12. the table shows the number of tickets of each price sold for a show.\n| price of ticket | $1 | $2 | $5 |…

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 $4$.