
Question content area top

Part 1

Listed below are the measured radiation rates 

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

corresponding to these cell​ phones: 

Motorola VE 465Motorola VE465​, 

OnePlus 5OnePlus 5​, 

Samsung Galaxy Upper S 8Samsung Galaxy S8​, 

Google PixelGoogle Pixel​, 

iPhone 7iPhone 7​, 

and 

iPhone Upper XiPhone X. 

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?

1.091.09

1.371.37

1.521.52

0.930.93

1.381.38

0.970.97
 
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.)

We are tasked with calculating the range, variance, and standard deviation for the given sample of cell phone radiation rates. Let's work through each step.

### Given Data:
Radiation rates for the following phones:
- 1.09 W/kg (Motorola VE 465)
- 1.37 W/kg (OnePlus 5)
- 1.52 W/kg (Samsung Galaxy S8)
- 0.93 W/kg (Google Pixel)
- 1.38 W/kg (iPhone 7)
- 0.97 W/kg (iPhone X)

### Step 1: Calculate the Range
The range is calculated as the difference between the highest and lowest values in the dataset.

- Maximum value = 1.52 W/kg
- Minimum value = 0.93 W/kg

**Range = Maximum - Minimum = 1.52 - 0.93 = 0.59 W/kg**

So, the **range** is **0.59 W/kg**.

### Step 2: Calculate the Variance
The formula for the sample variance is:
\[
s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1}
\]
where \(x_i\) represents each data point, \(\bar{x}\) is the sample mean, and \(n\) is the number of data points.

1. First, find the mean (\(\bar{x}\)):
\[
\bar{x} = \frac{1.09 + 1.37 + 1.52 + 0.93 + 1.38 + 0.97}{6} = \frac{7.26}{6} = 1.21 \, \text{W/kg}
\]

2. Now, calculate the squared deviations from the mean:
\[
\begin{align*}
(1.09 - 1.21)^2 &= (-0.12)^2 = 0.0144 \\
(1.37 - 1.21)^2 &= (0.16)^2 = 0.0256 \\
(1.52 - 1.21)^2 &= (0.31)^2 = 0.0961 \\
(0.93 - 1.21)^2 &= (-0.28)^2 = 0.0784 \\
(1.38 - 1.21)^2 &= (0.17)^2 = 0.0289 \\
(0.97 - 1.21)^2 &= (-0.24)^2 = 0.0576 \\
\end{align*}
\]

3. Sum these squared deviations:
\[
0.0144 + 0.0256 + 0.0961 + 0.0784 + 0.0289 + 0.0576 = 0.301
\]

4. Finally, divide by \(n - 1 = 6 - 1 = 5\):
\[
\text{Variance} (s^2) = \frac{0.301}{5} = 0.0602 \, \text{W}^2/\text{kg}^2
\]

So, the **variance** is **0.0602 W²/kg²**.

### Step 3: Calculate the Standard Deviation
The standard deviation is the square root of the variance:
\[
s = \sqrt{0.0602} \approx 0.2453 \, \text{W/kg}
\]

So, the **standard deviation** is approximately **0.245 W/kg**.

### Conclusion: Are the Statistics Helpful?
- **Range** tells us the span of radiation levels between the highest and lowest values, which gives a rough sense of the variation.
- **Variance** and **standard deviation** provide insight into how spread out the radiation levels are around the mean. A lower standard deviation indicates that the values are closer to the mean.

However, these statistics alone may not be very helpful in selecting a cell phone for purchase. Other factors such as price, features, and health guidelines on radiation levels should be considered alongside these statistics.

---

Do you have any questions or would you like further clarification?

Here are 5 related questions to deepen your understanding:
1. How would the range change if a new phone with 1.60 W/kg radiation was included?
2. Why is the sample variance divided by \(n - 1\) instead of \(n\)?
3. How could you interpret the standard deviation in terms of radiation safety?
4. If one of the radiation values was measured incorrectly, how would you detect the error using the statistics?
5. What other metrics (besides variance) could help in selecting a cell phone?

**Tip:** The range provides a quick snapshot of data spread but can be heavily affected by extreme values.

Listed below are the measured radiation rates (in W/kg) corresponding to these cell phones: Motorola VE465, OnePlus 5, Samsung Galaxy S8, Google Pixel, iPhone 7, and iPhone X. 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?

Data: 1.09, 1.37, 1.52, 0.93, 1.38, 0.97.

Learn how to calculate the range, variance, and standard deviation for radiation rates of popular cell phones like the Motorola VE465, OnePlus 5, Samsung Galaxy S8, Google Pixel, iPhone 7, and iPhone X. Includes step-by-step calculations with sample data from the FCC.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation