Calculate Range, Variance, and Standard Deviation of Cell Phone Radiation Levels

Math Problem Statement

Listed below are the measured radiation rates

left parenthesis in StartFraction Upper W Over kg EndFraction right parenthesisin Wkg

corresponding to these cell phones:

LG Upper G 3LG G3,

HTC Desire 310HTC Desire 310,

iPhone Upper XiPhone X,

iPhone 7iPhone 7,

Google PixelGoogle Pixel,

and

Samsung Galaxy Upper S 8Samsung Galaxy S8.

The data are from the Federal Communications Commission. Find the range, variance, and standard deviation for the given sample data. Are any of the resulting statistics helpful in selecting a cell phone for purchase?

0.480.48

0.650.65

0.970.97

1.381.38

0.930.93

1.521.52 Question content area bottom

Part 1

The range is

enter your response here

▼ StartFraction Upper W squared Over kg EndFraction .W2kg.

StartFraction Upper W Over kg EndFraction .Wkg.

StartRoot StartFraction Upper W Over kg EndFraction EndRoot .Wkg.

left parenthesis StartFraction Upper W Over kg EndFraction right parenthesis squared .Wkg2.

(Type an integer or decimal rounded to three decimal places as needed.)

Solution

Let's calculate the range, variance, and standard deviation of the given sample dataLet's calculate the range, variance, and standard deviation of the given sample

Step 1: Calculate the Range

The range is the difference between the highest and lowest values in the data set.

Given dataLet's calculate the range, variance, and standard deviation of the given sample

Step 1: Calculate the Range

The range is the difference between the highest and lowest values in the data set.

Given $0.48, 0.65, 0.97, 1.38, 0.93, 1.52$ .

Range:

$\text{Range} = \text{Maximum value} - \text{Minimum value} = 1.52 - 0.48 = 1.04$

Step 2: Calculate the Variance

Variance is a measure of how much the values in the data set deviate from the mean. The formula for the sample variance is:

$s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1}$

where:

$x_i$ represents each value in the data set,
$\bar{x}$ is the mean of the data,
$n$ is the number of values in the data set.

First, calculate the mean ( $\bar{x}$ ):

$\bar{x} = \frac{0.48 + 0.65 + 0.97 + 1.38 + 0.93 + 1.52}{6} = \frac{5.93}{6} = 0.9883$

Next, find each deviation from the mean, square it, and sum all these squared deviations:

$\sum (x_i - \bar{x})^2 = (0.48 - 0.9883)^2 + (0.65 - 0.9883)^2 + (0.97 - 0.9883)^2 + (1.38 - 0.9883)^2 + (0.93 - 0.9883)^2 + (1.52 - 0.9883)^2$

Calculating each term:

= (0.65 - 0.9883)^2 = 0.1145, \\ = (0.97 - 0.9883)^2 = 0.0003, \\ = (1.38 - 0.9883)^2 = 0.1529, \\ = (0.93 - 0.9883)^2 = 0.0035, \\ = (1.52 - 0.9883)^2 = 0.2832.$$ $$\sum (x_i - \bar{x})^2 = 0.2588 + 0.1145 + 0.0003 + 0.1529 + 0.0035 + 0.2832 = 0.8132$$ Now, calculate the variance: $$s^2 = \frac{0.8132}{6 - 1} = \frac{0.8132}{5} = 0.1626$$ ### Step 3: Calculate the Standard Deviation The standard deviation ($$s$$) is the square root of the variance: $$s = \sqrt{0.1626} = 0.4032$$ ### Summary of Results 1. **Range**: $$1.04 \, \text{W/kg}$$ 2. **Variance**: $$0.1626 \, \text{(W/kg)}^2$$ 3. **Standard Deviation**: $$0.4032 \, \text{W/kg}$$ ### Conclusion The statistics give you an understanding of the spread and consistency of radiation rates among the listed cell phones. A smaller variance and standard deviation would suggest less variability in radiation levels, but in the context of choosing a phone, these metrics might not be the sole deciding factors. It's important to consider the specific radiation limits set by health standards and other features of the phones as well. Do you need further explanation on any part, or have additional questions? Here are some follow-up questions for deeper exploration: 1. What other factors should be considered when choosing a phone besides radiation levels? 2. How does the standard deviation help in understanding the data set? 3. What is the acceptable range of radiation levels for cell phones according to health standards? 4. Would it be better to choose a phone with the lowest radiation level, or is the average more important? 5. Can high radiation levels affect phone performance or health? **Tip:** Always compare radiation rates against recommended safety thresholds to make an informed decision.

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Math Problem Analysis

Mathematical Concepts

Descriptive Statistics
Range
Variance
Standard Deviation

Formulas

Range = Max value - Min value
Variance (s^2) = Σ(x_i - x̄)^2 / (n - 1)
Standard Deviation (s) = √Variance

Theorems

Suitable Grade Level

Grades 9-12

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