To find the standard deviation of the mileage (in miles per gallon) for the sample of convertibles, we'll follow these steps:
### Step 1: List the given MPG values for the convertibles
The mileage (MPG) values for the convertibles are:
- BMW 328i: 21
- Toyota Camry Solara: 21
- Ford Mustang V6: 20
- Volkswagen Eos: 25
- MINI Cooper: 25
- Saab 9-3: 24
So, we have the MPG values: [21, 21, 20, 25, 25, 24].
### Step 2: Calculate the Mean (Average) MPG
The mean or average MPG is calculated as follows:
[tex]\[
\text{Mean} = \frac{\sum \text{MPG values}}{\text{Number of values}} = \frac{21 + 21 + 20 + 25 + 25 + 24}{6} = \frac{136}{6} \approx 22.67
\][/tex]
### Step 3: Calculate the Variance
The variance for a sample is given by:
[tex]\[
\text{Variance} = \frac{\sum (x_i - \text{mean})^2}{n - 1}
\][/tex]
Where [tex]\( x_i \)[/tex] are the MPG values, the mean is 22.67, and [tex]\( n = 6 \)[/tex].
Calculate each squared difference:
[tex]\[
(21 - 22.67)^2 = (-1.67)^2 = 2.7889
\][/tex]
[tex]\[
(21 - 22.67)^2 = (-1.67)^2 = 2.7889
\][/tex]
[tex]\[
(20 - 22.67)^2 = (-2.67)^2 = 7.1289
\][/tex]
[tex]\[
(25 - 22.67)^2 = 2.33^2 = 5.4289
\][/tex]
[tex]\[
(25 - 22.67)^2 = 2.33^2 = 5.4289
\][/tex]
[tex]\[
(24 - 22.67)^2 = 1.33^2 = 1.7689
\][/tex]
Sum of squared differences:
[tex]\[
2.7889 + 2.7889 + 7.1289 + 5.4289 + 5.4289 + 1.7689 = 25.3334
\][/tex]
The variance is:
[tex]\[
\text{Variance} = \frac{25.3334}{6 - 1} = \frac{25.3334}{5} \approx 5.07
\][/tex]
### Step 4: Calculate the Standard Deviation
Standard deviation is the square root of the variance:
[tex]\[
\text{Standard Deviation} = \sqrt{5.07} \approx 2.25
\][/tex]
### Step 5: Round the Standard Deviation to One Decimal Place
[tex]\[
\text{Standard Deviation (rounded)} \approx 2.3
\][/tex]
### Summary of Results
1. Mean MPG: 22.67
2. Variance: 5.07
3. Standard Deviation: 2.25
4. Rounded Standard Deviation: 2.3
So, the standard deviation of the mileage for the sample of convertibles, rounded to at least one decimal place, is approximately 2.3.
