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This module explains how to construct a confidence interval estimate for an unknown population mean when the population standard deviation is known, using the Standard Normal distribution. This module has been revised from the original module m16962 written by authors S. Dean and Dr. B. Illowsky in the textbook collection Collaborative Statistics. Most of the content is identical to the original module; it has been revised to include step by step solutions for all examples. In addition, this revision of the module now includes the calculation of the sample size needed to obtain a specified margin of error and confidence level.

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 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 depends on the confidence level (abbreviated CL ). The confidence level is the probability that the confidence interval estimate that we will calculate will contain the true population parameter. Most often, itis the choice of the person constructing the confidence interval to choose a confidence level of 90% or higher because he wants to be reasonably certain of hisconclusions.

There is another probability called alpha ( α ). α is related to the confidence level CL. α is the probability that the sample produced a point estimate that is not within the appropriate margin of error of the unknown population parameter.

  • Suppose we have collected data from a sample. We know the sample average but we do not know the average for the entire population.
  • The sample mean is 7 and the error bound for the mean is 2.5.

x = 7 and EBM = 2.5.

The confidence interval is ( 7 - 2.5 , 7 + 2.5 ) ; calculating the values gives ( 4.5 , 9.5 ) .

If the confidence level (CL) is 95%, then we say that "We estimate with 95% confidence that the true value of the population mean is between 4.5 and 9.5."

A confidence interval for a population mean with a known standard deviation is based on the fact that the sample means follow an approximately normaldistribution. Suppose that our sample has a mean of x = 10 and we have constructed the 90% confidence interval (5, 15)where EBM = 5 .

To get a 90% confidence interval, we must include the central 90% of the probability of the normal distribution. If we include the central 90%, we leave out a total of 10% in both tails, or 5% in each tail, of the normal distribution.

Normal distribution curve with values of 5 and 15 on the x-axis. Vertical upward lines from points 5 and 15 extend to the curve. The confidence interval area between these two points is equal to 0.90.

To capture the central 90%, we must go out 1.645 "standard deviations" on either side of the calculated sample mean. 1.645 is the z-score from a Standard Normalprobability distribution that puts an area of 0.90 in the center, an area of 0.05 in the far left tail, and an area of 0.05 in the far right tail.

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Source:  OpenStax, Collaborative statistics: custom version modified by r. bloom. OpenStax CNX. Nov 15, 2010 Download for free at http://legacy.cnx.org/content/col10617/1.4
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