Question
The table contains data on the cost of attending a play at 10 different theaters in the United States. Find the interquartile range for this data set. Explain what the interquartile range represents for this data set.
Cost of Attending a Play
Theater
Q
R
S
T
U
V
W
X
Y
Z
Cost ($)
60
63
43
45
30
51
62
40
38
41
Question content area bottom
Part 1
The interquartile range is
enter your response here.
Cost of Attending a Play
Theater
Q
R
S
T
U
V
W
X
Y
Z
Cost ($)
60
63
43
45
30
51
62
40
38
41
Question content area bottom
Part 1
The interquartile range is
enter your response here.
Answers
GPT-4o mini
To find the interquartile range (IQR) for the given data set, you need to follow these steps:
1. **Organize the Data**: Sort the costs in ascending order:
- 30, 38, 40, 41, 43, 45, 51, 60, 62, 63
2. **Determine the Quartiles**:
- **Q1 (First Quartile)**: This is the median of the first half of the data (values below the median).
- **Q2 (Second Quartile/Median)**: This is the median of the entire data set.
- **Q3 (Third Quartile)**: This is the median of the second half of the data (values above the median).
3. **Calculate the Median (Q2)**:
- There are 10 data points, which means the median will be the average of the 5th and 6th values.
- Median (Q2) = (43 + 45) / 2 = 44
4. **Calculate Q1**:
- The first half of the data is: 30, 38, 40, 41, 43.
- Q1 (median of the first half) = 40
5. **Calculate Q3**:
- The second half of the data is: 45, 51, 60, 62, 63.
- Q3 (median of the second half) = 60
6. **Calculate the Interquartile Range (IQR)**:
- IQR = Q3 - Q1
- IQR = 60 - 40 = 20
### Conclusion
The **interquartile range** is **20**.
### Explanation of the IQR:
The interquartile range represents the range of the middle 50% of the data. In this context, it indicates the spread of the costs of attending a play at the theaters, meaning that half of the theaters charge between $40 and $60 for attending a play. The IQR provides insight into the variability of the data set without being influenced by outliers or extreme values.
1. **Organize the Data**: Sort the costs in ascending order:
- 30, 38, 40, 41, 43, 45, 51, 60, 62, 63
2. **Determine the Quartiles**:
- **Q1 (First Quartile)**: This is the median of the first half of the data (values below the median).
- **Q2 (Second Quartile/Median)**: This is the median of the entire data set.
- **Q3 (Third Quartile)**: This is the median of the second half of the data (values above the median).
3. **Calculate the Median (Q2)**:
- There are 10 data points, which means the median will be the average of the 5th and 6th values.
- Median (Q2) = (43 + 45) / 2 = 44
4. **Calculate Q1**:
- The first half of the data is: 30, 38, 40, 41, 43.
- Q1 (median of the first half) = 40
5. **Calculate Q3**:
- The second half of the data is: 45, 51, 60, 62, 63.
- Q3 (median of the second half) = 60
6. **Calculate the Interquartile Range (IQR)**:
- IQR = Q3 - Q1
- IQR = 60 - 40 = 20
### Conclusion
The **interquartile range** is **20**.
### Explanation of the IQR:
The interquartile range represents the range of the middle 50% of the data. In this context, it indicates the spread of the costs of attending a play at the theaters, meaning that half of the theaters charge between $40 and $60 for attending a play. The IQR provides insight into the variability of the data set without being influenced by outliers or extreme values.