Chapter 3, Problem 1.1SR

The annual incomes of a sample of middle-management employees at Westinghouse are $62,900, $69,100, $58,300, and $76,800.

1. (a) Give the formula for the sample mean.
2. (b) Find the sample mean.
3. (c) Is the mean you computed in (b) a statistic or a parameter? Why?
4. (d) What is your best estimate of the population mean?

To determine

Mention the formula for the sample mean.

Answer to Problem 1.1SR

The formula for sample mean is $\overline{x}=\frac{{\displaystyle \sum x}}{n}$.

Explanation of Solution

Sample mean:

The sample mean is the sum of all data values in the sample, divided by total number of data values from the sample. The formula for sample mean is,

$\text{Sample mean}=\frac{\text{Sum of all the values in the sample}}{\text{Number of values in the sample}}$

In notations the formula for sample mean is,

$\overline{x}=\frac{{\displaystyle \sum x}}{n}$

In the formula, $\overline{x}$ denotes the sample mean, ${\displaystyle \sum x}$ denotes sum of all sample values and n is the size of sample.

To determine

Find the sample mean for a sample of annual incomes of middle-management employees at Westinghouse.

Answer to Problem 1.1SR

The sample mean for a sample of annual incomes of middle-management employees at Westinghouse is $66,775.

Explanation of Solution

Calculation:

The number of data values in the sample is 4.

Add all values of sample for ‘${\displaystyle \sum x}$’ and substitute 4 for ‘n’ in sample mean formula.

$\begin{array}{c}\overline{x}=\frac{\$62,900+\$69,100+\$58,300+\$76,800}{4}\\ =\frac{\$267,100}{4}\\ =\$66,775\end{array}$

Hence, the sample mean for a sample of annual incomes of middle-management employees at Westinghouse is $66,775.

To determine

Identify whether the mean is a statistic or parameter.

Answer to Problem 1.1SR

The mean calculated is a statistic as it is calculated for the sample values.

Explanation of Solution

Parameter:

The characteristic that is measurable for a population in the study is denoted as parameter.

Statistic:

The characteristic that is measurable for a sample in the study is denoted as statistic.

The mean is calculated for a sample of annual incomes middle-management employees at Westinghouse. Since mean is calculated for the sample data values it would represent the measurable characteristic for the sample. This shows that, the mean calculated is a statistic.

Hence, the mean calculated is a statistic as it is calculated for the sample values.

To determine

Find the best estimate of the population mean.

Answer to Problem 1.1SR

The best estimate of the population mean is sample mean value $66,775.

Explanation of Solution

The best estimate for any population is its sample. Thus, the best estimate for the population mean would be its sample mean.

The sample mean for annual incomes of middle-management employees at Westinghouse is $66,775. The value $66,775 would be the best estimate for population mean.

Hence, the best estimate of the population mean is sample mean value $66,775.