IBDP Maths analysis and approaches Topic: SL 4.1-Concepts of population, sample, random sample HL Paper 1

Ans:

1. \(\overline{x}_{new}=\frac{\sum_{i=1}^{n}(x_i+k)}{n}=\frac{\sum_{i=1}^{n}x_i+\sum_{i=1}^{n}k}{n}=\frac{\sum_{i=1}^{n}x_i}{n}+\frac{\sum_{i=1}^{n}k}{n}=\overline{x}+\frac{kn}{n}=\overline{x}+k\)
   \(s_{new}=\sqrt{\frac{\sum_{i=1}^{n} ((x_i+k)-(\overline{x}-k))^2 }{n}}=\sqrt{\frac{\sum_{i=1}^{n}(x_i-\overline{x})^2)}{n}}=s\)
2. \(\overline{x}_{new}=\frac{\sum_{i=1}^{n}(kx_i)}{n}=\frac{k\sum_{i=1}^{n}(x_i)}{n}=k\frac{\sum_{i=1}^{n}(x_i)}{n}=k\overline{x}\)
   \(s_{new}=\sqrt{\frac{\sum_{i=1}^{n}(kx_i-k\overline{x})^2}{n}}=\sqrt{\frac{k^2\sum_{i=1}^{n}(x_i-\overline{x})^2}{n}}=k\sqrt{\frac{\sum_{i=1}^{n}(x_i-\overline{x})^2}{n}}=ks\)
   \(s_{new}=ks\)⇒\(s^2_{new}=k^2s^2\)
   so the original variance is multiplied by \(k^2\).