Question

use the accompanying radiation levels \\(\left(\text{in } \frac{\text{w}}{\text{kg}}\
ight)\\) for 50 different cell phones. find the percentile \\(p_{70}\\).
\\(p_{70} = \square \frac{\text{w}}{\text{kg}}\\) (type an integer or decimal rounded to two decimal places as needed.)
data points (partial) shown: 0.27, 0.29, 0.31, 0.52, 0.55, 0.56, 0.58, 0.69, 0.73, 0.87, 0.88, 0.89, 0.90, 0.93, 0.94, 0.98, 0.99, 1.03, 1.11, 1.11, 1.12, 1.12, 1.13, 1.14, 1.14, 1.14, 1.15, 1.17, 1.19, 1.19, 1.21, 1.21, 1.24, 1.24, 1.27, 1.28, 1.28, 1.32, 1.33, 1.33, 1.39, 1.39, 1.41, 1.45, 1.46, 1.49, 1.49, 1.50, 1.53, 1.60

Explanation:

Answer:

To find the 70th percentile (\(P_{70}\)) for the given data set of 50 values:

1. Order the data (already partially ordered, but confirm full sorting).
2. Calculate the index:

\[
i = \frac{70}{100} \times n = 0.7 \times 50 = 35
\]
Since \(i = 35\) is an integer, \(P_{70}\) is the average of the 35th and 36th values in the sorted list.

1. Sort the data (full sorted list, truncated for clarity; after sorting, the 35th and 36th values are \(1.28\) and \(1.28\) or similar? Wait, rechecking the data:

Wait, the data provided (rows):
Row 1: 0.27, 0.29, 0.31, 0.52, 0.55, 0.56, 0.58, 0.69, 0.73, 0.87
Row 2: 0.88, 0.89, 0.90, 0.93, 0.94, 0.98, 0.99, 1.03, 1.11, 1.11
Row 3: 1.12, 1.12, 1.13, 1.14, 1.14, 1.14, 1.15, 1.17, 1.19, 1.19
Row 4: 1.21, 1.21, 1.24, 1.24, 1.27, 1.28, 1.28, 1.32, 1.33, 1.33
Row 5: 1.39, 1.39, 1.41, 1.45, 1.46, 1.49, 1.49, 1.50, 1.53, 1.60

Wait, total 50 values. Let's count positions:

• Rows 1–4: 10 + 10 + 10 + 10 = 40 values.
• Row 5: 10 values (positions 41–50).

Wait, no—rows are 5 rows of 10? Wait, original data (as per image):

First column (10 values): 0.27, 0.29, 0.31, 0.52, 0.55, 0.56, 0.58, 0.69, 0.73, 0.87
Second column: 0.88, 0.89, 0.90, 0.93, 0.94, 0.98, 0.99, 1.03, 1.11, 1.11
Third column: 1.12, 1.12, 1.13, 1.14, 1.14, 1.14, 1.15, 1.17, 1.19, 1.19
Fourth column: 1.21, 1.21, 1.24, 1.24, 1.27, 1.28, 1.28, 1.32, 1.33, 1.33
Fifth column: 1.39, 1.39, 1.41, 1.45, 1.46, 1.49, 1.49, 1.50, 1.53, 1.60

Wait, no—each row has 10 values? Wait, the image shows 5 rows (vertical) with 10 values each? Wait, no, the data is 5 columns (horizontal) with 10 rows? Wait, the user’s image has:

First column (top to bottom): 0.27, 0.29, 0.31, 0.52, 0.55, 0.56, 0.58, 0.69, 0.73, 0.87 (10 values)
Second column: 0.88, 0.89, 0.90, 0.93, 0.94, 0.98, 0.99, 1.03, 1.11, 1.11 (10)
Third column: 1.12, 1.12, 1.13, 1.14, 1.14, 1.14, 1.15, 1.17, 1.19, 1.19 (10)
Fourth column: 1.21, 1.21, 1.24, 1.24, 1.27, 1.28, 1.28, 1.32, 1.33, 1.33 (10)
Fifth column: 1.39, 1.39, 1.41, 1.45, 1.46, 1.49, 1.49, 1.50, 1.53, 1.60 (10)

Total: 5 columns × 10 rows = 50 values. Now, sort all 50 values:

After sorting, the values are in order. Let's list them:

0.27, 0.29, 0.31, 0.52, 0.55, 0.56, 0.58, 0.69, 0.73, 0.87,
0.88, 0.89, 0.90, 0.93, 0.94, 0.98, 0.99, 1.03, 1.11, 1.11,
1.12, 1.12, 1.13, 1.14, 1.14, 1.14, 1.15, 1.17, 1.19, 1.19,
1.21, 1.21, 1.24, 1.24, 1.27, 1.28, 1.28, 1.32, 1.33, 1.33,
1.39, 1.39, 1.41, 1.45, 1.46, 1.49, 1.49, 1.50, 1.53, 1.60

Now, index \(i = 35\) (since \(n = 50\), \(i = 0.7 \times 50 = 35\)). The 35th value (1-based index) is in the 4th row (positions 31–40: 1.21, 1.21, 1.24, 1.24, 1.27, 1.28, 1.28, 1.32, 1.33, 1.33? Wait, no:

Wait, positions (1-based):

1: 0.27
2: 0.29
...
10: 0.87
11: 0.88
...
20: 1.11
21: 1.12
...
30: 1.19
31: 1.21
32: 1.21
33: 1.24
34: 1.24
35: 1.27
36: 1.28
37: 1.28
38: 1.32
39: 1.33
40: 1.33
41: 1.39
...
50: 1.60

Wait, no—wait, the 4th column (10 values) is: 1.21, 1.21, 1.24, 1.24, 1.27, 1.28, 1.28, 1.32, 1.33, 1.33 (positions 31–40: 31=1.21, 32=1.21, 33=1.24, 34=1.24, 35=1.27, 36=1.28, 37=1.28, 38=1.32, 39=1.33, 40=1.33).

Thus, the 35th value is \(1.27\), and the 36th value is \(1.28\). Wait, no—wait, index \(i = 35\) (1-based), so the 35th value is at position 35, and 36th at 36.

Wait, let's recount:

• Rows 1–3: 30 values (10×3).
• Row 4 (positions 31–40): 1.21 (31), 1.21 (32), 1.24 (33), 1.24 (34), 1.27 (35), 1.28 (36), 1.28 (37), 1.32 (38), 1.33 (39), 1.33 (40).

Ah, there we go. So position 35: \(1.27\), position 36: \(1.28\).

Since \(i = 35\) is an integer, \(P_{70} = \frac{\text{35th value} + \text{36th value}}{2} = \frac{1.27 + 1.28}{2} = \frac{2.55}{2} = 1.275\), rounded to two decimal places: \(1.28\). Wait, but wait—maybe I miscounted the data. Let's check the original data again.

Wait, the fourth column (row 4) has: 1.21, 1.21, 1.24, 1.24, 1.27, 1.28, 1.28, 1.32, 1.33, 1.33. So positions 31–40:

31: 1.21
32: 1.21
33: 1.24
34: 1.24
35: 1.27
36: 1.28
37: 1.28
38: 1.32
39: 1.33
40: 1.33

Yes. So 35th value: 1.27, 36th: 1.28. Average: \((1.27 + 1.28)/2 = 1.275 \approx 1.28\) (rounded to two decimal places).

Wait, but maybe the data is sorted differently. Let's confirm the full sorted list:

All values:

0.27, 0.29, 0.31, 0.52, 0.55, 0.56, 0.58, 0.69, 0.73, 0.87,
0.88, 0.89, 0.90, 0.93, 0.94, 0.98, 0.99, 1.03, 1.11, 1.11,
1.12, 1.12, 1.13, 1.14, 1.14, 1.14, 1.15, 1.17, 1.19, 1.19,
1.21, 1.21, 1.24, 1.24, 1.27, 1.28, 1.28, 1.32, 1.33, 1.33,
1.39, 1.39, 1.41, 1.45, 1.46, 1.49, 1.49, 1.50, 1.53, 1.60

Count the number of values: 10×5=50. Correct.

Now, the 70th percentile index: \(i = 0.7 \times 50 = 35\) (1-based). So we take the average of the 35th and 36th values.

35th value: index 35 (1-based) is 1.27 (as above).
36th value: index 36 is 1.28.

Thus, \(P_{70} = \frac{1.27 + 1.28}{2} = 1.275 \approx 1.28\) (rounded to two decimal places).

Wait, but maybe I made a mistake in the position. Let's list all 50 values with indices:

1: 0.27
2: 0.29
3: 0.31
4: 0.52
5: 0.55
6: 0.56
7: 0.58
8: 0.69
9: 0.73
10: 0.87
11: 0.88
12: 0.89
13: 0.90
14: 0.93
15: 0.94
16: 0.98
17: 0.99
18: 1.03
19: 1.11
20: 1.11
21: 1.12
22: 1.12
23: 1.13
24: 1.14
25: 1.14
26: 1.14
27: 1.15
28: 1.17
29: 1.19
30: 1.19
31: 1.21
32: 1.21
33: 1.24
34: 1.24
35: 1.27
36: 1.28
37: 1.28
38: 1.32
39: 1.33
40: 1.33
41: 1.39
42: 1.39
43: 1.41
44: 1.45
45: 1.46
46: 1.49
47: 1.49
48: 1.50
49: 1.53
50: 1.60

Yes, so index 35 is 1.27, index 36 is 1.28. Average: \( (1.27 + 1.28)/2 = 1.275 \), which rounds to 1.28 (two decimal places).

Thus, the 70th percentile \(P_{70}\) is \(\boldsymbol{1.28}\) (or 1.275, but rounded to two decimals is 1.28).