Chapter 8, Problem 1CR

Define (a) parameter, (b) estimator, (c) sampling error, and (d) sampling distribution.

To determine

Define parameter.

Explanation of Solution

Parameter:

Parameter is a measurement or characteristic of the population. It is usually unknown and represented by a lower case Greek letter (Example: $\mu ,\sigma \text{ or }\pi$).

For instance, for a normal distribution with mean $\mu$ and variance ${\sigma }^2$ the parameters are, $\mu$ and $\sigma$.

To determine

Define estimator.

Explanation of Solution

Estimator:

A statistic derived from a sample to obtain the value of the population parameter is called an estimator.

An estimate is the value of the estimator in a particular sample. Since the random samples vary, an estimator is a random variable.

The sample estimator is usually denoted by a Roman letter (Example: $\overline{x},s\text{ or }p$) or a Greek letter with a “hat” (Example: $\widehat{\mu },\widehat{\sigma }\text{ or }\widehat{\pi }$).

To determine

Define sampling error.

Explanation of Solution

Sampling Error:

The difference between an estimate and the corresponding population parameter is called the sampling error.

For example, for the population mean $\mu$, the estimate is $\overline{x}$. Then, the sampling error is,

$\text{Sampling Error = }\overline{x}-\mu$.

The reason for the sampling error is that the value of the estimate $\overline{x}$, varies from sample to sample.

To determine

Define sampling distribution.

Explanation of Solution

Sampling Distribution:

The probability distribution of all possible values that the statistic can assume for a sample of size n is called the sampling distribution of an estimator.

A sampling distribution is described using the mean, variance and shape of the distribution.