Question

the table below shows the annual salaries for the 24 members of a professional sports team in terms of millions of dollars.
0.5 0.5 0.6 0.7 0.75 0.8
1.0 1.0 1.1 1.25 1.3 1.4
1.4 1.8 2.5 3.7 3.8 4
4.2 4.6 5.1 6 6.3 7.2
the team signs an additional player to a contract worth 10 million dollars per year. which statement about the median and mean is true?
(1) both will increase.
(2) only the median will increase.
(3) only the mean will increase.
(4) neither will change.

Explanation:

Step1: Calculate the original mean

The sum of the original 24 - member salaries is \(0.5\times2 + 0.6+0.7+0.75 + 0.8+1.0\times2+1.1+1.25+1.3+1.4\times2+1.8+2.5+3.7+3.8+4+4.2+4.6+5.1+6+6.3+7.2\)
\[

$$\begin{align*} &(0.5\times2)+0.6 + 0.7+0.75+0.8+(1.0\times2)+1.1+1.25+1.3+(1.4\times2)+1.8+2.5+3.7+3.8+4+4.2+4.6+5.1+6+6.3+7.2\\ =&1+0.6 + 0.7+0.75+0.8 + 2+1.1+1.25+1.3+2.8+1.8+2.5+3.7+3.8+4+4.2+4.6+5.1+6+6.3+7.2\\ =&(1 + 2)+(0.6+0.7+0.75+0.8)+(1.1+1.25+1.3)+2.8+(1.8+2.5)+(3.7+3.8)+4+(4.2+4.6)+(5.1+6)+(6.3+7.2)\\ =&3+(2.85)+(3.65)+2.8+(4.3)+(7.5)+4+(8.8)+(11.1)+(13.5)\\ =&3+2.85+3.65+2.8+4.3+7.5+4+8.8+11.1+13.5\\ =&(3+2.85+3.65)+2.8+(4.3+7.5)+4+(8.8+11.1)+13.5\\ =&9.5+2.8 + 11.8+4+19.9+13.5\\ =&(9.5+2.8)+11.8+4+19.9+13.5\\ =&12.3+11.8+4+19.9+13.5\\ =&(12.3+11.8)+4+19.9+13.5\\ =&24.1+4+19.9+13.5\\ =&(24.1+4)+19.9+13.5\\ =&28.1+19.9+13.5\\ =&48+13.5\\ =&61.5 \end{align*}$$

\]
The original mean \(\bar{x}_1=\frac{61.5}{24}=2.5625\) million - dollars.

Step2: Calculate the new mean

The new sum of salaries is \(61.5 + 10=71.5\) million - dollars, and the new number of data - points is \(n = 25\). The new mean \(\bar{x}_2=\frac{71.5}{25}=2.86\) million - dollars. So the mean increases.

Step3: Calculate the original median

Since \(n = 24\) (an even number), the median is the average of the 12th and 13th ordered data - values. The 12th value is \(1.4\) and the 13th value is \(1.4\), so the original median \(M_1=\frac{1.4 + 1.4}{2}=1.4\) million - dollars.

Step4: Calculate the new median

Since \(n = 25\) (an odd number), the median is the 13th ordered data - value. When we add the value \(10\) and re - order the data, the 13th value is still \(1.4\) million - dollars. So the median does not change.

Answer:

(3) Only the mean will increase.