## Confidence Intervals for the Mean

When a sample is taken from a population, we can use sample statistics, such as  and s as point estimates for the population parameters, μ and σ.

A one-off sample such as this could be a very inaccurate estimator however and it is best to find a range or interval of values and say with some level of probability that the population parameters will lie within this range. This range of values is called a confidence interval for the mean.

### Degree of Confidence

The degree of confidence is given as a percentage probability changed into a z-score, using the standard normal distribution tables. Check out where the z-scores are found in the tables for the following degrees of confidence.

Any degree of confidence can be found in this manner. e.g. For an 80% level of confidence look up 0.40 in the tables.

### Confidence Interval for the Mean

The confidence interval, based on the Central Limit Theorem, for the mean μ is given by:

 ± z .

Remember that  is called the standard error of the mean.

The value of z will depend upon the degree of confidence required. The z values for 90, 95 and 99% are shown above.

If the population standard deviation (σ) is not known the sample standard deviation (s) can be used as an estimate.

With a 95% confidence interval, this means that if we know the population standard deviation, σ, we could take a random sample of size n, find its mean  and know that 95% of the times we did this, the interval  ± 1.96.  would contain the population mean μ

Example

A dairy factory produces blocks of cheese for export whose weights are known to be normally distributed with a standard deviation of 1.4 kg. A random sample of 25 blocks of cheese has a mean weight of 22.3 kg. Find 99% confidence intervals for the population mean.

The 99% confidence interval is:

± 2.576 .
= 22.3 ± 2.576 .
= 22.3 ± 0.721

= (21.579 , 23.021)

The 99% confidence interval for μ is (21.579 kg, 23.021 kg)

This is sometimes written 21.579 < μ < 23.021

### Meaning of Confidence Intervals

The answer in the example above means that 99% of such confidence intervals will contain the population mean.