Calculate a Confidence Interval for a Mean When You Know Sigma

In inferential statistics, one of the major goals is to estimate an unknown population parameter. You start with a statistical sample, and from this, you can determine a range of values for the parameter. This range of values is called a confidence interval.

Confidence Intervals

Confidence intervals are all similar to one another in a few ways. First, many two-sided confidence intervals have the same form:

Estimate ± Margin of Error

Second, the steps for calculating confidence intervals are very similar, regardless of the type of confidence interval you are trying to find. The specific type of confidence interval that will be examined below is a two-sided confidence interval for a population mean when you know the population standard deviation. Also, assume that you are working with a population that is normally distributed.

Confidence Interval for a Mean With a Known Sigma

Below is a process to find the desired confidence interval. Although all of the steps are important, the first one is particularly so:

1. Check conditions: Begin by ensuring that the conditions for your confidence interval have been met. Assume that you know the value of the population standard deviation, denoted by the Greek letter sigma σ. Also, assume a normal distribution.
2. Calculate estimate: Estimate the population parameter—in this case, the population mean—by use of a statistic, which in this problem is the sample mean. This involves forming a simple random sample from the population. Sometimes, you can suppose that your sample is a simple random sample, even if it does not meet the strict definition.
3. Critical value: Obtain the critical value z^* that corresponds with your confidence level. These values are found by consulting a table of z-scores or by using the software. You can use a z-score table because you know the value of the population standard deviation, and you assume that the population is normally distributed. Common critical values are 1.645 for a 90-percent confidence level, 1.960 for a 95-percent confidence level, and 2.576 for a 99-percent confidence level.
4. Margin of error: Calculate the margin of error z^* σ /√n, where n is the size of the simple random sample that you formed.
5. Conclude: Finish by putting together the estimate and margin of error. This can be expressed as either Estimate ± Margin of Error or as Estimate - Margin of Error to Estimate + Margin of Error. Be sure to clearly state the level of confidence that is attached to your confidence interval.

Example

To see how you can construct a confidence interval, work through an example. Suppose you know that the IQ scores of all incoming college freshman are normally distributed with standard deviation of 15. You have a simple random sample of 100 freshmen, and the mean IQ score for this sample is 120. Find a 90-percent confidence interval for the mean IQ score for the entire population of incoming college freshmen.

Work through the steps that were outlined above:

1. Check conditions: The conditions have been met since you have been told that the population standard deviation is 15 and that you are dealing with a normal distribution.
2. Calculate estimate: You have been told that you have a simple random sample of size 100. The mean IQ for this sample is 120, so this is your estimate.
3. Critical value: The critical value for confidence level of 90 percent is given by z^* = 1.645.
4. Margin of error: Use the margin of error formula and obtain an error of z^* σ /√n = (1.645)(15) /√(100) = 2.467.
5. Conclude: Conclude by putting everything together. A 90-percent confidence interval for the population’s mean IQ score is 120 ± 2.467. Alternatively, you could state this confidence interval as 117.5325 to 122.4675.

Practical Considerations

Confidence intervals of the above type are not very realistic. It is very rare to know the population standard deviation but not know the population mean. There are ways that this unrealistic assumption can be removed.

While you have assumed a normal distribution, this assumption does not need to hold. Nice samples, which exhibit no strong skewness or have any outliers, along with a large enough sample size, allow you to invoke the central limit theorem. As a result, you are justified in using a table of z-scores, even for populations that are not normally distributed.