Question

In Exercises 5–20, find the range, variance, and standard deviation for the given sample data. Include appropriate units (such as “minutes”) in your results. (The same data were used in Section 3-1, where we found measures of center. Herewe find measures of variation.) Then answer the given questions.

Cell Phone Radiation Listed below are the measured radiation absorption rates (in W/kg) corresponding to these cell phones: iPhone 5S, BlackBerry Z30, Sanyo Vero, Optimus V, Droid Razr, Nokia N97, Samsung Vibrant, Sony Z750a, Kyocera Kona, LG G2, and Virgin Mobile Supreme. The data are from the Federal Communications Commission. If one of each model of cell phone is measured for radiation and the results are used to find the measures of variation, are the results typical of the population of cell phones that are in use?

1.18 1.41 1.49 1.04 1.45 0.74 0.89 1.42 1.45 0.51 1.38

Short answer

The calculated measures of variation have the following values:

• The sample range is computed as 0.98 W/kg.
• The sample variance is computed as 0.114${\left(\text{W/kg}\right)}^2$.
• The sample standard is computed as 0.337 W/kg.

The measures of variation are not considered to be typical of the entire population of cell phones in use as some brands are more preferred in comparison to other brands.

Step-by-step solution

Given information

A sample of radiation absorption rates (in W/kg) of 11 brands of cell phones is given.

The sample size (n) is 11.

Computation of measures of variation

For good statistical analysis, it is essential to compute the measures of variation to quantify theamount of variability present in the data. Some popular measures are range, variance, and standard deviation.

The range is calculated as

$$\begin{gathered} \text{Rang}e=\text{Maximum}v\text{alue}-\text{Minimum}\text{value} \\ =1.49-0.51 \\ =0.98 \end{gathered}$$

Therefore, the sample range is equal to 0.98 W/kg.

The sample variance is calculated using the following formula:

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

Here,

x represents the sample values, and

$\overline{x}$is the sample mean.

The sample mean is calculated as

$$\begin{gathered} \overline{x}=\frac{{\sum }_{1=1}^n{x}_i}{n} \\ =\frac{1.18+1.41+...+1.38}{11} \\ =\frac{12.96}{11} \\ \approx 1.178 \end{gathered}$$

.

Thus, the sample mean is 1.178 W/kg.

The sample variance is calculated as

$$\begin{gathered} s^2=\frac{{\sum }_{i=1}^n{\left(x_i-\overline{x}\right)}^2}{n-1} \\ =\frac{{\left(1.18-1.178\right)}^2+{\left(1.41-1.178\right)}^2+...+{\left(1.38-1.178\right)}^2}{11-1} \\ =\frac{1.139}{10} \\ =0.114 \end{gathered}$$

.

Therefore, the sample variance is equal to 0.114${\left(\text{W/kg}\right)}^2$ .

The sample standard deviation is given as

$$\begin{gathered} s=\sqrt{s^2} \\ =\sqrt{0.114} \\ \approx 0.337 \end{gathered}$$

Therefore, the sample standard deviation is equal to 0.337 W/kg.

Interpretation of the results

The given sample comprises each of the 11 brands that are commonly available in the market. This sample cannot be considered representative of the entire population of cell phones as all brands are given equal weightage, but in general, some brands are more used by the population in comparison to others.

Therefore, the measures of variation are not typical of all the cell phones that are in use.