Tuesday, June 2, 2020

06/02/2020 (2019: AMC 8, Problem 10)

Q: The diagram shows the number of students at soccer practice each weekday during last week. After computing the mean and median values, Coach discovers that there were actually 21 participants on Wednesday. Which of the following statements describes the change in the mean and median after the correction is made?

[asy] unitsize(2mm); defaultpen(fontsize(8bp)); real d = 5; real t = 0.7; real r; int[] num = {20,26,16,22,16}; string[] days = {"Monday","Tuesday","Wednesday","Thursday","Friday"}; for (int i=0; i<30; i=i+2) { draw((i,0)--(i,-5*d),gray); }for (int i=0; i<5; ++i) { r = -1*(i+0.5)*d; fill((0,r-t)--(0,r+t)--(num[i],r+t)--(num[i],r-t)--cycle,gray); label(days[i],(-1,r),W); }for(int i=0; i<32; i=i+4) { label(string(i),(i,1)); }label("Number of students at soccer practice",(14,3.5)); [/asy]

$\textbf{(A) }$ The mean increases by $1$ and the median does not change.

$\textbf{(B) }$ The mean increases by $1$ and the median increases by $1$.

$\textbf{(C) }$ The mean increases by $1$ and the median increases by $5$.

$\textbf{(D) }$ The mean increases by $5$ and the median increases by $1$.

$\textbf{(E) }$ The mean increases by $5$ and the median increases by $5$.

If we are going by the numbers in the table, then the data set would be {20, 26, 16, 22, 16}. That means the median is 20 (because it's the middle number) and the mean is (20 + 26 + 16 + 22 + 16) ÷ 5 = 20.
If there are 21 people on Wednesday, then the data set would be {20, 26, 21, 22, 16}. That means the median is 21 (because it's the middle number) and its mean is (20 + 26 + 21 + 22 + 16) ÷ 5  = 21.
The median went from 20 to 21 (so it increased by 1) and the mean went from 20 to 21 (so it also increased by 1). That means B is the answer.
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