Question 1173181
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.