Question

25. Let X be the temperature in at which a certain chemical

reaction takes place, and let Y be the temperature in (so
).
a. If the median of the X distribution is , show that
is the median of the Y distribution.
b. How is the 90th percentile of the Y distribution related to
the 90th percentile of the X distribution? Verify your
conjecture.
c. More generally, if , how is any particular
percentile of the Y distribution related to the corresponding percentile of the X distribution?

Answer 1

a. If the median of the X distribution is \(M\), then the median of the Y distribution (\(Y\)) will also be \(M\). This is because the median is the middle value of a dataset, and the relationship between \(X\) and \(Y\) is given.

b. The 90th percentile of the Y distribution (\(Y\)) will also be \(90^\text{th}\) percentile of the X distribution (\(X\)). Percentiles are relative positions in a dataset, and if the values in \(Y\) are related to the values in \(X\), the percentile positions will be maintained.

c. More generally, if \(P_X\) is the \(p^\text{th}\) percentile of the X distribution, then the \(p^\text{th}\) percentile of the Y distribution (\(P_Y\)) will also be \(P_X\). This holds true because the relationship between \(X\) and \(Y\) is specified, so the percentile positions are consistent between the two distributions.