Question
A random survey of seventh- and eighth-grade students was conducted to find out how many hours is spent doing homework each night. Select all the conclusions that may be made by comparing the interquartile ranges of each data set.
A double box plot showing Hours of Homework with seventh graders on the top and eighth graders on the bottom. For the seventh graders, the left end mark is at 2 point 5, the left edge of the box is at 3, the line inside the box is at 3 point 5, the right edge of the box is at 4, and the right end mark is at 4 point 5. For the eighth graders, the left end mark is at 3, the left edge of the box is at 3 point 5, the line inside the box is at 4, the right edge of the box is at 4 point 5, and the right end mark is at 5.
The interquartile range of both sets is equal to 1 hour.
The interquartile range for eighth-graders is greater than the interquartile range for seventh-graders.
The interquartile range for seventh-graders is greater than the interquartile range for eighth-graders.
The data sets exhibit similar variability, or spread.
The data for seventh-graders are more skewed than the data for eighth-graders.
A double box plot showing Hours of Homework with seventh graders on the top and eighth graders on the bottom. For the seventh graders, the left end mark is at 2 point 5, the left edge of the box is at 3, the line inside the box is at 3 point 5, the right edge of the box is at 4, and the right end mark is at 4 point 5. For the eighth graders, the left end mark is at 3, the left edge of the box is at 3 point 5, the line inside the box is at 4, the right edge of the box is at 4 point 5, and the right end mark is at 5.
The interquartile range of both sets is equal to 1 hour.
The interquartile range for eighth-graders is greater than the interquartile range for seventh-graders.
The interquartile range for seventh-graders is greater than the interquartile range for eighth-graders.
The data sets exhibit similar variability, or spread.
The data for seventh-graders are more skewed than the data for eighth-graders.
Answers
GPT-4o mini
To analyze the interquartile ranges (IQRs) of the given data sets for seventh and eighth graders, we first calculate the IQR for both groups:
### Seventh Graders:
- First Quartile (Q1): 3
- Third Quartile (Q3): 4
- IQR = Q3 - Q1 = 4 - 3 = **1 hour**
### Eighth Graders:
- First Quartile (Q1): 3.5
- Third Quartile (Q3): 4.5
- IQR = Q3 - Q1 = 4.5 - 3.5 = **1 hour**
### Conclusions:
1. **The interquartile range of both sets is equal to 1 hour.**
- **This conclusion is correct.**
2. **The interquartile range for eighth-graders is greater than the interquartile range for seventh-graders.**
- **This conclusion is incorrect.** (Both IQRs are equal.)
3. **The interquartile range for seventh-graders is greater than the interquartile range for eighth-graders.**
- **This conclusion is incorrect.** (Both IQRs are equal.)
4. **The data sets exhibit similar variability, or spread.**
- **This conclusion is correct.** (Since the IQRs are equal, their spread is similar.)
5. **The data for seventh-graders are more skewed than the data for eighth-graders.**
- **This conclusion cannot be determined solely from the IQR; additional analysis such as examining the shape of the data or the presence of outliers would be required to assess skewness.**
### Summary of Valid Conclusions:
- The interquartile range of both sets is equal to 1 hour.
- The data sets exhibit similar variability, or spread.
The conclusions that may be made are therefore the first and fourth statements.
### Seventh Graders:
- First Quartile (Q1): 3
- Third Quartile (Q3): 4
- IQR = Q3 - Q1 = 4 - 3 = **1 hour**
### Eighth Graders:
- First Quartile (Q1): 3.5
- Third Quartile (Q3): 4.5
- IQR = Q3 - Q1 = 4.5 - 3.5 = **1 hour**
### Conclusions:
1. **The interquartile range of both sets is equal to 1 hour.**
- **This conclusion is correct.**
2. **The interquartile range for eighth-graders is greater than the interquartile range for seventh-graders.**
- **This conclusion is incorrect.** (Both IQRs are equal.)
3. **The interquartile range for seventh-graders is greater than the interquartile range for eighth-graders.**
- **This conclusion is incorrect.** (Both IQRs are equal.)
4. **The data sets exhibit similar variability, or spread.**
- **This conclusion is correct.** (Since the IQRs are equal, their spread is similar.)
5. **The data for seventh-graders are more skewed than the data for eighth-graders.**
- **This conclusion cannot be determined solely from the IQR; additional analysis such as examining the shape of the data or the presence of outliers would be required to assess skewness.**
### Summary of Valid Conclusions:
- The interquartile range of both sets is equal to 1 hour.
- The data sets exhibit similar variability, or spread.
The conclusions that may be made are therefore the first and fourth statements.