To understand how the change affects Nathan's box plot, we need to calculate the quartiles and the median for both the original and the corrected data set.
The original data set:
26, 22, 21, 42, 21, 36, 20, 26
The corrected data set (replacing the erroneous 42 with 24):
26, 22, 21, 24, 21, 36, 20, 26
Step 1: Order both sets from smallest to largest:
Original: 20, 21, 21, 22, 26, 26, 36, 42
Corrected: 20, 21, 21, 22, 24, 26, 26, 36
Step 2: Find the median.
For both sets, there are 8 numbers, so the median will be the average of the 4th and 5th numbers.
Original: (22 + 26)/2 = 24
Corrected: (22 + 24)/2 = 23
The median has changed, so the statement "The median remained the same" is false.
Step 3: Find the quartiles.
The lower quartile (Q1) is the median of the lower half of the data, excluding the median if it's part of the data. The upper quartile (Q3) is the median of the upper half of the data, again excluding the median if it's part of the data.
For the original data:
Q1 (lower quartile) is the median of 20, 21, 21, 22 which is (21 + 21)/2 = 21
Q3 (upper quartile) is the median of 26, 26, 36, 42 which is (26 + 36)/2 = 31
For the corrected data:
Q1 (lower quartile) is now the median of 20, 21, 21, 22 which is the same (21 + 21)/2 = 21
Q3 (upper quartile) is the median of 24, 26, 26, 36 which is (26 + 26)/2 = 26
The lower quartile (Q1) remained the same, so "The lower quartile decreased by 1" is false. However, the upper quartile (Q3) did decrease by 5, from 31 to 26, so "The upper quartile decreased by 5" is true.
Step 4: Find the maximum value.
Original: The maximum value is 42.
Corrected: The maximum value is 36.
The maximum decreased by 42 - 36 = 6, not by 18, so "The maximum decreased by 18" is false.
Step 5: Determine if the value 24 was used to calculate the upper quartile in the corrected data set.
From our calculations for the corrected data set, we see that the value 24 was indeed below the calculated upper quartile value of 26, so "The value 24 was used to calculate the upper quartile" is true since it fell within the upper half of the data set that we used to calculate Q3.
In summary, the changes to the box plot are as follows:
- The median did not remain the same. (False)
- The upper quartile decreased by 5. (True)
- The maximum decreased by 6, not 18. (False)
- The lower quartile did not decrease by 1; it remained the same. (False)
- The value 24 was used to calculate the upper quartile. (True)
