To determine the mean and mode of the given data set, let's follow these steps:
### Step 1: List the Data
First, rewrite the data in a single list to make it easier to work with:
59, 16, 57, 46, 58, 21, 36, 41, 23, 64, 20, 66, 40, 51, 28, 46
### Step 2: Calculate the Mean
The mean is the average of the data points. It is calculated by summing all the data points and then dividing by the number of data points.
1. Sum all the data points:
[tex]\[ 59 + 16 + 57 + 46 + 58 + 21 + 36 + 41 + 23 + 64 + 20 + 66 + 40 + 51 + 28 + 46 \][/tex]
First, add the numbers step-by-step or use a calculator:
[tex]\[ 59 + 16 = 75 \][/tex]
[tex]\[ 75 + 57 = 132 \][/tex]
[tex]\[ 132 + 46 = 178 \][/tex]
[tex]\[ 178 + 58 = 236 \][/tex]
[tex]\[ 236 + 21 = 257 \][/tex]
[tex]\[ 257 + 36 = 293 \][/tex]
[tex]\[ 293 + 41 = 334 \][/tex]
[tex]\[ 334 + 23 = 357 \][/tex]
[tex]\[ 357 + 64 = 421 \][/tex]
[tex]\[ 421 + 20 = 441 \][/tex]
[tex]\[ 441 + 66 = 507 \][/tex]
[tex]\[ 507 + 40 = 547 \][/tex]
[tex]\[ 547 + 51 = 598 \][/tex]
[tex]\[ 598 + 28 = 626 \][/tex]
[tex]\[ 626 + 46 = 672 \][/tex]
The sum of the data points is 672.
2. Count the number of data points, which is 16.
3. Calculate the mean:
[tex]\[ \text{Mean} = \frac{\text{Sum of data points}}{\text{Number of data points}} = \frac{672}{16} = 42 \][/tex]
### Step 3: Calculate the Mode
The mode is the data point that appears most frequently in the set.
1. List the frequencies of each unique data point:
[tex]\[ 59: 1, 16: 1, 57: 1, 46: 2, 58: 1, 21: 1, 36: 1, 41: 1, 23: 1, 64: 1, 20: 1, 66: 1, 40: 1, 51: 1, 28: 1, 46: 2 \][/tex]
2. Identify the data point that appears most frequently:
[tex]\[ 46 \][/tex]
### Step 4: Conclude the Mean and Mode
- Mean: 42
- Mode: 46
### Step 5: Select the Best Answer from the Choices Provided
The best answer is:
[tex]\[ \boxed{b. \text{mean: 42, mode: 46}} \][/tex]
