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Chapter 4: Problem 15

Let \(X\) be the temperature in \({ }^{\circ} \mathrm{C}\) at which a certain chemical reaction takes place, and let \(Y\) be the temperature in \({ }^{\circ} \mathrm{F}\) (so \(Y=1,8 X+32)\) a. If the median of the \(X\) distribution is \(\tilde{\mu}\), show that \(1.8 \tilde{\mu}+32\) is the median of the \(Y\) distribution. b. How is the 90th percentile of the \(Y\) distribution related to the 90 th percentile of the \(X\) distribution? Verify your conjecture. c. More generally, if \(Y=a X+b\), how is any particular percentile of the \(Y\) distribution related to the corresponding percentile of the \(X\) distribution?

Short Answer

Expert verified
a: Median is transformed by \(1.8\tilde{\mu}+32\). b: 90th percentile in \(Y\) is \(1.8\) of 90th percentile in \(X\), plus 32. c: Any \(k\)th percentile in \(Y\) is \(aP_{kX} + b\).

Step by step solution

01

Understanding the relationship between X and Y

We are given the equation for converting temperatures from Celsius to Fahrenheit: \Y = 1.8X + 32.\ Here, \(Y\) represents temperatures in Fahrenheit, and \(X\) represents temperatures in Celsius.
02

Median of X distribution (Part a)

The median of \(X\) is denoted as \(\tilde{\mu}\). By the definition of a linear transformation, if \(X\) is transformed by a linear equation to form \(Y\), then the median of \(Y\) denoted as \(\tilde{u}\) will be affected the same way as that transformation. Therefore, \(\tilde{u} = 1.8\tilde{\mu} + 32\). Hence, \(1.8\tilde{\mu} + 32\) is the median of the \(Y\) distribution.
03

Understanding Percentiles (Part b)

Percentiles are measures that indicate the relative standing of a value within a data. The 90th percentile of \(X\), denoted \(P_{90}\) for \(X\), when mapped to \(Y\), will transform linearly just like the median did. Hence, the 90th percentile for \(Y\), \(P_{90}\) for \(Y\), will be \(1.8P_{90X} + 32\).
04

General Relationship of Percentiles (Part c)

For a linear transformation \(Y = aX + b\), any percentile \(P_k\) of \(X\), when transformed, will change linearly to form \(Y\). This means \(P_k\) of \(Y\) will be \(aP_{kX} + b\). This holds true for any percentile: median, quartiles, deciles, etc.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Temperature Conversion
Temperature conversion is a common linear transformation in statistics and science. A common example is converting temperatures from Celsius to Fahrenheit.
The conversion formula is \( Y = 1.8X + 32 \) where \( X \) is the temperature in Celsius, and \( Y \) is the temperature in Fahrenheit.
The formula represents a linear relationship, where \( 1.8 \) is the slope, indicating that an increase of 1 degree Celsius equates to an increase of 1.8 degrees Fahrenheit. The \( + 32 \) part adjusts the y-intercept, setting the Fahrenheit equivalent when Celsius is zero.
Understanding this conversion is crucial in fields where temperature needs to be consistent across different measurement systems. It also highlights broader principles of linear functions and how scale and reference points can be altered without changing underlying relationships.
Medians in Statistics
The median is a measure of central tendency in a statistical data set. It is the middle number in an ordered data set and provides a different perspective than the arithmetic mean.
In the context of linear transformations, such as converting temperature from Celsius to Fahrenheit, the median is transformed similarly to individual data points.
When you have a linear relationship like \( Y = 1.8X + 32 \), if \( \tilde{\mu} \) is the median of \( X \), then the median of \( Y \), which we'll call \( \tilde{u} \), is \( 1.8\tilde{\mu} + 32 \).
Linear transformations do not alter the relative standing of the median within a data set; they just shift and rescale its actual value according to the formula. This understanding aids in preserving the central tendency when converting data to different scales or units.
Linear Equations in Statistics
Linear equations form the backbone of many transformations in statistics, particularly linear transformations. A linear equation in the form \( Y = aX + b \) describes a straight-line relationship between two variables, \( X \) and \( Y \).
For every value \( X \), \( Y \) computed is simply \( X \) scaled by \( a \) and shifted by \( b \). This relationship is key in statistical transformations, like converting percentiles from one distribution to another.
Any specific percentile, say the 90th percentile of \( X \), will transform to \( Y \) using this linear equation. This means the 90th percentile of \( X \) when transformed to \( Y \) will be \( aP_{90X} + b \).
This relationship holds for any percentile and shows how linear transformations preserve the order and relative distances of data points, making them invaluable in data analysis and interpretation.

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Most popular questions from this chapter

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