Reverse transformation: find original statistics

2. The mark, \(x\), scored by each student who sat a statistics examination is coded using $$y = 1.4 x - 20$$ The coded marks have mean 60.8 and standard deviation 6.60 Find the mean and the standard deviation of \(x\). \includegraphics[max width=\textwidth, alt={}, center]{8270bcae-494c-4248-8229-a72e9e84eab0-04_99_97_2613_1784}

Test scores, \(X\), have mean 54 and variance 144. The scores are scaled using the formula \(Y = a + bX\), where \(a\) and \(b\) are constants and \(b > 0\). The scaled scores, \(Y\), have mean 50 and variance 100. Find the values of \(a\) and \(b\). [4]

A researcher has collected data on the heights of a sample of adults but has encoded the actual values using a linear transformation of the form \(aX + b\), where \(X\) represents the original height in centimetres. Given the following information about the encoded data: The mean of the encoded heights is 5.4 cm The standard deviation of the encoded heights is 2.0 cm The researcher knows that the transformation used was \(0.2X - 30\)

1. Find the mean of the original heights in the sample. [2]
2. Find the standard deviation of the original heights in the sample. [2]
3. If an encoded height value is 6.8, what was the original height in centimetres? [1]