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To solve the problem, we need to create a frequency distribution table from the data provided in the image, and then use it to plot the frequency polygon.

### Step 1: Organize the Data
We are given the following sales dataTo solve the problem, we need to create a frequency distribution table from the data provided in the image, and then use it to plot the frequency polygon.

### Step 1: Organize the Data
We are given the following sales 

```
42, 51, 55, 61, 65
45, 51, 57, 61, 66
47, 52, 58, 63, 68
47, 53, 58, 63, 68
48, 54, 58, 65, 70
49, 55, 61, 65, 70
```

### Step 2: Create the Frequency Distribution Table

**Steps to make the frequency distribution:**
1. Find the range (difference between the largest and smallest values).
2. Decide the number of classes (usually between 5 and 10).
3. Calculate the class width.
4. Create the class intervals.
5. Count the number of occurrences (frequency) in each class.

Let me calculate that.Here is the frequency distribution table based on the given dataHere is the frequency distribution table based on the given 

| Class Interval | Frequency |
|----------------|-----------|
| [42, 48)       | 4         |
| [48, 54)       | 6         |
| [54, 60)       | 7         |
| [60, 66)       | 8         |
| [66, 72)       | 5         |

### Step 3: Frequency Polygon
To create the frequency polygon, we plot the midpoints of each class interval against their respective frequencies. Let me calculate the midpoints and explain how you can plot it by hand.Here are the midpoints of the class intervals and their corresponding frequencies for constructing the frequency polygon:

| Midpoint | Frequency |
|----------|-----------|
| 45.0     | 4         |
| 51.0     | 6         |
| 57.0     | 7         |
| 63.0     | 8         |
| 69.0     | 5         |

### How to plot the frequency polygon:
1. On the horizontal axis (x-axis), mark the midpoints of each class interval: \( 45.0, 51.0, 57.0, 63.0, 69.0 \).
2. On the vertical axis (y-axis), plot the corresponding frequencies.
3. Join the points with straight lines.

You can add extra points at the start and end (with frequency zero) to close the polygon shape.

Would you like me to provide a detailed sketch of the frequency polygon?

### Follow-up Questions:
1. What would happen to the class width if you increase the number of classes?
2. How can you adjust the class intervals to avoid overlapping data values?
3. What other types of graphs can be drawn from a frequency distribution?
4. Can you describe how the range affects the shape of the frequency polygon?
5. How would the frequency distribution change if the data were spread across a larger range?

**Tip:** Always check the number of classes and class width to ensure that your frequency distribution provides meaningful insight.

Supóngase que durante un mes se llevó un registro diario de las ventas de corbatas realizadas por una casa comercial. Los datos recopilados son los siguientes: 42, 51, 55, 61, 65; 45, 51, 57, 61, 66; 47, 52, 58, 63, 68; 47, 53, 58, 63, 68; 48, 54, 58, 65, 70; 49, 55, 61, 65, 70. Elaborar la tabla de distribución de frecuencias, y a partir de ella graficar a mano el polígono de frecuencias y marcar a cuál de los tres siguientes corresponde.

Learn how to create a frequency distribution table using tie sales data and plot a frequency polygon. This problem covers key statistical concepts including class intervals, frequencies, and constructing polygons from a frequency table.

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