[latex]\frac{\sigma }{\sqrt{n}}[/latex]=[latex]\frac{1}{\sqrt{100}}=0.1[/latex].
The Empirical Rule, which applies to bell-shaped distributions, says that in approximately 95% of the samples, the sample mean, \(\overline{x}\), will be within two standard deviations of the population mean μ. For our iTunes example, two standard deviations is (2)(0.1) = 0.2. The sample mean \(\overline{x}\) is likely to be within 0.2 units of μ.
Because \(\overline{x}\) is within 0.2 units of μ, which is unknown, then μ is likely to be within 0.2 units of \(\overline{x}\) in 95% of the samples. The population mean μ is contained in an interval whose lower number is calculated by taking the sample mean and subtracting two standard deviations (2)(0.1) and whose upper number is calculated by taking the sample mean and adding two standard deviations. In other words, μ is between \(\overline{x}\text{ }-\text{ 0}\text{.2}\) and \(\overline{x}\text{ }+\text{ 0}\text{.2}\) in 95% of all the samples.
For the iTunes example, suppose that a sample produced a sample mean \(\overline{x}\text{ }=\text{ 2}\). Then the unknown population