Error in an estimate

When we use a summary statistic from a random sample to estimate a population parameter, the estimate will usually not be exactly the same as the parameter.

The difference between the estimate and the target parameter is called the error in the estimate.

For example, if the sample mean, , is used to estimate a population mean, µ, the error in this estimate is

error  =   - µ

If a sample proportion, p , is used to estimate a population proportion, π, the estimate's error is

error  =  p - π

Silkworm poisoning

Silkworms must be killed after spinning their cocoons since the silk is damaged when the moths break free. Heat is often used to kill the caterpillars, but researchers performed experiments to investigate the toxic action of arsenic. As part of their research, the survival time of fourth-instar silkworm larvae weighing between 0.41 and 0.45 grams was examined after they were given 0.10 mg of sodium arsenate per gram of body weight.

What is the mean survival time of the larvae with this dose of sodium arsenate?

In other words, we want to estimate the population mean of the survival times.

The experiment was conducted on 80 fourth-instar silkworm larvae and their survival times (seconds) were recorded. The stacked dot plot below shows the survival times.

We are not interested in the specific 80 silkworms, but want to understand the underlying 'population' distribution of survival times. Assuming that these silkworms are a random sample from this wider population, ...

... we estimate that the population mean survival time is 272.6 seconds.

However our estimate of 272.6 seconds is only based on a sample of 80 observations so there is likely to be an error in this estimate.

How accurate is the estimate?

In the example above, we used a sample statistic to estimate of an unknown population parameter.

How big is the error likely to be?

The remainder of this chapter introduces some methods to describe the accuracy of estimates (and equivalently, the likely size of the resulting errors).