To find the range and the standard deviation of the calories in the donuts, we first need to list all the calorie values and determine the necessary calculations.

Step 1: List all the calorie values

• One type of donut has 390 calories.
• Three types of donuts have 550 calories each.
• Four types of donuts have 420 calories each.
• Seven types of donuts have 410 calories each.
• Five types of donuts have 560 calories each.

So the list of calorie values is: $390, 550, 550, 550, 420, 420, 420, 420, 410, 410, 410, 410, 410, 410, 410, 560, 560, 560, 560, 560$

Step 2: Calculate the Range

The range is the difference between the highest and lowest calorie values.

• Minimum calorie value: 390
• Maximum calorie value: 560

Range = $560 - 390 = 170$ calories

Step 3: Calculate the Mean

First, find the mean (average) of the calorie values.

$\text{Mean} = \frac{\text{Sum of all calorie values}}{\text{Total number of donuts}}$

The sum of the calorie values is: $390 + 3 \times 550 + 4 \times 420 + 7 \times 410 + 5 \times 560$

Let's calculate this: $390 + (3 \times 550) + (4 \times 420) + (7 \times 410) + (5 \times 560) = 390 + 1650 + 1680 + 2870 + 2800 = 9390$

The total number of donuts: $1 + 3 + 4 + 7 + 5 = 20$

So, the mean is: $\text{Mean} = \frac{9390}{20} = 469.5 \text{ calories}$

Step 4: Calculate the Variance

Variance is calculated using the formula: $\text{Variance} = \frac{\sum (x_i - \mu)^2}{N}$ Where:

• $x_i$ = each calorie value
• $\mu$ = mean (469.5)
• $N$ = total number of donuts (20)

Let's compute the variance: $\text{Variance} = \frac{(390 - 469.5)^2 + 3 \times (550 - 469.5)^2 + 4 \times (420 - 469.5)^2 + 7 \times (410 - 469.5)^2 + 5 \times (560 - 469.5)^2}{20}$

Now, calculate each squared difference: $(390 - 469.5)^2 = (-79.5)^2 = 6320.25$ $(550 - 469.5)^2 = (80.5)^2 = 6480.25$ $(420 - 469.5)^2 = (-49.5)^2 = 2450.25$ $(410 - 469.5)^2 = (-59.5)^2 = 3540.25$ $(560 - 469.5)^2 = (90.5)^2 = 8190.25$

Now multiply by their respective frequencies and sum them up: $\text{Variance} = \frac{6320.25 + 3 \times 6480.25 + 4 \times 2450.25 + 7 \times 3540.25 + 5 \times 8190.25}{20}$ $\text{Variance} = \frac{6320.25 + 19440.75 + 9801 + 24781.75 + 40951.25}{20}$ $\text{Variance} = \frac{101295}{20} = 5064.75$

Step 5: Calculate the Standard Deviation

Standard deviation is the square root of the variance: $\text{Standard Deviation} = \sqrt{5064.75} \approx 71.2 \text{ calories}$

Final Answers:

• Range: 170 calories
• Standard Deviation: 71.2 calories

Would you like more details or have any other questions?

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Here are 5 related questions to explore:

1. How does increasing the calorie count of one type of donut affect the standard deviation?
2. What is the effect on the mean if the number of low-calorie donuts increases?
3. How would the range change if a new donut type with 580 calories is introduced?
4. What happens to the variance if all donuts are changed to have the same calorie value?
5. How does removing the highest calorie donuts affect the overall standard deviation?

Tip: The range provides a quick insight into the spread of your data, but standard deviation gives a more detailed understanding of the data's variability.