7.9:
Estimating Population Mean with Known Standard Deviation
Consider an example wherein a truck container is to be redesigned to accommodate longer oakwood logs.
The container is designed based on obsolete measurements, so engineers require a new mean length of the logs.
As getting measurements of all the oakwood trees or logs in the world is impossible, samples can be drawn from the available stock.
This is the sample mean, which is the best point estimate of the population mean when its standard deviation is small.
However, the confidence interval can provide a more reliable estimate of the population mean, which requires calculating the margin of error using the following equation.
If the population and samples both assume the normal distribution, and the sample size is more than 30, a critical value can be obtained using the z distribution.
However, determining the population mean with these assumptions requires prior knowledge of population standard deviation, which is an unrealistic situation.
In the example of oakwood logs, previous forestry studies may provide this standard deviation to calculate the margin of error.
7.9:
Estimating Population Mean with Known Standard Deviation
To construct a confidence interval for a single unknown population mean μ, where the population standard deviation is known, we need sample mean 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 is the point estimate of the unknown population mean μ.
The confidence interval estimate will have the form as follows:
(point estimate – error bound, point estimate + error bound)
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.
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.
Steps to Calculate the Confidence Interval:
To construct a confidence interval estimate for an unknown population mean, we need data from a random sample. The steps to construct and interpret the confidence interval are:
- Calculate the sample mean from the sample data. (Assumption: the population standard deviation σ is known)
- Find the z-score that corresponds to the confidence level e.g. 95%.
- Calculate the error bound EBM.
- Construct the confidence interval.
- Write a sentence that interprets the estimate in the context of the situation in the problem.
This text is adapted from Openstax, Introductory Statistics, Section 8.1 A single population mean using the normal distribution.