The table below shows the number of hours some business people in two states spend in meetings each week:
State A 21 23 24 22 24 25 23 23 22
State B 24 22 20 23 23 50 20 46 21
Part A: Create a five-number summary and calculate the interquartile range for the two sets of data. (6 points)
Part B: Are the box plots symmetric? Justify your answer. (4 points)
To answer Part A, we need to calculate the five-number summary and then find the interquartile range for the two sets of data: State A and State B.
The five-number summary consists of the minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum values.
For State A, the data is: 21, 23, 24, 22, 24, 25, 23, 23, 22
1. Minimum value: 21
2. Q1: The median of the lower half of the data set. Since there are 9 data points, the middle value is the 5th data point: 23.
3. Q2: The median of the entire data set. Since there are 9 data points, we take the average of the 4th and 5th data points: (22 + 24) / 2 = 23.
4. Q3: The median of the upper half of the data set. Again, the middle value is the 5th data point: 23.
5. Maximum value: 25
So the five-number summary for State A is: 21, 23, 23, 23, 25.
For State B, the data is: 24, 22, 20, 23, 23, 50, 20, 46, 21
1. Minimum value: 20
2. Q1: The middle value of the lower half, which is the 4th data point: 23.
3. Q2: The average of the 5th and 6th data points: (23 + 50) / 2 = 36.5
4. Q3: The middle value of the upper half, which is the 7th data point: 46.
5. Maximum value: 50
So the five-number summary for State B is: 20, 23, 36.5, 46, 50.
To calculate the interquartile range (IQR), we subtract Q1 from Q3 for each set of data:
For State A: IQR = Q3 - Q1 = 23 - 23 = 0.
For State B: IQR = Q3 - Q1 = 46 - 23 = 23.
Therefore, the interquartile range for State A is 0, and the interquartile range for State B is 23.
Moving on to Part B, to determine if the box plots are symmetric, we need to look at the shape and position of the box plots.
A box plot is symmetric if the median is in the middle of the box, and both halves of the box are roughly the same length.
For State A, the median is 23, which is not in the middle of the box. The lower half of the box (Q1 to the median) is shorter than the upper half (median to Q3), suggesting that the box plot is not symmetric.
For State B, the median is 36.5, which is closer to the middle of the box. Both halves of the box appear to be roughly the same length, indicating symmetry.
In summary, the box plot for State A is not symmetric, while the box plot for State B is symmetric.