Question

1. - / 25 points 0/100 submissions used find the range, the standard deviation, and the variance for the given sample. round non - integer results to the nearest tenth. 10, 11, 13, 15, 31 range standard deviation variance

Explanation:

Part 1: Dataset = {2, 3, 4, 7, 8, 8, 24}

Step1: Calculate the range

Range = Max value - Min value = $24 - 2 = 22$

Step2: Calculate the mean

$\bar{x} = \frac{2+3+4+7+8+8+24}{7} = \frac{56}{7} = 8$

Step3: Calculate squared deviations

$(2-8)^2=36,\ (3-8)^2=25,\ (4-8)^2=16,\ (7-8)^2=1,\ (8-8)^2=0,\ (8-8)^2=0,\ (24-8)^2=256$
Sum of squared deviations: $36+25+16+1+0+0+256 = 334$

Step4: Calculate sample variance

Sample variance $s^2 = \frac{\sum(x_i-\bar{x})^2}{n-1} = \frac{334}{7-1} \approx 55.7$

Step5: Calculate sample standard deviation

Sample standard deviation $s = \sqrt{55.7} \approx 7.5$

Part 2: Dataset = {10, 11, 13, 15, 31}

Step1: Calculate the range

Range = Max value - Min value = $31 - 10 = 21$

Step2: Calculate the mean

$\bar{x} = \frac{10+11+13+15+31}{5} = \frac{80}{5} = 16$

Step3: Calculate squared deviations

$(10-16)^2=36,\ (11-16)^2=25,\ (13-16)^2=9,\ (15-16)^2=1,\ (31-16)^2=225$
Sum of squared deviations: $36+25+9+1+225 = 296$

Step4: Calculate sample variance

Sample variance $s^2 = \frac{\sum(x_i-\bar{x})^2}{n-1} = \frac{296}{5-1} = 74.0$

Step5: Calculate sample standard deviation

Sample standard deviation $s = \sqrt{74.0} \approx 8.6$

Answer:

For dataset {2, 3, 4, 7, 8, 8, 24}:

• range: 22
• standard deviation: 7.5
• variance: 55.7

For dataset {10, 11, 13, 15, 31}:

• range: 21
• standard deviation: 8.6
• variance: 74.0