To determine the 5-number summary of the given machine belt diameters and the skewness of the data, we'll consider each part step-by-step.
1. Minimum: This is the smallest number in the data set.
- Given diameters are: [tex]\(6, 4, 10, 2, 6, 5, 7, 3, 5, 7, 3\)[/tex].
- The minimum diameter is [tex]\(2\)[/tex] inches.
2. First Quartile ([tex]\(Q_1\)[/tex]): This is the median of the first half of the data (or 25th percentile).
- For this, we need the sorted data set: [tex]\(2, 3, 3, 4, 5, 5, 6, 6, 7, 7, 10\)[/tex].
- The first quartile ([tex]\(Q_1\)[/tex]) is [tex]\(3.5\)[/tex] inches.
3. Median: This is the middle number of the data set.
- The sorted data set is: [tex]\(2, 3, 3, 4, 5, 5, 6, 6, 7, 7, 10\)[/tex].
- Since we have 11 numbers (odd number), the median is the 6th number, which is [tex]\(5\)[/tex] inches.
4. Third Quartile ([tex]\(Q_3\)[/tex]): This is the median of the second half of the data (or 75th percentile).
- Considering the sorted data: [tex]\(2, 3, 3, 4, 5, 5, 6, 6, 7, 7, 10\)[/tex].
- The third quartile ([tex]\(Q_3\)[/tex]) is [tex]\(6.5\)[/tex] inches.
5. Maximum: This is the largest number in the data set.
- The maximum diameter is [tex]\(10\)[/tex] inches.
6. Skewness:
- To determine skewness, we look at where the median is positioned relative to the minimum and maximum.
- Given the data set's minimum, median, and maximum values, the skewness is determined as `'R'` (right-skewed).
So, putting this all together:
- Minimum [tex]\(= 2\)[/tex] inches
- [tex]\(Q_1 = 3.5\)[/tex] inches
- Median [tex]\(= 5\)[/tex] inches
- [tex]\(Q_3 = 6.5\)[/tex] inches
- Maximum [tex]\(= 10\)[/tex] inches
The measurements skew to the right, so the answer is [tex]\( R \)[/tex].
