Let's go step-by-step to solve this problem:

Part (a): Find the sample standard deviation

The formula for the sample standard deviation $s$ is:

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

1. Calculate the sample mean $\bar{x}$: $\bar{x} = \frac{\sum x_i}{n}$ Given the gas mileages: $23, 23, 21, 20, 36, 23, 18, 17, 20, 20, 17, 14, 15, 16, 16, 13$ Sum = $256$, $n = 16$: $\bar{x} = \frac{256}{16} = 16$

2. Compute the squared deviations: $(x_i - \bar{x})^2$

3. Sum up the squared deviations and divide by $n-1$.

4. Take the square root to find $s$.

Performing the above calculation step-by-step gives the sample standard deviation $s$ rounded to two decimal places.

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**Part (B) np

Given data

data = np.array([23, 23, 21, 20, 36, 23, 18, 17, 20, 20, 17, 14, 15, 16, 16, 13])

Part (a): Sample standard deviation

mean = np.mean(data) std_dev = np.std(data, ddof=1) # Sample standard deviation

Part (b): Range prediction using Chebyshev's inequality (k = 2 for 75%)

k = 2 lower_bound = mean - k * std_dev upper_bound = mean + k * std_dev range_chebyshev = (lower_bound, upper_bound)

Part (c): Actual percentage within the range

in_range = np.sum((data >= lower_bound) & (data <= upper_bound)) percentage_actual = (in_range / len(data)) * ### Results:

(a) The sample standard deviation is approximately 5.44.

(b) Using Chebyshev's inequality, at least 75% of the gas mileages should fall within the range: $(8.62, 30.38)$

(c) The actual percentage of SUVs within this range is 93.75%.

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Explanation:

• Chebyshev's inequality guarantees that at least $1 - \frac{1}{k^2}$ of the data lies within $k$ standard deviations from the mean. Here $k=2$, meaning at least 75% of the data should fall within this range.
• The actual percentage is higher, showing that the data distribution is tighter than the broad bounds of Chebyshev's rule.

Which rule is more accurate? In this case, the empirical rule (specific to normal-like distributions) would likely provide a closer estimate than Chebyshev's, which is more general.

Would you like further clarification or any additional calculations? Here are five related questions for exploration:

1. How does the empirical rule range compare with the Chebyshev range in this data set?
2. Can we calculate the mean absolute deviation for these gas mileages?
3. How would the results change if the dataset included an extreme outlier?
4. What are the implications of Chebyshev's inequality in non-normal distributions?
5. How would these results differ if the standard deviation were calculated using population data?

Tip: Always use the empirical rule for normal-like data when precision is critical, but apply Chebys