Question

use the accompanying radiation levels \\(\left(\frac{\text{w}}{\text{kg}}\
ight)\\) for 50 different cell phones. find the percentile \\(p_{50}\\). \\(p_{50} = \square \frac{\text{w}}{\text{kg}}\\) (type an integer or a decimal. do not round) ... data values (e.g., 0.24, 0.28, 0.31, 0.47, 0.55, 0.58, 0.63, 0.66, 0.72, 0.87, 0.90, 0.90, 0.92, 0.92, 0.93, 0.97, 0.99, 1.00, 1.10, 1.11, 1.13, 1.14, 1.15, 1.16, 1.17, 1.18, 1.19, 1.20, 1.21, 1.22, 1.23, 1.26, 1.28, 1.29, 1.32, 1.32, 1.32, 1.32, 1.37, 1.37, 1.38, 1.44, 1.44, 1.45, 1.47, 1.48, 1.48, 1.49, 1.52) as shown ...

Explanation:

Step1: Order the data

First, we need to order the given radiation levels (in \(\frac{W}{kg}\)) from smallest to largest. Let's list out all the data points:

0.24, 0.28, 0.31, 0.47, 0.55, 0.58, 0.63, 0.66, 0.72, 0.87,

0.90, 0.90, 0.92, 0.92, 0.93, 0.97, 0.99, 1.00, 1.10, 1.11,

1.13, 1.14, 1.15, 1.16, 1.17, 1.18, 1.19, 1.20, 1.21, 1.22,

1.23, 1.26, 1.28, 1.29, 1.32, 1.32, 1.32, 1.32, 1.37, 1.37,

1.38, 1.44, 1.44, 1.45, 1.47, 1.48, 1.48, 1.49, 1.52

Step2: Find the position of \(P_{50}\)

The formula for the position of the \(n\)-th percentile (\(P_n\)) is \(i=\frac{n}{100}\times N\), where \(N\) is the number of data points. For \(P_{50}\) (the median), \(n = 50\) and \(N=50\). So, \(i=\frac{50}{100}\times50 = 25\). Since \(i\) is an integer, the \(50\)-th percentile is the average of the \(i\)-th and \((i + 1)\)-th values in the ordered data. Wait, actually, when \(N\) is even, the median (which is \(P_{50}\)) is the average of the \(\frac{N}{2}\)-th and \((\frac{N}{2}+ 1)\)-th values. Here, \(N = 50\), so \(\frac{N}{2}=25\) and \(\frac{N}{2}+ 1=26\).

Step3: Identify the 25th and 26th values

Looking at the ordered data:

- The first 10 values: 0.24, 0.28, 0.31, 0.47, 0.55, 0.58, 0.63, 0.66, 0.72, 0.87 (10 values)
- Next 10 values: 0.90, 0.90, 0.92, 0.92, 0.93, 0.97, 0.99, 1.00, 1.10, 1.11 (20 values total)
- Next 10 values: 1.13, 1.14, 1.15, 1.16, 1.17, 1.18, 1.19, 1.20, 1.21, 1.22 (30 values total)
- Next 10 values: 1.23, 1.26, 1.28, 1.29, 1.32, 1.32, 1.32, 1.32, 1.37, 1.37 (40 values total)
- Next 10 values: 1.38, 1.44, 1.44, 1.45, 1.47, 1.48, 1.48, 1.49, 1.52 (Wait, no, the last group should be 10 values. Wait, let's count again. Wait, the data is 50 points. Let's list the indices:

Index 1: 0.24

Index 2: 0.28

Index 3: 0.31

Index 4: 0.47

Index 5: 0.55

Index 6: 0.58

Index 7: 0.63

Index 8: 0.66

Index 9: 0.72

Index 10: 0.87

Index 11: 0.90

Index 12: 0.90

Index 13: 0.92

Index 14: 0.92

Index 15: 0.93

Index 16: 0.97

Index 17: 0.99

Index 18: 1.00

Index 19: 1.10

Index 20: 1.11

Index 21: 1.13

Index 22: 1.14

Index 23: 1.15

Index 24: 1.16

Index 25: 1.17

Index 26: 1.18

Ah, there we go. So the 25th value is 1.17 and the 26th value is 1.18.

Step4: Calculate the median (\(P_{50}\))

Since \(P_{50}\) is the median, and for even \(N\), it's the average of the two middle values. So, \(P_{50}=\frac{1.17 + 1.18}{2}=\frac{2.35}{2}=1.175\)

Wait, wait, no. Wait, let's check the data again. Wait, maybe I made a mistake in counting. Let's list all 50 data points properly:

First column (10 points): 0.24, 0.28, 0.31, 0.47, 0.55, 0.58, 0.63, 0.66, 0.72, 0.87

Second column (10 points): 0.90, 0.90, 0.92, 0.92, 0.93, 0.97, 0.99, 1.00, 1.10, 1.11

Third column (10 points): 1.13, 1.14, 1.15, 1.16, 1.17, 1.18, 1.19, 1.20, 1.21, 1.22

Fourth column (10 points): 1.23, 1.26, 1.28, 1.29, 1.32, 1.32, 1.32, 1.32, 1.37, 1.37

Fifth column (10 points): 1.38, 1.44, 1.44, 1.45, 1.47, 1.48, 1.48, 1.49, 1.52? Wait, no, fifth column should have 10 points. Wait, the last row in the image: 1.37, 1.22, 1.11, 0.87? Wait, maybe the data is presented in rows. Let's re - examine the image:

Looking at the numbers:

First set (top to bottom, left column?): 0.24, 0.28, 0.31, 0.47, 0.55, 0.58, 0.63, 0.66, 0.72, 0.87 (10)

Second set: 0.90, 0.90, 0.92, 0.92, 0.93, 0.97, 0.99, 1.00, 1.10, 1.11 (20)

Third set: 1.13, 1.14, 1.15, 1.16, 1.17, 1.18, 1.19, 1.20, 1.21, 1.22 (30)

Fourth set: 1.23, 1.26, 1.28, 1.29, 1.32, 1.32, 1.32, 1.32, 1.37, 1.37 (40)

Fifth set: 1.38, 1.44, 1.44, 1.45, 1.47, 1.48, 1.48, 1.49, 1.52? Wait, no, the fifth set should have 10 points. Wait, maybe the last row has 1.37, 1.22, 1.11, 0.87? No, the image shows:

Looking at the right - most column (maybe first column) down: 0.24, 0.28, 0.31, 0.47, 0.55, 0.58, 0.63, 0.66, 0.72, 0.87

Then next column: 0.90, 0.90, 0.92, 0.92, 0.93, 0.97, 0.99, 1.00, 1.10, 1.11

Then next: 1.13, 1.14, 1.15, 1.16, 1.17, 1.18, 1.19, 1.20, 1.21, 1.22

Then next: 1.23, 1.26, 1.28, 1.29, 1.32, 1.32, 1.32, 1.32, 1.37, 1.37

Then next: 1.38, 1.44, 1.44, 1.45, 1.47, 1.48, 1.48, 1.49, 1.52? Wait, that's 9 points. Oh, maybe I missed one. Wait, the last row in the image (bottom) has 1.37, 1.22, 1.11, 0.87? No, the numbers are:

Wait, the vertical list:

First column (10): 0.24, 0.28, 0.31, 0.47, 0.55, 0.58, 0.63, 0.66, 0.72, 0.87

Second column (10): 0.90, 0.90, 0.92, 0.92, 0.93, 0.97, 0.99, 1.00, 1.10, 1.11

Third column (10): 1.13, 1.14, 1.15, 1.16, 1.17, 1.18, 1.19, 1.20, 1.21, 1.22

Fourth column (10): 1.23, 1.26, 1.28, 1.29, 1.32, 1.32, 1.32, 1.32, 1.37, 1.37

Fifth column (10): 1.38, 1.44, 1.44, 1.45, 1.47, 1.48, 1.48, 1.49, 1.52? Wait, no, maybe the last number in the fifth column is missing, but according to the problem, there are 50 data points. So when \(N = 50\), the median ( \(P_{50}\)) is the average of the 25th and 26th values.

From the ordered list:

The 25th value is the 5th value in the third column: 1.17

The 26th value is the 6th value in the third column: 1.18

So, \(P_{50}=\frac{1.17 + 1.18}{2}=\frac{2.35}{2}=1.175\)

Wait, but let's check again. Wait, maybe the data is ordered correctly. Let's count the number of data points:

First column: 10

Second: 10 (total 20)

Third: 10 (total 30)

Fourth: 10 (total 40)

Fifth: 10 (total 50)

Yes, 5 columns of 10 each. So the 25th value is in the third column, 5th position (since 20 + 5=25), which is 1.17, and the 26th value is 20+6 = 26, which is 1.18.

So the average of 1.17 and 1.18 is \(\frac{1.17+1.18}{2}=1.175\)

Answer:

\(1.175\)