Q 3.63. - Elementary Statistics

Question

Age of U.S Residents. The U.S census Bureau publishes information about ages of people in the United States in Current Population Reports. A sample of five U.S residents has the following ages in year.

Part (a) Determine the range of the ages.

Part (b) Find the sample standard deviation of these ages by using the defining formula. Definition 3.6 on page 108.

Part (c) Find the sample standard deviation of these ages by using the computing formula. Formula 3.1 on page 110.

Part (d)Compare your work in part (b) and (c).

Step-by-Step Solution

Answer

Part (a) 45

Part (b) 19.8997

Part (c) 19.8997

Part (d) Parts (a) and (b) have the same answers.

Part (a) Step 1. Given information.

The data points are as follows:

21,54,9,45,51.

Part (a) Step 2. The data range provided.

A data set's range is given by range = max-min.

Range 

 $$\begin{gathered} =54-9 \\ =45 \end{gathered}$$

As a result, the data set's range is 45.

Part (b) Step 1. Given information.

The data points are as follows:

21,54,9,45,51.

Part (b) Step 2. The given data's sample standard deviation.

A data set's range is given by range = max-min.

The formula computes the standard deviation.

$s = \sqrt{\frac{\sum _{}{x_i}^2-{\left(\sum _{}{x}_i\right)}^2/n}{n-1}}$

Determine the mean of the given data set.

$$\begin{gathered} \overline{x}=\frac{21+54+45+9+51+}{5} \\ \overline{x}=\frac{180}{5} \\ \overline{x}=36 \end{gathered}$$

$$\begin{gathered} s=\sqrt{\frac{{\left(21-36\right)}^2+{\left(54-36\right)}^2+{\left(45-36\right)}^2+{\left(9-36\right)}^2+{\left(51-36\right)}^2}{5-1}} \\ s\approx 19.8997 \end{gathered}$$

As a result, the required value is 19.8997

Part (c) Step 1. Given information.

The data points are as follows:

21,54,9,45,51.

Part (c) Step 2. The given data's sample standard deviation.

A data set's range is given by range = max-min.

The formula computes the standard deviation.

$$\begin{gathered} s = \sqrt{\frac{\sum _{}{x_i}^2-{\left(\sum _{}{x}_i\right)}^2/n}{n-1}} \\ s=\sqrt{\frac{8064-{\left(180\right)}^2/5}{5-1}} \\ s\approx 19.8997 \end{gathered}$$

As a result, the required value is 19.8997

Part (d) Step 1. The comparison of answers from parts (a) and (b).

Because the formula does not matter when calculating the standard deviation, the answers in parts (a) and (b) are the same.