Statistics
Linear Transformation of Data
nta_pyq_2025_apr
Grade 11
Question:
Let $x_1, x_2, \ldots, x_{10}$ be ten observations such that $\displaystyle\sum_{i=1}^{10}(x_i - 2) = 30$, $\displaystyle\sum_{i=1}^{10}(x_i - \beta)^2 = 98$, $\beta > 2$, and their variance is $\frac{4}{5}$. If $\mu$ and $\sigma^2$ are respectively the mean and the variance of $2(x_1 - 1) + 4\beta, 2(x_2 - 1) + 4\beta, \ldots, 2(x_{10} - 1) + 4\beta$, then $\frac{\beta\mu}{\sigma^2}$ is equal to:
Step-by-Step Solution
Key Concept: Find mean and $\beta$ from the given conditions, then apply linear transformation rules: if $y = ax + b$, then $\mu_y = a\mu_x + b$ and $\sigma^2_y = a^2 \sigma^2_x$.
$\sum(x_i - 2) = 30 \Rightarrow$ Mean $= 5$, $\sum x_i = 50$. From variance condition: $\sum x_i^2 = 258$, and $10\beta^2 - 100\beta + 160 = 0 \Rightarrow \beta = 8$ (since $\beta > 2$). New data: $2x_i + 30$, so $\mu = 2(5) + 30 = 40$, $\sigma^2 = 4 \times \frac{4}{5} = \frac{16}{5}$. $\frac{\beta\mu}{\sigma^2} = \frac{8 \times 40}{16/5} = 100$.
Correct Answer: 100