Reverse transformation: find original statistics

1 Test scores, \(X\), have mean 54 and variance 144. The scores are scaled using the formula \(Y = a + b X\), 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 \(b\).
\(235 \%\) of a random sample of \(n\) students walk to college. This result is used to construct an approximate \(98 \%\) confidence interval for the population proportion of students who walk to college. Given that the width of this confidence interval is 0.157 , correct to 3 significant figures, find \(n\).

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 \(x\).
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3. 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 \(a X + b\), 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 \(0.2 X - 30\)

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