Let x1, x2, ..., xn be n observations. Let \[y_i = a x_i + b\] for i = 1, 2, 3, ..., n, where a and b are constants. If the mean of \[x_i 's\] is 48 and their standard deviation is 12, the mean of \[y_i 's\] is 55 and standard deviation of \[y_i 's\] is 15, the values of a and b are
Options
a = 1.25, b = −5
a = −1.25, b = 5
a = 2.5, b = −5
a = 2.5, b = 5
Solution
It is given that \[y_i = a x_i + b\] for i = 1, 2, 3, ..., n, where a and b are constants.
\[y_i = a x_i + b\]
\[ \Rightarrow \frac{\sum_{} y_i}{n} = \frac{\sum_{} \left( a x_i + b \right)}{n}\]
\[ \Rightarrow \frac{\sum_{} y_i}{n} = a\frac{\sum_{} x_i}{n} + \frac{\sum_{} b}{n}\]
\[ \Rightarrow y_i = a x_i + b \]
\[ \Rightarrow 55 = 48a + b . . . . . \left( 1 \right)\]
Now,
Standard deviation of yi = Standard deviation of \[a x_i + b\]
\[\Rightarrow \sigma_{y_i} = a \times \sigma_{x_i} \]
\[ \Rightarrow 15 = 12a\]
\[ \Rightarrow a = \frac{15}{12} = 1 . 25\]
