Question

in a neighborhood donut shop, one type of donut has 360 calories, two types of donuts have 350 calories, six types of donuts have 450 calories, six types of donuts have 580 calories, and three types of donuts have 440 calories.
find the range.
calories
enter an integer or decimal number more..
find the standard deviation. round your answer to the nearest tenth, if necessary.
calories
question help: video 1 video 2

Explanation:

Step1: Identify min and max calories

Minimum calorie value: $350$, Maximum calorie value: $580$

Step2: Calculate the range

Range = Max - Min
$\text{Range} = 580 - 350$

Step3: Calculate total number of donut types

Sum all donut type counts
$n = 1 + 2 + 6 + 6 + 3 = 18$

Step4: Calculate the mean calorie value

Weighted average of calories
$\bar{x} = \frac{(1\times360)+(2\times350)+(6\times450)+(6\times580)+(3\times440)}{18}$
$\bar{x} = \frac{360 + 700 + 2700 + 3480 + 1320}{18} = \frac{8560}{18} \approx 475.56$

Step5: Calculate sum of squared deviations

Sum weighted squared differences from mean
$\sum (f_i(x_i-\bar{x})^2) = 1\times(360-475.56)^2 + 2\times(350-475.56)^2 + 6\times(450-475.56)^2 + 6\times(580-475.56)^2 + 3\times(440-475.56)^2$
$\sum (f_i(x_i-\bar{x})^2) \approx 1\times13354.11 + 2\times15765.31 + 6\times653.31 + 6\times10907.71 + 3\times1264.51$
$\sum (f_i(x_i-\bar{x})^2) \approx 13354.11 + 31530.62 + 3919.86 + 65446.26 + 3793.53 = 117044.38$

Step6: Calculate sample variance

Divide sum by $n-1$
$s^2 = \frac{117044.38}{18-1} \approx 6884.96$

Step7: Calculate standard deviation

Take square root of variance
$s = \sqrt{6884.96} \approx 83.0$

Answer:

Range: $230$ calories
Standard deviation: $83.0$ calories