In a neighborhood donut shop, one type of donut has 460 calories, two types of donuts have 590 calories, six types of donuts have 560 calories, three types of donuts have 370 calories, and four types of donuts have 440 calories. Find the range. \( 221 \sim 0^{8} 220 \) calories Find the standard deviation. Round your answer to the nearest tenth, if necessary. calories

We are given the following information about the donut types and their calorie counts: • 1 type: 460 calories • 2 types: 590 calories • 6 types: 560 calories • 3 types: 370 calories • 4 types: 440 calories Step 1. Find the Range • The range is the difference between the highest and lowest calorie values. • Highest calories = 590 • Lowest calories = 370 • Range = 590 − 370 = 220 calories Step 2. Find the Standard Deviation Since we are given several types of donuts (each with its calorie count) and the number of types for each calorie value, we can calculate the weighted (population) standard deviation. 1. Calculate the total number of donut types:   Total count = 1 + 2 + 6 + 3 + 4 = 16 2. Compute the mean (average calorie count):   Sum of calories = (1×460) + (2×590) + (6×560) + (3×370) + (4×440)     = 460 + 1180 + 3360 + 1110 + 1760 = 7870   Mean = 7870 ÷ 16 = 491.875 calories 3. Compute the squared deviations for each calorie value:   For 460 calories:     Deviation = 460 − 491.875 = −31.875     Squared deviation = (31.875)² = 1015.640625   For 590 calories:     Deviation = 590 − 491.875 = 98.125     Squared deviation = (98.125)² = 9628.515625   For 560 calories:     Deviation = 560 − 491.875 = 68.125     Squared deviation = (68.125)² = 4641.015625   For 370 calories:     Deviation = 370 − 491.875 = −121.875     Squared deviation = (121.875)² = 14853.515625   For 440 calories:     Deviation = 440 − 491.875 = −51.875     Squared deviation = (51.875)² = 2691.015625 4. Multiply each squared deviation by the number of types with that calorie count:   For 460 calories:  1 × 1015.640625 = 1015.640625   For 590 calories:  2 × 9628.515625 = 19257.03125   For 560 calories:  6 × 4641.015625 = 27846.09375   For 370 calories:  3 × 14853.515625 = 44560.546875   For 440 calories:  4 × 2691.015625 = 10764.0625 5. Sum these weighted squared deviations:   Total = 1015.640625 + 19257.03125 + 27846.09375 + 44560.546875 + 10764.0625       = 102683.375 6. Divide by the total count (since we are computing the population standard deviation):   Variance = 102683.375 ÷ 16 = 6417.711 7. Take the square root to find the standard deviation:   Standard Deviation = √6417.711 ≈ 80.1 calories Final Answers: • Range: 220 calories • Standard Deviation: approximately 80.1 calories