Answer :
To find the five-number summary of a data set, we need to identify five key numbers: the minimum, first quartile (Q1), median, third quartile (Q3), and the maximum.
Let’s go through these steps for the data set: 22, 29, 33, 38, 44, 47, 51, 56, 64, 69.
1. Minimum: This is the smallest number in the data set.
- For our data set, the minimum value is 22.
2. Maximum: This is the largest number in the data set.
- For our data set, the maximum value is 69.
3. Median: This is the middle number when the data set is arranged in ascending order. For an even number of observations, it's the average of the two middle numbers.
- Our data set has 10 numbers, so the median will be the average of the 5th and 6th numbers: [tex]\( (44 + 47) / 2 = 45.5 \)[/tex].
- Thus, the median is 45.5.
4. First Quartile (Q1): This is the median of the first half of the data.
- The first half is 22, 29, 33, 38, 44. The median of these numbers is 33 (the third number in this subset).
- However, the calculation would yield a more precise value of 34.25 (this accounts for interpolation between exact quartile positions).
5. Third Quartile (Q3): This is the median of the second half of the data.
- The second half is 47, 51, 56, 64, 69. The median of these numbers is also (which precise calculation yields) 54.75 (accounting for interpolation).
Combining these, the five-number summary for this data set is:
- Minimum: 22
- Q1: 34.25
- Median: 45.5
- Q3: 54.75
- Maximum: 69
This doesn't match any of the given answer choices exactly. However, based on this detailed analysis, it confirms there may have been a misalignment in the expected result compared to the options.
Let’s go through these steps for the data set: 22, 29, 33, 38, 44, 47, 51, 56, 64, 69.
1. Minimum: This is the smallest number in the data set.
- For our data set, the minimum value is 22.
2. Maximum: This is the largest number in the data set.
- For our data set, the maximum value is 69.
3. Median: This is the middle number when the data set is arranged in ascending order. For an even number of observations, it's the average of the two middle numbers.
- Our data set has 10 numbers, so the median will be the average of the 5th and 6th numbers: [tex]\( (44 + 47) / 2 = 45.5 \)[/tex].
- Thus, the median is 45.5.
4. First Quartile (Q1): This is the median of the first half of the data.
- The first half is 22, 29, 33, 38, 44. The median of these numbers is 33 (the third number in this subset).
- However, the calculation would yield a more precise value of 34.25 (this accounts for interpolation between exact quartile positions).
5. Third Quartile (Q3): This is the median of the second half of the data.
- The second half is 47, 51, 56, 64, 69. The median of these numbers is also (which precise calculation yields) 54.75 (accounting for interpolation).
Combining these, the five-number summary for this data set is:
- Minimum: 22
- Q1: 34.25
- Median: 45.5
- Q3: 54.75
- Maximum: 69
This doesn't match any of the given answer choices exactly. However, based on this detailed analysis, it confirms there may have been a misalignment in the expected result compared to the options.
