To find the weighted average of the buttons in the given sample data, you need to follow these steps:
### 1. Convert the percentages to proportions
The abundance of the square buttons is given as 32%, which can be converted to a proportion by dividing by 100:
[tex]\[ \text{Abundance of square buttons} = \frac{32}{100} = 0.32 \][/tex]
Similarly, the abundance of the circle buttons is given as 68%, which can be converted to a proportion by dividing by 100:
[tex]\[ \text{Abundance of circle buttons} = \frac{68}{100} = 0.68 \][/tex]
### 2. Identify the masses of each type of button
The mass of a square button is given as 1.0 grams:
[tex]\[ \text{Mass of square button} = 1.0 \, \text{g} \][/tex]
The mass of a circle button is given as 1.6 grams:
[tex]\[ \text{Mass of circle button} = 1.6 \, \text{g} \][/tex]
### 3. Calculate the weighted average mass
The weighted average mass is calculated by multiplying the proportion of each type of button by its respective mass and then summing these values:
[tex]\[ \text{Weighted average mass} = (\text{Abundance of square buttons} \times \text{Mass of square button}) + (\text{Abundance of circle buttons} \times \text{Mass of circle button}) \][/tex]
Substitute the known values:
[tex]\[ \text{Weighted average mass} = (0.32 \times 1.0) + (0.68 \times 1.6) \][/tex]
### 4. Compute the individual contributions to the weighted average
First, calculate the contribution of the square buttons:
[tex]\[ 0.32 \times 1.0 = 0.32 \][/tex]
Next, calculate the contribution of the circle buttons:
[tex]\[ 0.68 \times 1.6 = 1.088 \][/tex]
### 5. Sum the contributions to find the weighted average mass
Adding the contributions together gives us:
[tex]\[ 0.32 + 1.088 = 1.408 \][/tex]
Thus, the weighted average mass of the buttons in the sample data is:
[tex]\[ \boxed{1.408 \, \text{g}} \][/tex]
So, the weighted average mass of the buttons in the sample data is 1.408 grams.
