Answer:
Let's break down this problem into three parts as you've indicated:
a) Construct a relative frequency histogram for these data and describe the distribution.
b) Construct a box plot for these data.
c) Based on the IQR, are there any outliers? If so, which data points? Clearly show your work to justify your answer.
**a) Construct a relative frequency histogram and describe the distribution:**
A relative frequency histogram shows the frequency of data points within specific intervals or bins. To construct one, we need to first decide on the number of bins and then count how many data points fall into each bin. We'll use a simple example with 5 bins for this dataset.
Here are the bins and frequencies:
- Bin 1: 0 - 500, Frequency = 13
- Bin 2: 501 - 1000, Frequency = 5
- Bin 3: 1001 - 1500, Frequency = 2
- Bin 4: 1501 - 2000, Frequency = 2
- Bin 5: 2001 - 2500, Frequency = 3
Now, we'll create a relative frequency histogram. The relative frequency is calculated by dividing the frequency of each bin by the total number of data points (25 in this case).
- Relative Frequency for Bin 1: 13/25 = 0.52
- Relative Frequency for Bin 2: 5/25 = 0.20
- Relative Frequency for Bin 3: 2/25 = 0.08
- Relative Frequency for Bin 4: 2/25 = 0.08
- Relative Frequency for Bin 5: 3/25 = 0.12
Now, let's create a histogram with these relative frequencies:
```
Frequency
|
0.6 | *
| * *
0.5 | * * * * *
| * * * * *
0.4 | * * * * *
| * * * * *
0.3 | * * * * *
| * * * * *
0.2 | * * * * *
| * * * * *
0.1 | * * * * *
+--------------------------
Bin 1 Bin 2 Bin 3 Bin 4 Bin 5
```
**Description of the distribution:**
The distribution of chair prices appears to be right-skewed, as most of the data points are concentrated on the lower end of the price range, with a few higher-priced outliers. This is evident from the histogram where the majority of chairs fall in the first bin (0-500) and the frequencies decrease as we move to higher price ranges.
**b) Construct a box plot for these data:**
To create a box plot, we need to find the following statistics: minimum, maximum, median (Q2), lower quartile (Q1), and upper quartile (Q3).
- Minimum: 19
- Maximum: 2765
- Median (Q2): Middle value when data is sorted, which is 310.
- Lower Quartile (Q1): Median of the lower half of the data, which is the median of [19, 25, 28, 55, 100, 125, 145, 250, 295, 345, 395, 545, 595], resulting in Q1 = 100.
- Upper Quartile (Q3): Median of the upper half of the data, which is the median of [995, 1099, 1266, 1675, 2200, 2275, 2500, 2765], resulting in Q3 = 1967.5.
Now, we can construct the box plot:
```
| +---------+---------+---------+---------+----------+----------+-----------+
| | | | | | | | |
| * +---------+---------+---------+---------+----------+----------+-----------+
| | | | | | | | |
| | +---* | * | | |
| | | | | | | | |
| | | | | | | | |
| | | | | | | | |
| | | | | | | | |
| * | | | | | | | |
| | | | | | | | |
| | | | | | | | |
| | | | | | | | |
| | | | | | | | |
| | | | | | | | |
| | | | | | | | |
| | | | | | | | |
| | | | | | | | |
| +---------+---------+---------+---------+----------+----------+-----------+
| | | | | | | | |
| Q1 | | Median (Q2) | Q3 |
| | Interquartile Range (IQR) |
```
**c) Based on the IQR, are there any outliers? If so, which data points?**
To determine if there are any outliers, we can use the IQR method. The IQR is the difference between the upper quartile (Q3) and the lower quartile (Q1):
IQR = Q3 - Q1 = 1967.5 - 100 = 1867.5
Now, we can define the lower and upper bounds for potential outliers:
- Lower Bound = Q1 - 1.5 * IQR = 100 - 1.5 * 1867.5 = -2666.25
- Upper Bound = Q3 + 1.5 * IQR = 1967.5 + 1.5 * 1867.5 = 4634.75
Any data points below the lower bound or above the upper bound can be considered outliers.
Looking at the data points, we have:
- 19 (below the lower bound)
- 2765 (above the upper bound)
So, there are two outliers in the dataset: 19 and 2765.
Explanation: