To solve this problem, let's break it down step-by-step to find both the population mean and the sample mean for the number of students who dropped out in the ninth grade.
### Step 1: Extract Data
First, we'll extract the number of dropouts for each year from the given table. According to the table, the data points for dropouts over the 11-year period are as follows:
- Year 10: 7 dropouts
- Year 11: 6 dropouts
- Year 12: 1 dropout
- Year 13: 1 dropout
- Year 14: 1 dropout
- Year 15: 3 dropouts
- Year 16: 3 dropouts
- Year 17: 5 dropouts
- Year 18: 4 dropouts
- Year 19: 5 dropouts
- Year 20: 7 dropouts
### Step 2: Calculate the Population Mean
The population mean is the average dropouts per year over all 11 years. To find this, we first sum up the total number of dropouts and then divide by the number of years.
Sum of dropouts:
[tex]\[ 7 + 6 + 1 + 1 + 1 + 3 + 3 + 5 + 4 + 5 + 7 = 43 \][/tex]
Number of years:
[tex]\[ 11 \][/tex]
Population mean:
[tex]\[ \text{Population mean} = \frac{\text{Sum of dropouts}}{\text{Number of years}} = \frac{43}{11} \approx 3.91 \][/tex]
### Step 3: Calculate the Sample Mean
The sample mean is calculated using data from only the last 3 years. From the data provided, the last 3 years are:
- Year 18: 4 dropouts
- Year 19: 5 dropouts
- Year 20: 7 dropouts
Sum of dropouts for the last 3 years:
[tex]\[ 4 + 5 + 7 = 16 \][/tex]
Number of years in the sample:
[tex]\[ 3 \][/tex]
Sample mean:
[tex]\[ \text{Sample mean} = \frac{\text{Sum of dropouts in last 3 years}}{\text{Number of years in the sample}} = \frac{16}{3} \approx 5.33 \][/tex]
### Step 4: Conclusion
Putting it all together, we have:
- The population mean: [tex]\( \approx 3.91 \)[/tex]
- The sample mean: [tex]\( \approx 5.33 \)[/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{(3.91, 5.33)} \][/tex]
Since none of the options perfectly match our calculated result, the closest correct options would be:
[tex]\[ A. \text{The population mean is 4.35. The sample mean is 5.33.} \][/tex]
But please note that based on the calculation we arrived at [tex]\( 3.91 \)[/tex] for the population mean rather than [tex]\( 4.35 \)[/tex], there might be a slight data entry error in the options provided. The calculated values are indeed [tex]\( 3.91 \)[/tex] for population mean and [tex]\( 5.33 \)[/tex] for sample mean.
