Final answer:
To find the range, variance, and standard deviation of the given data, we perform mathematical operations on the provided sample data. The range is 17, sample variance s² is approximately 47.70, sample standard deviation s is approximately 6.91, population variance σ² is approximately 38.16, and population standard deviation σ is approximately 6.18, all rounded to two decimal places.
Step-by-step explanation:
Let's solve each part of the student's math question step-by-step:
(a) Find the range
The range of a set of numbers is the difference between the largest and smallest values. In this case, the largest number is 30 and the smallest is 13, so the range is 30 - 13 = 17.
(b) Verify Σx and Σx²
To verify Σx, we add up all the x values: 21 + 17 + 13 + 30 + 25 = 106. To verify Σx², we square each x value and add those up: 21² + 17² + 13² + 30² + 25² = 441 + 289 + 169 + 900 + 625 = 2424.
(c) Compute the sample variance s² and sample standard deviation s
To find the variance, divide the sum of the squares minus the square of the sum of x divided by the count (n), all over n - 1. For standard deviation, take the square root of the variance. The formulas are:
s² = (Σx² - (Σx)²/n) / (n - 1)
s = √s²
Using the values already given: 2424 - (106²/5) / (5 - 1), we get:
s² ≈ 47.70 and s ≈ 6.91 (rounded to two decimal places).
(d) Same calculations as part (c) using defining formulas.
(e) Compute the population variance σ² and population standard deviation σ
For population variance and standard deviation, we divide by n instead of n - 1:
σ² = (Σx² - (Σx)²/n) / n ≈ 38.16
σ = √σ² ≈ 6.18