Suppose you were trying to determine the mean rent of a two-bedroom apartment in your town.
You might look in the classified section of the newspaper, write down several rents listed, and average them together.
You would have obtained a point estimate of the true mean. If you are trying to determine the percentage of times you make a basket when shooting a basketball, you might count the number of shots you make and divide that by the number of shots you attempted.
In this case, you would have obtained a point estimate for the true proportion.
We use sample data to make generalizations about an unknown population. This part of statistics is called inferential
The sample data help us to make an estimate of a population parameter. We realize that the point estimate is most likely not the exact value of the population parameter, but close to it.
After calculating point estimates, we construct interval estimates, called confidence intervals.
In this article, you will learn to construct and interpret confidence intervals. You will also learn a new distribution, the Student’s-t, and how it is used with these intervals.
Throughout the chapter, it is important to keep in mind that the confidence interval is a random variable. It is the population parameter that is fixed.
A confidence interval is another type of estimate but, instead of being just one number, it is an interval of numbers.
The interval of numbers is a range of values calculated from a given set of sample data. The confidence interval is likely to include an unknown population parameter.
Single Population Mean Using the Normal Distribution
A confidence interval for a population mean with a known standard deviation is based on the fact that the sample means follow an approximately normal distribution. Suppose that our sample has a mean of x = 10 and we have constructed the 90% confidence interval (5, 15) where EBM = 5.
Calculating the Confidence Interval
To construct a confidence interval for a single unknown population mean μ, where the population standard deviation is known, we need x as an estimate for μ and we need the margin of error.
Here, the margin of error (EBM) is called the error bound for a population mean (abbreviated EBM).
The sample mean x is the point estimate of the unknown population mean μ.
The confidence interval estimate will have the form:
(point estimate – error bound, point estimate + error bound) or, in symbols,( x – EBM, x+EBM )
The margin of error (EBM) depends on the confidence level (abbreviated CL).
The confidence level is often considered the probability that the calculated confidence interval estimate will contain the true population parameter.
However, it is more accurate to state that the confidence level is the percent of confidence intervals that contain the true population parameter when repeated samples are taken.
Most often, it is the choice of the person constructing the confidence interval to choose a confidence level of 90% or higher because that person wants to be reasonably certain of his or her conclusions.
There is another probability called alpha (α). α is related to the confidence level, CL. α is the probability that the interval does not contain the unknown population parameter.
Mathematically, α + CL = 1.