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Answer :
To find the five-number summary for the given data set, we need to determine the minimum, first quartile (Q1), median, third quartile (Q3), and maximum. Here’s a step-by-step solution:
1. Organize the Data:
First, we need to arrange the data in ascending order:
[tex]\[
198, 199, 203, 205, 208, 208, 213, 214, 217, 222, 224, 228, 228, 232, 237
\][/tex]
2. Minimum:
The smallest number in the sorted data set is 198.
3. First Quartile (Q1):
Q1 is the median of the first half of the data. With 15 data points, we consider the first 7 data points for Q1:
[tex]\[
198, 199, 203, 205, 208, 208, 213
\][/tex]
Since there are 7 numbers, Q1 is the fourth value, which is 205.
4. Median (Q2):
The median is the middle number of the data set. Since there are 15 values, the median is the eighth value:
[tex]\[
214
\][/tex]
5. Third Quartile (Q3):
Q3 is the median of the second half of the data. For Q3, we consider the last 7 data points:
[tex]\[
214, 217, 222, 224, 228, 228, 232, 237
\][/tex]
The median of this subset (values beyond the eighth one) is the fourth value:
[tex]\[
228
\][/tex]
6. Maximum:
The largest number in the data set is 237.
Putting it all together, the five-number summary is:
- Minimum: 198
- Q1: 205
- Median: 214
- Q3: 228
- Maximum: 237
So, the correct five-number summary is:
Option D: 198, 205, 214, 228, 237
1. Organize the Data:
First, we need to arrange the data in ascending order:
[tex]\[
198, 199, 203, 205, 208, 208, 213, 214, 217, 222, 224, 228, 228, 232, 237
\][/tex]
2. Minimum:
The smallest number in the sorted data set is 198.
3. First Quartile (Q1):
Q1 is the median of the first half of the data. With 15 data points, we consider the first 7 data points for Q1:
[tex]\[
198, 199, 203, 205, 208, 208, 213
\][/tex]
Since there are 7 numbers, Q1 is the fourth value, which is 205.
4. Median (Q2):
The median is the middle number of the data set. Since there are 15 values, the median is the eighth value:
[tex]\[
214
\][/tex]
5. Third Quartile (Q3):
Q3 is the median of the second half of the data. For Q3, we consider the last 7 data points:
[tex]\[
214, 217, 222, 224, 228, 228, 232, 237
\][/tex]
The median of this subset (values beyond the eighth one) is the fourth value:
[tex]\[
228
\][/tex]
6. Maximum:
The largest number in the data set is 237.
Putting it all together, the five-number summary is:
- Minimum: 198
- Q1: 205
- Median: 214
- Q3: 228
- Maximum: 237
So, the correct five-number summary is:
Option D: 198, 205, 214, 228, 237
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