Certainly! Let's go through the steps to find the standard deviation for the given set of population data. Here is a step-by-step solution:
### Step 1: Calculate the Mean (μ)
First, we need to calculate the mean (average) of the data set.
The data set consists of the following numbers: 14, 23, 31, 29, 33.
The formula for the mean is:
[tex]\[ \mu = \frac{\sum{x}}{N} \][/tex]
where [tex]\(\sum{x}\)[/tex] is the sum of all the data values and [tex]\(N\)[/tex] is the number of data values.
[tex]\[ \sum{x} = 14 + 23 + 31 + 29 + 33 = 130 \][/tex]
[tex]\[ N = 5 \][/tex]
Now, calculate the mean:
[tex]\[ \mu = \frac{130}{5} = 26 \][/tex]
### Step 2: Calculate Each Deviation from the Mean and Square It
Next, we will calculate the deviation of each data point from the mean and then square each deviation.
1. [tex]\( (14 - 26)^2 = (-12)^2 = 144 \)[/tex]
2. [tex]\( (23 - 26)^2 = (-3)^2 = 9 \)[/tex]
3. [tex]\( (31 - 26)^2 = (5)^2 = 25 \)[/tex]
4. [tex]\( (29 - 26)^2 = (3)^2 = 9 \)[/tex]
5. [tex]\( (33 - 26)^2 = (7)^2 = 49 \)[/tex]
### Step 3: Calculate the Variance (σ²)
Variance is the average of these squared deviations. The formula for variance is:
[tex]\[ \sigma^2 = \frac{\sum (x - \mu)^2}{N} \][/tex]
Sum of the squared deviations:
[tex]\[ 144 + 9 + 25 + 9 + 49 = 236 \][/tex]
Now, calculate the variance:
[tex]\[ \sigma^2 = \frac{236}{5} = 47.2 \][/tex]
### Step 4: Calculate the Standard Deviation (σ)
Finally, the standard deviation is the square root of the variance. The formula for standard deviation is:
[tex]\[ \sigma = \sqrt{\sigma^2} \][/tex]
So, the standard deviation is:
[tex]\[ \sigma = \sqrt{47.2} \approx 6.870225614927067 \][/tex]
This is approximately 6.9 when rounded to one decimal place.
### Conclusion
The standard deviation of the number of cars sold at the dealership over the given weeks is approximately [tex]\( \mathbf{6.9} \)[/tex]. Therefore, the correct answer is:
[tex]\[ \boxed{6.9} \][/tex]
