Question

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

price of ticket	$1	$2	$5	$10	$20
number of tickets sold	12	14	9	7	8

a. calculate the mean price.
b. if each ticket price is increased by $2, and the frequency distribution remains unchanged, describe the effect on the mean and find the median price of the tickets.

Explanation:

Step1: Calculate the total price

First, find the total price for each ticket - price category by multiplying the price by the number of tickets sold and then sum them up.
$1\times12 + 2\times14+5\times9 + 10\times7+20\times8=12 + 28+45 + 70+160 = 315$

Step2: Calculate the total number of tickets

Sum up the number of tickets sold in each category.
$12 + 14+9 + 7+8=50$

Step3: Calculate the mean price

The mean price $\bar{x}=\frac{\text{Total price}}{\text{Total number of tickets}}$.
$\bar{x}=\frac{315}{50}=6.3$

Step4: Analyze the effect of increasing ticket - price on the mean

If each ticket price is increased by $2$, the new mean will be the old mean plus $2$. So the new mean is $6.3 + 2=8.3$.

Step5: Find the median

Arrange the data in ascending order of price and consider the frequencies. The total number of data - points $n = 50$ (an even number). To find the median, we first find the cumulative frequencies.
The cumulative frequencies are: $12$ (for $\$1$ tickets), $12 + 14=26$ (for $\$2$ tickets), $26+9 = 35$ (for $\$5$ tickets), $35 + 7=42$ (for $\$10$ tickets), and $42+8 = 50$ (for $\$20$ tickets).
The median is the average of the $25^{th}$ and $26^{th}$ ordered values. Since the $25^{th}$ and $26^{th}$ values are both in the $\$2$ - ticket category, the median doesn't change when each ticket price is increased by $\$2$ (because the order of the data based on frequency remains the same). The median of the original data is $\frac{2 + 2}{2}=2$, and the median of the new data (after increasing each price by $\$2$) is $\frac{4 + 4}{2}=4$.

Answer:

a. The mean price is $6.3$.
b. The mean price increases by $2$ (new mean is $8.3$), and the median of the original data is $2$ and the median of the new data (after increasing each price by $\$2$) is $4$.