To find the standard deviation of the number of cars sold at a dealership over several weeks, follow these steps:
1. List the Data Points:
The number of cars sold each week is: [tex]\(14, 23, 31, 29, 33\)[/tex].
2. Calculate the Mean (μ):
Mean (μ) is found by summing all the data points and dividing by the number of data points.
[tex]\[
\mu = \frac{14 + 23 + 31 + 29 + 33}{5} = \frac{130}{5} = 26
\][/tex]
3. Calculate the Squared Differences from the Mean:
For each data point, subtract the mean and square the result.
[tex]\[
(14 - 26)^2 = (-12)^2 = 144
\][/tex]
[tex]\[
(23 - 26)^2 = (-3)^2 = 9
\][/tex]
[tex]\[
(31 - 26)^2 = 5^2 = 25
\][/tex]
[tex]\[
(29 - 26)^2 = 3^2 = 9
\][/tex]
[tex]\[
(33 - 26)^2 = 7^2 = 49
\][/tex]
These squared differences are: [tex]\(144, 9, 25, 9, 49\)[/tex].
4. Calculate the Variance (σ²):
Variance is the average of these squared differences.
[tex]\[
\sigma^2 = \frac{144 + 9 + 25 + 9 + 49}{5} = \frac{236}{5} = 47.2
\][/tex]
5. Calculate the Standard Deviation (σ):
The standard deviation is the square root of the variance.
[tex]\[
\sigma = \sqrt{47.2} \approx 6.8702
\][/tex]
Thus, the standard deviation of the number of cars sold is approximately [tex]\(6.87\)[/tex].
Reviewing the options given:
- 6.9
- 12.4
- 15.4
- 47.2
The closest option to our calculated standard deviation is 6.9.
Therefore, the correct answer is 6.9.
