91Ó°ÊÓ

Watch out for these little buggers. Each of these exercises involves some feature that is somewhat tricky. Find the (a) mean, (b) median, (c) mode, \((d)\) midrange, and then answer the given question. Listed below are the measured radiation absorption rates (in W/kg) corresponding to these cell phones: iPhone \(5 \mathrm{S}\), BlackBerry \(\mathrm{Z} 30\), 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 (FCC). The media often report about the dangers of cell phone radiation as a cause of cancer. The FCC has a standard that a cell phone absorption rate must be \(1.6 \mathrm{W} / \mathrm{kg}\) or less. If you are planning to purchase a cell phone, are any of the measures of center the most important statistic? Is there another statistic that is most relevant? If so, which one? $$\begin{array}{cccccccccc}1.18 & 1.41 & 1.49 & 1.04 & 1.45 & 0.74 & 0.89 & 1.42 & 1.45 & 0.51 & 1.38\end{array}$$

Step by step solution

01

- List All Values

Write down all the radiation absorption rates: 1.18, 1.41, 1.49, 1.04, 1.45, 0.74, 0.89, 1.42, 1.45, 0.51, 1.38.

02

- Calculate the Mean

To find the mean, add all the values together and divide by the number of values.Sum = 1.18 + 1.41 + 1.49 + 1.04 + 1.45 + 0.74 + 0.89 + 1.42 + 1.45 + 0.51 + 1.38 = 13.96.Mean = Sum / Number of values = 13.96 / 11 = 1.269.

03

- Arrange Values in Ascending Order

Order the values from smallest to largest: 0.51, 0.74, 0.89, 1.04, 1.18, 1.38, 1.41, 1.42, 1.45, 1.45, 1.49.

04

- Find the Median

The median is the middle value in the ordered list. Since there are 11 values, the median is the 6th value.Median = 1.38.

05

- Identify the Mode

The mode is the value that appears most frequently.Mode = 1.45 (appears twice).

06

- Calculate the Midrange

The midrange is the average of the maximum and minimum values.Midrange = (Maximum + Minimum) / 2 = (1.49 + 0.51) / 2 = 1.00.

07

- Answer the Given Question

In deciding which cell phone to purchase considering the FCC standard, the most relevant statistic is likely the maximum value. This represents the worst-case scenario and ensures compliance with the safety standard as the most critical criterion. While measures of center like mean, median, and mode provide useful information, the maximum value directly relates to the safety limit derived by the FCC.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mean Calculation

The mean, or average, is a fundamental concept in statistics. It helps us understand the central tendency of a dataset. To calculate the mean absorption rate, follow these steps:

1. Sum all the values: 1.18, 1.41, 1.49, 1.04, 1.45, 0.74, 0.89, 1.42, 1.45, 0.51, 1.38.
2. The total sum is: 1.18 + 1.41 + 1.49 + 1.04 + 1.45 + 0.74 + 0.89 + 1.42 + 1.45 + 0.51 + 1.38 = 13.96
3. Divide the total sum by the number of values (11 in this case): \[ \text{Mean} = \frac{13.96}{11} = 1.269 \]

So, the mean absorption rate is approximately 1.269 W/kg. This gives us a single value representing the central point of the data, but it doesn't capture the variability in absorption rates.

Median Calculation

The median is the middle value in a list of numbers and is a good measure of the center of a dataset, especially when the data is skewed. To find the median:

1. Arrange all values in ascending order: 0.51, 0.74, 0.89, 1.04, 1.18, 1.38, 1.41, 1.42, 1.45, 1.45, 1.49.
2. Identify the middle value in the sorted list. Since there are 11 values, the median is the 6th value.
3. The 6th value is 1.38.

Hence, the median absorption rate is 1.38 W/kg. This value divides the dataset into two equal halves and is not affected by extreme values (outliers).

Mode Calculation

The mode is the value that appears most frequently in a dataset. This measure can be useful to see what value is most common:

1. List all values and count the frequency of each.
2. The frequencies are as follows:

• 1.45 appears twice
  
• All other values appear once
  

3. Since 1.45 appears more often than any other value, it is the mode.

Thus, the mode of the absorption rates is 1.45 W/kg. The mode is particularly useful in categorical data where we wish to know the most common category.

Midrange Calculation

The midrange gives us the average of the maximum and minimum values in a dataset. It provides a rough estimate of the central tendency. To calculate the midrange:

1. Identify the maximum (1.49) and minimum (0.51) values in the dataset.
2. Use the formula:

\[ \text{Midrange} = \frac{\text{Maximum Value} + \text{Minimum Value}}{2} \]

So, the midrange is:

\[ \text{Midrange} = \frac{1.49 + 0.51}{2} = 1.00 \]

The midrange of 1.00 W/kg is a simple measure but can be heavily influenced by outliers.

FCC Safety Standards

The FCC (Federal Communications Commission) sets safety standards for radiation absorption from cell phones, currently capped at 1.6 W/kg. It is crucial to consider these standards when choosing a cell phone:

1. The mean, median, and mode give central tendencies, but the maximum value indicates the worst-case exposure.
2. If any phone exceeds the FCC limit, it should be avoided to ensure safety.
3. It's generally better to evaluate the maximum value of radiation absorption rates, as it ensures no cell phone will surpass the safety limit of 1.6 W/kg.

While central tendency measures help understand general data trends, the maximum value in context of FCC standards is the most crucial statistic to consider for safety.