Sure, let's go through the step-by-step solution for each part of the problem.
### Part (a): Determine the Mean of the Population
To find the mean of the population, we need to sum up all the values and then divide by the number of values.
Given values: 8, 3, 7, 3, and 4.
1. Sum the values:
[tex]\( \text{Sum} = 8 + 3 + 7 + 3 + 4 \)[/tex]
[tex]\( \text{Sum} = 25 \)[/tex]
2. Count the number of values (n):
Here, [tex]\( n = 5 \)[/tex]
3. Calculate the mean:
[tex]\( \text{Mean} = \frac{\text{Sum}}{n} \)[/tex]
[tex]\( \text{Mean} = \frac{25}{5} \)[/tex]
[tex]\( \text{Mean} = 5.0 \)[/tex]
Thus, the mean of the population is [tex]\( 5.0 \)[/tex].
### Part (b): Determine the Variance
To find the variance, we follow these steps:
1. Calculate each value's deviation from the mean and square it:
Deviations from the mean:
- For 8: [tex]\( (8 - 5.0)^2 = 3.0^2 = 9.0 \)[/tex]
- For 3: [tex]\( (3 - 5.0)^2 = (-2.0)^2 = 4.0 \)[/tex]
- For 7: [tex]\( (7 - 5.0)^2 = 2.0^2 = 4.0 \)[/tex]
- For 3: [tex]\( (3 - 5.0)^2 = (-2.0)^2 = 4.0 \)[/tex]
- For 4: [tex]\( (4 - 5.0)^2 = (-1.0)^2 = 1.0 \)[/tex]
2. Sum these squared deviations:
[tex]\( \text{Sum of squared deviations} = 9.0 + 4.0 + 4.0 + 4.0 + 1.0 = 22.0 \)[/tex]
3. Calculate the variance (by dividing the sum of squared deviations by the number of values):
[tex]\( \text{Variance} = \frac{\text{Sum of squared deviations}}{n} \)[/tex]
[tex]\( \text{Variance} = \frac{22.0}{5} \)[/tex]
[tex]\( \text{Variance} = 4.4 \)[/tex]
If necessary, variance should be rounded to 1 decimal place.
Thus, the variance of the population is [tex]\( 4.4 \)[/tex].
To summarize:
- The mean of the population is [tex]\( 5.0 \)[/tex].
- The variance of the population is [tex]\( 4.4 \)[/tex].
