Question 1173181: Problem 2. Below is a list of vertical jumps (in cm) for 24 athletes who compete in various different sports.
28 45 55 46 70 72 62 32 75 68 74 54 50 62 63 31 55 68 73 55 60 69 68 92
Part A. Using # of classes = 4, create a frequency table for the given data set, and use your frequency table to construct a histogram.
Part B. Using the data set above, calculate the quartiles Qi where i = 0, 1, 2, 3, 4, and construct a box-and-whiskers plot for this data set.
Part C. Calculate the 15th percentile of this data set.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Absolutely! Let's break down each part of this data analysis problem.
**Part A: Frequency Table and Histogram**
1. **Determine the Range:**
* Maximum value: 92
* Minimum value: 28
* Range: 92 - 28 = 64
2. **Determine Class Width:**
* Number of classes: 4
* Approximate class width: 64 / 4 = 16
* To make it easier, let's round up to a class width of 17.
3. **Determine Class Limits:**
* Starting with the minimum value (28), create the class limits:
* Class 1: 28 - (28 + 16) 44
* Class 2: 45 - (45 + 16) 61
* Class 3: 62 - (62 + 16) 78
* Class 4: 79 - (79 + 16) 95
4. **Create the Frequency Table:**
| Class | Class Limits | Tally | Frequency |
| --------- | ------------ | ---------- | --------- |
| Class 1 | 28 - 44 | II | 2 |
| Class 2 | 45 - 61 | IIIIIIII | 8 |
| Class 3 | 62 - 78 | IIIIIIIIII | 10 |
| Class 4 | 79 - 95 | IIII | 4 |
5. **Construct the Histogram:**
* X-axis: Class Limits
* Y-axis: Frequency
* Draw bars with heights corresponding to the frequencies of each class.
Here's a text-based representation of what the histogram would look like:
```
Frequency
10 | **********
9 |
8 | ********
7 |
6 |
5 |
4 | ****
3 |
2 | **
1 |
---------------------------------
28-44 45-61 62-78 79-95 Class Limits
```
**Part B: Quartiles and Box-and-Whiskers Plot**
1. **Order the Data:**
* 28, 31, 32, 45, 46, 50, 54, 55, 55, 55, 60, 62, 62, 63, 68, 68, 68, 69, 70, 72, 73, 74, 75, 92
2. **Calculate Quartiles:**
* n = 24 (number of data points)
* Q1 (25th percentile): (n + 1) * 0.25 = 25 * 0.25 = 6.25, so Q1 is between the 6th and 7th values: (50 + 54) / 2 = 52
* Q2 (50th percentile, median): (n + 1) * 0.5 = 25 * 0.5 = 12.5, so Q2 is between the 12th and 13th values: (62 + 62) / 2 = 62
* Q3 (75th percentile): (n + 1) * 0.75 = 25 * 0.75 = 18.75, so Q3 is between the 18th and 19th values: (69 + 70) / 2 = 69.5
* Q0 (minimum): 28
* Q4 (maximum): 92
3. **Construct the Box-and-Whiskers Plot:**
* Draw a number line that covers the range of your data.
* Draw a box from Q1 (52) to Q3 (69.5).
* Draw a line inside the box at Q2 (62).
* Draw a line (whisker) from Q1 to the minimum value (28).
* Draw a line (whisker) from Q3 to the maximum value (92).
**Part C: 15th Percentile**
1. **Calculate the Position:**
* (n + 1) * 0.15 = 25 * 0.15 = 3.75
2. **Find the Value:**
* The 15th percentile is between the 3rd and 4th values.
* 3rd value: 32
* 4th value: 45
* Interpolate: 32 + 0.75 * (45 - 32) = 32 + 0.75 * 13 = 32 + 9.75 = 41.75
Therefore, the 15th percentile is approximately 41.75 cm.
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