91Ó°ÊÓ

Let's denote the set of numbers as \(x_1, x_2, \dots, x_n\). The definitions of the means we need to compute are: - Arithmetic Mean (\(\bar{x}\)): \(\bar{x} = \frac{\sum_{i=1}^n x_i}{n}\) - RMS-Average (\(x_{rms}\)): \(x_{rms} = \sqrt{\frac{\sum_{i=1}^n x_i^2}{n}}\) - Geometric Mean (\(x_{gm}\)): \(x_{gm} = \sqrt[n]{\prod_{i=1}^n x_i}\) - Harmonic Mean (\(x_{hm}\)): \(x_{hm} = \frac{n}{\sum_{i=1}^n \frac{1}{x_i}}\)

Using the formulas defined above, we will calculate the mean values for each set of numbers (a, b, c, and d): Set a: \(4, 4, 4, 4, 4, 4, 4\) - Arithmetic Mean: \(\bar{x} = \frac{4+4+4+4+4+4+4}{7} = 4\) - RMS-Average: \(x_{rms} = \sqrt{\frac{4^2+4^2+4^2+4^2+4^2+4^2+4^2}{7}} = 4\) - Geometric Mean: \(x_{gm} = \sqrt[7]{4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4} = 4\) - Harmonic Mean: \(x_{hm} = \frac{7}{\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}} = 4\) Set b: \(4, 3, 4, 5, 4, 3, 5\) - Arithmetic Mean: \(\bar{x} = \frac{4+3+4+5+4+3+5}{7} = \frac{28}{7} = 4\) - RMS-Average: \(x_{rms} = \sqrt{\frac{4^2+3^2+4^2+5^2+4^2+3^2+5^2}{7}} \approx 4.22\) - Geometric Mean: \(x_{gm} = \sqrt[7]{4 \cdot 3 \cdot 4 \cdot 5 \cdot 4 \cdot 3 \cdot 5} \approx 3.98\) - Harmonic Mean: \(x_{hm} = \frac{7}{\frac{1}{4}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{4}+\frac{1}{3}+\frac{1}{5}} \approx 3.76\) Set c: \(4, 1, 4, 7, 4, 1, 7\) - Arithmetic Mean: \(\bar{x} = \frac{4+1+4+7+4+1+7}{7} = \frac{28}{7} = 4\) - RMS-Average: \(x_{rms} = \sqrt{\frac{4^2+1^2+4^2+7^2+4^2+1^2+7^2}{7}} \approx 4.49\) - Geometric Mean: \(x_{gm} = \sqrt[7]{4 \cdot 1 \cdot 4 \cdot 7 \cdot 4 \cdot 1 \cdot 7} \approx 3.21\) - Harmonic Mean: \(x_{hm} = \frac{7}{\frac{1}{4}+\frac{1}{1}+\frac{1}{4}+\frac{1}{7}+\frac{1}{4}+\frac{1}{1}+\frac{1}{7}} \approx 2.15\) Set d: \(1, 2, 3, 4, 5, 6, 7\) - Arithmetic Mean: \(\bar{x} = \frac{1+2+3+4+5+6+7}{7} = \frac{28}{7} = 4\) - RMS-Average: \(x_{rms} = \sqrt{\frac{1^2+2^2+3^2+4^2+5^2+6^2+7^2}{7}} \approx 4.72\) - Geometric Mean: \(x_{gm} = \sqrt[7]{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7} \approx 3.48\) - Harmonic Mean: \(x_{hm} = \frac{7}{\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}} \approx 2.81\)

Now that we have calculated the mean values for each set of numbers, let's compare them: - Set a: Arithmetic, RMS-Average, Geometric, and Harmonic means are all equal to 4. This is because all the values in the set are the same. - Set b, c, and d: In all three sets, the arithmetic mean remains at 4, while the RMS-Average, Geometric, and Harmonic means vary. It can be observed that the RMS-Average is always greater than or equal to the Arithmetic mean, and the Harmonic mean is always smaller than or equal to the Geometric mean. This solution calculates and compares the Arithmetic Mean, RMS-Average, Geometric Mean, and Harmonic Mean for each given set of numbers.