Question

1. the table shows the number of tickets of each price sold for a show.

a.) calculate the mean price.
the mean price =

b.) calculate the median price.
the median price =

c.) if each ticket price is increased by $2 and the number of tickets sold remains
the same, the mean will (increase/decrease).
if each ticket price is increased by $2 and the number of tickets sold remains
the same, the new median price will be

Explanation:

Step1: Calculate total price and total number of tickets

Total price: \(1\times12 + 2\times14+5\times9 + 10\times7+20\times8\)
\(=12+28 + 45+70+160\)
\(=315\)
Total number of tickets: \(12 + 14+9+7+8=50\)

Step2: Calculate the mean

Mean formula: \(\text{Mean}=\frac{\text{Total price}}{\text{Total number of tickets}}\)
\(\text{Mean}=\frac{315}{50}=6.3\)

Step3: Calculate the median

Arrange the data in ascending order. The total number of data points \(n = 50\) (even). The median is the average of the \(\frac{n}{2}\) -th and \((\frac{n}{2}+1)\) -th values.
Count the cumulative frequencies:
For \(\$1\): cumulative frequency \(f_1=12\)
For \(\$2\): cumulative frequency \(f_2=12 + 14=26\)
For \(\$5\): cumulative frequency \(f_3=26+9 = 35\)
The \(25\) -th and \(26\) -th values are in the \(\$2\) group. So the median is \(\$2\)

Step4: Analyze the effect of increasing each ticket price by \(\$2\)

Let the original prices be \(x_1,x_2,\cdots,x_n\) and the new prices be \(y_i=x_i + 2\)
The original mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}\)
The new mean \(\bar{y}=\frac{\sum_{i=1}^{n}(x_i + 2)}{n}=\frac{\sum_{i = 1}^{n}x_i+2n}{n}=\bar{x}+2\)
So the mean will increase.
The original median is \(M\). The new median \(M'=M + 2\). Since the original median \(M = 2\), the new median \(M'=4\)

Answer:

A. \(6.3\)
B. \(2\)
C. increase; \(4\)