Question

an orchestra of 25 members bought tickets, at all different prices, to a concert, and the mean price paid was $82. a new musician joined the orchestra and spent $90 on his ticket to the same concert. what is the new mean price paid when the new musician is included? round to the nearest cent.

Explanation:

Step1: Find the total amount paid by 25 members

The formula for the mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$. Given $n = 25$ and $\bar{x}=82$, we can find the sum $\sum_{i = 1}^{25}x_{i}$. Since $\bar{x}=\frac{\sum_{i = 1}^{25}x_{i}}{25}=82$, then $\sum_{i = 1}^{25}x_{i}=25\times82 = 2050$.

Step2: Find the total amount paid by 26 members

The new musician paid $90$. So the total amount paid by 26 members is $\sum_{i = 1}^{26}x_{i}=2050 + 90=2140$.

Step3: Calculate the new mean

The new mean for $n = 26$ members is $\bar{x}=\frac{\sum_{i = 1}^{26}x_{i}}{26}=\frac{2140}{26}\approx82.31$.

Answer:

$82.31$