How a linear change of the data affects mean and standard deviation

A data set \(x_1,x_2,\ldots,x_n\) has mean \(m\) and standard deviation \(s\). Each value is changed to \(y_i=ax_i+b\), where \(a>0\) and \(b\) are constants.

1. Show that the mean of the \(y\)-values is \(am+b\) and that their standard deviation is \(as\).[3]
2. The monthly salaries of a company's staff have mean \(\$3200\) and standard deviation \(\$450\). Every salary is increased by \(4\%\) and then a fixed allowance of \(\$50\) is added. Use (a) to find the new mean and standard deviation.[3]
3. A student says that adding \(10\) to every value in a data set increases its standard deviation by \(10\). Explain why the student is wrong.[2]