Question

the table shows information about donations made to an animal shelter.
donation amount | number of people
$500 | 3
$100 | 7
$50 | 5
$10 | 10
in calculating the mean donation amount, which steps would you need to take? check all that apply.
□ multiply each donation amount by the number of people.
□ add the total amount of money.
□ add the total number of people.
□ 2,550 ÷ 25 = $102
□ 25 ÷ 2,550 = $0.01

Explanation:

Step1: Recall the formula for mean

The formula for the mean of a grouped data is \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}f_{i}}{\sum_{i = 1}^{n}f_{i}}\), where \(x_{i}\) is the value of the \(i -\)th group and \(f_{i}\) is the frequency of the \(i -\)th group.

Step2: Analyze each option

• Multiply each donation amount by the number of people:

This is equivalent to calculating \(x_{i}f_{i}\) for each row in the table. For example, for the row with donation amount \(x = 500\) and number of people \(f=3\), we calculate \(500\times3\). This is a necessary step as per the mean formula \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}f_{i}}{\sum_{i = 1}^{n}f_{i}}\).

• Add the total amount of money:

Adding the total amount of money is equivalent to calculating \(\sum_{i = 1}^{n}x_{i}f_{i}\). After multiplying each \(x_{i}\) by \(f_{i}\) (e.g., \(500\times3 = 1500\), \(100\times7=700\), \(50\times5 = 250\), \(10\times10=100\)), we add \(1500 + 700+250 + 100=2550\). This is part of the numerator of the mean formula.

• Add the total number of people:

Adding the total number of people is equivalent to calculating \(\sum_{i = 1}^{n}f_{i}\). Here, \(3 + 7+5 + 10=25\). This is the denominator of the mean formula.

• \(2550\div25=\$102\):

Since \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}f_{i}}{\sum_{i = 1}^{n}f_{i}}\), substituting \(\sum_{i = 1}^{n}x_{i}f_{i}=2550\) and \(\sum_{i = 1}^{n}f_{i}=25\), we get \(\bar{x}=\frac{2550}{25}=102\). This is the final step of calculating the mean.

• \(25\div2550=\$0.01\):

The formula for the mean is \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}f_{i}}{\sum_{i = 1}^{n}f_{i}}\), not \(\frac{\sum_{i = 1}^{n}f_{i}}{\sum_{i = 1}^{n}x_{i}f_{i}}\). So this step is incorrect.

Answer:

Multiply each donation amount by the number of people, Add the total amount of money, Add the total number of people, \(2550\div25 = \$102\)