In a skating competition, the marks of judges are randomly selected to count. Then the highest and lowest marks are dropped, and the skater receives the mean of the remaining marks.
There are some of the marks that 11 judges gave a skater for a free-skating program.
6.75, 6.25, 6.50, 6.00, 5.75, 6.50, 7.25, 7.00, 6.75, 6.50, 6.25
a) Remove four marks that would lower the score the most. Calculate the new mean and median.
b) Remove four marks that could raise the mean score the most. Calculate the new mean and median.
to lower the mean, remove the highest scores.
To raise the mean, remove the lowest scores.
Thank you
To remove the marks that would lower the score the most, we need to remove the highest and lowest marks, as well as two more marks that are closest to the extremes. Let's calculate the new mean and median after removing these marks.
a) Remove the marks that would lower the score the most:
Original marks: 6.75, 6.25, 6.50, 6.00, 5.75, 6.50, 7.25, 7.00, 6.75, 6.50, 6.25
After removing the highest and lowest marks, we have:
6.75, 6.25, 6.50, 6.00, 5.75, 6.50, 6.75, 6.50, 6.25
Now, let's remove two more marks that are closest to the extremes. In this case, we can remove one mark from the lower end and one mark from the higher end. The closest marks to the extremes are 5.75 and 7.25, so we will remove those.
After removing these two marks, we have:
6.75, 6.25, 6.50, 6.00, 6.50, 6.75, 6.50, 6.25
To calculate the new mean, we sum up all the remaining marks and divide by the number of remaining marks:
(6.75 + 6.25 + 6.50 + 6.00 + 6.50 + 6.75 + 6.50 + 6.25) / 8 = 52 / 8 = 6.50
To calculate the new median, we first arrange the remaining marks in ascending order:
6.00, 6.25, 6.25, 6.50, 6.50, 6.50, 6.75, 6.75
Since there are 8 remaining marks, the median is the average of the two middle values, which are 6.50 and 6.50:
(6.50 + 6.50) / 2 = 6.50
So, the new mean is 6.50 and the new median is also 6.50.
b) To remove the marks that could raise the mean score the most, we need to remove the highest and lowest marks, as well as two more marks closest to the mean. Let's calculate the new mean and median after removing these marks.
Original marks: 6.75, 6.25, 6.50, 6.00, 5.75, 6.50, 7.25, 7.00, 6.75, 6.50, 6.25
After removing the highest and lowest marks, we have:
6.75, 6.25, 6.50, 6.00, 5.75, 6.50, 7.00, 6.75, 6.50
Now, let's remove two more marks that are closest to the mean. In this case, we can remove one mark above the mean and one mark below the mean. The closest marks to the mean (which is approximately 6.52) are 6.50 and 6.50, so we will remove those.
After removing these two marks, we have:
6.75, 6.25, 6.00, 5.75, 7.00, 6.75
To calculate the new mean, we sum up all the remaining marks and divide by the number of remaining marks:
(6.75 + 6.25 + 6.00 + 5.75 + 7.00 + 6.75) / 6 = 38.5 / 6 ≈ 6.42
To calculate the new median, we first arrange the remaining marks in ascending order:
5.75, 6.00, 6.25, 6.75, 6.75, 7.00
Since there are 6 remaining marks, the median is the middle value, which is 6.75.
So, the new mean is approximately 6.42 and the new median is 6.75.