91Ó°ÊÓ

a. Let \(a\) and \(b\) be constants and let \(y_{i}=a x_{i}+b\) for \(i=1,2, \ldots, n\). What are the relationships between \(\bar{x}\) and \(\bar{y}\) and between \(s_{x}^{2}\) and \(s_{y}^{2}\) ? b. The Australian army studied the effect of high temperatures and humidity on human body temperature (Neural Network Training on Human Body Core Temperature Data, Technical Report DSTO TN-0241, Combatant Protection Nutrition Branch, Aeronautical and Maritime Research Laboratory). They found that, at \(30^{\circ} \mathrm{C}\) and \(60 \%\) relative humidity, the sample average body temperature for nine soldiers was \(38.21^{\circ} \mathrm{C}\), with standard deviation \(.318^{\circ} \mathrm{C}\). What are the sample average and the standard deviation in \({ }^{\circ} \mathrm{F}\) ?

Step by step solution

01

Understanding the Linear Transformation

Given the linear relationship, \(y_i = ax_i + b\), we can express \(\bar{y}\) in terms of \(\bar{x}\). The average \(\bar{y}\) is calculated as \(\bar{y} = \frac{1}{n} \sum y_i = \frac{1}{n} \sum (a x_i + b)\). This simplifies to \(a\bar{x} + b\) using properties of sums.

02

Determining the Relationship Between Variances

The variance \(s_y^2\) can be found from \(s_x^2\) since variance is affected by scaling but not by shifting. Given, \(y_i = ax_i + b\), the variance \(s_y^2 = a^2 s_x^2\). The constant \(b\) does not affect the variance.

03

Conversion from Celsius to Fahrenheit

For part b, use the conversion formula \(F = \frac{9}{5}C + 32\) to convert the average temperature \(38.21\degree C\) to Fahrenheit. Substitute to find \(\bar{F} = \frac{9}{5} \times 38.21 + 32 = 100.778\degree F\).

04

Calculating the Standard Deviation in Fahrenheit

Use the relationship between Celsius and Fahrenheit to convert the standard deviation. Since standard deviation measures spread and not location, conversion just involves multiplying by the slope (9/5) without adding the 32. Thus, \(s_F = \frac{9}{5} \times 0.318 = 0.5724\degree F\).

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Key Concepts

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

Linear Transformations

Linear transformations in statistics involve using a linear equation to convert one set of numbers to another. In this context, it's expressed as \(y_i = ax_i + b\), where \(x_i\) is your original variable, and \(y_i\) is the transformed variable.
This transformation can adjust data through multiplying by \(a\) (scaling) and adding \(b\) (shifting). Such transformations are useful in standardizing data or converting units.

• Scaling (by \(a\)): Multiplies every data point, affecting the mean and variance.
• Shifting (by \(b\)): Adds a constant to each data point, changing the mean but not the variance.
  

The effect on the mean after transformation is that \(\bar{y} = a\bar{x} + b\), demonstrating both scaling and shifting. Variance is affected only by scaling: \(s^2_y = a^2 s^2_x\).
Understanding these effects helps in predicting how data distributions change with transformations.

Sample Mean

The sample mean is a key measure of central tendency, representing the average of a set of data points. When a linear transformation \(y_i = ax_i + b\) is applied, the mean \(\bar{y}\) of the transformed data can be determined from the original mean \(\bar{x}\).
Using the linear transformation properties, we evaluate \(\bar{y}\) as:

• \(\bar{y} = a\bar{x} + b\)

Here, scaling by \(a\) directly influences the magnitude of the mean, while adding \(b\) shifts the entire distribution. This is crucial for interpreting transformed data in practical scenarios like normalizing scores or adjusting monetary values.
By understanding the adjustments to the sample mean under different conditions, analysts can accurately interpret results regardless of transformation.

Variance Transformation

Variance measures data spread, and its behavior under a linear transformation \(y_i = ax_i + b\) is specific. When dealing with variability, it's essential to note that only the scaling factor \(a\) affects variance:

• Transforming variance: \(s^2_y = a^2 s^2_x\)

Shifting by \(b\) does not alter the variance because variance is about spread, not about absolute position. Thus, all shifts affect only mean but leave variance intact.
Such understanding is particularly pertinent when data requires conversion for consistency, like converting units in physics or financial models. It assures that only true variability, not arbitrary adjustments, is considered in analyses.

Temperature Conversion

Transforming temperatures between scales, such as Celsius to Fahrenheit, is a practical application of linear transformations. The formula \(F = \frac{9}{5}C + 32\) adjusts measurements between the two widely used scales.
The conversion entails:

• Scaling by \(\frac{9}{5}\): impacts spread, relevant for measures like standard deviation.
• Shifting by 32: adjusts base, altering mean but not spread.

The mean temperature and variability (standard deviation) in \(^{\circ}F\) is calculated by:

• Mean: \(\bar{F} = \frac{9}{5} \times 38.21 + 32 = 100.778^{\circ}F\)
• Standard Deviation: \(s_F = \frac{9}{5} \times 0.318 = 0.5724^{\circ}F\)

These computations allow consistent interpretation across different units, vital in scientific and medical studies, to facilitate comparisons and conclusions.