Questions: In a neighborhood donut shop, one type of donut has 580 calories, six types of donuts have 600 calories, two types of donuts have 420 calories, six types of donuts have 450 calories, and three types of donuts have 520 calories. Find the range. calories Find the standard deviation. Round your answer to the nearest tenth, if necessary. calories

Solution Steps

To find the range of the calorie counts, we use the formula:

\[ \text{Range} = \max(x_i) - \min(x_i) \]

Given the calorie counts, we have:

\[ \max(x_i) = 600 \quad \text{and} \quad \min(x_i) = 420 \]

Thus, the range is:

\[ \text{Range} = 600 - 420 = 180 \text{ calories} \]

The mean \( \mu \) of the dataset is calculated using the formula:

\[ \mu = \frac{\sum x_i}{n} \]

Where \( \sum x_i = 9280 \) and \( n = 18 \):

\[ \mu = \frac{9280}{18} \approx 515.6 \]

The variance \( \sigma^2 \) is calculated using the formula:

\[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{n} \]

Given that \( \sum (x_i - \mu)^2 = 5058.0 \):

\[ \sigma^2 = 5058.0 \]

The standard deviation \( \sigma \) is the square root of the variance:

\[ \sigma = \sqrt{5058.0} \approx 71.1 \]

Final Answer