Question

1. the mean weight of six schoolbags is 9 pounds. the weights of five schoolbags are 8 pounds, 10 pounds, 8.6 pounds, 8.8 pounds and 9.5 pounds. find the weight of the last schoolbag.
2. 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:

11.

Step1: Recall the mean formula

The mean $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$. Here $n = 6$ and $\bar{x}=9$. So the sum of the weights of six school - bags is $S=n\times\bar{x}=6\times9 = 54$ pounds.

Step2: Calculate the sum of the five known weights

The sum of the five given weights is $S_1=8 + 10+8.6+8.8+9.5=44.9$ pounds.

Step3: Find the weight of the last school - bag

Let the weight of the last school - bag be $x$. Then $S=S_1 + x$. So $x=S - S_1=54-44.9 = 9.1$ pounds.

Step1: Calculate the total money from ticket sales

We use the formula for the weighted - mean. The total money from ticket sales is $1\times12+2\times14 + 5\times9+10\times7+20\times8=12 + 28+45+70+160=315$. The total number of tickets sold is $n=12 + 14+9+7+8=50$.

Step2: Calculate the mean price

The mean price $\bar{x}=\frac{315}{50}=6.3$ dollars.

Step1: Analyze the effect on the mean

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

Step2: Arrange the data to find the median

First, arrange the number of tickets in ascending order of price: $12$ tickets at $\$1$, $14$ tickets at $\$2$, $9$ tickets at $\$5$, $7$ tickets at $\$10$, $8$ tickets at $\$20$. The total number of tickets $n = 50$ (an even number). The median is the average of the $25^{th}$ and $26^{th}$ ordered values. The first $12$ tickets are at $\$1$ and the next $14$ tickets are at $\$2$. So the $25^{th}$ and $26^{th}$ values are in the group of $\$2$ tickets. After increasing the price by $\$2$, the median price is $2 + 2=4$ dollars.

Answer:

$9.1$ pounds

12. a.