To calculate the mean and sample standard deviation for the given data:

1. Data:

Fuel Efficiency (mpg):
23, 24, 24, 27, 14, 25, 25, 31, 24, 23, 24, 23

2. Steps to Compute the Mean:

The mean is given by:

$\bar{x} = \frac{\sum x_i}{n}$

Where $\sum x_i$ is the sum of all data points, and $n$ is the number of data points.

$\sum x_i = 23 + 24 + 24 + 27 + 14 + 25 + 25 + 31 + 24 + 23 + 24 + 23 = 267$ $n = 12$

$\bar{x} = \frac{267}{12} \approx 22.25 \, \text{(rounded to the nearest hundredth)}.$

3. Steps to Compute the Sample Standard Deviation:

The sample standard deviation $s$ is given by:

$s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}$

Step 1: Compute Deviations from the Mean ($x_i - \bar{x}$):

$\bar{x} = 22.25$

&23 - 22.25 = 0.75, \quad 24 - 22.25 = 1.75, \quad 24 - 22.25 = 1.75, \\ &27 - 22.25 = 4.75, \quad 14 - 22.25 = -8.25, \quad 25 - 22.25 = 2.75, \\ &25 - 22.25 = 2.75, \quad 31 - 22.25 = 8.75, \quad 24 - 22.25 = 1.75, \\ &23 - 22.25 = 0.75, \quad 24 - 22.25 = 1.75, \quad 23 - 22.25 = 0.75. \end{aligned}$$ #### Step 2: Square Each Deviation: $$\begin{aligned} &0.75^2 = 0.5625, \quad 1.75^2 = 3.0625, \quad 1.75^2 = 3.0625, \\ &4.75^2 = 22.5625, \quad (-8.25)^2 = 68.0625, \quad 2.75^2 = 7.5625, \\ &2.75^2 = 7.5625, \quad 8.75^2 = 76.5625, \quad 1.75^2 = 3.0625, \\ &0.75^2 = 0.5625, \quad 1.75^2 = 3.0625, \quad 0.75^2 = 0.5625. \end{aligned}$$ $$\sum (x_i - \bar{x})^2 = 196.625$$ #### Step 3: Compute the Variance: $$s^2 = \frac{196.625}{12 - 1} = \frac{196.625}{11} \approx 17.875$$ #### Step 4: Compute the Standard Deviation: $$s = \sqrt{17.875} \approx 4.23 \, \text{(rounded to the nearest hundredth)}.$$ ### Final Results: - **Mean**: $$\bar{x} \approx 22.25 \, \text{mpg}$$ - **Sample Standard Deviation**: $$s \approx 4.23 \, \text{mpg}$$ Would you like additional clarification or details? 😊 --- #### Related Questions: 1. How is the sample standard deviation different from the population standard deviation? 2. Why is the denominator $$n-1$$ used in the sample standard deviation formula? 3. Can the mean and standard deviation be affected by outliers? If so, how? 4. How would you compute the variance for this dataset? 5. What does the standard deviation tell us about the spread of the data? --- #### Tip: When calculating deviations, keep intermediate steps precise to avoid rounding errors, and round only in the final results.