/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 35 In Exercise \(13.17,\) we consid... [FREE SOLUTION] | 91影视

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Chapter 13: Problem 35

In Exercise \(13.17,\) we considered a regression of \(y=\) oxygen consumption on \(x=\) time spent exercising. Summary quantities given there yield $$ \begin{array}{lll} n=20 & \bar{x}=2.50 & S_{x x}=25 \\ b=97.26 & a=592.10 & s_{e}=16.486 \end{array} $$ a. Calculate \(s_{a+b(2,0)}\), the estimated standard deviation of the statistic \(a+b(2.0)\). b. Without any further calculation, what is \(s_{d+k 3,0\\}}\) and what reasoning did you use to obtain it? c. Calculate the estimated standard deviation of the statistic \(a+b(2.8)\) d. For what value \(x^{*}\) is the estimated standard deviation of \(a+b x^{*}\) smallest, and why?

Short Answer

Expert verified
a) The estimated standard deviation of the statistic \(a+b(2.0)\) can be calculated using the given formula for the standard deviation of the slope intercept. b) The value of \(s_{d + k * 3.0}\) will be equal to \(s_{a + b * 2.0}\) due to transformation properties of standard deviation. c) The estimated standard deviation of the statistic \(a+b(2.8)\) is calculated in a similar manner as part a) by using the standard deviation formula. d) The minimum estimated standard deviation of \(a + b * x^{*}\) is achieved when \(x^{*}\) is equal to \(\overline{x}\), or 2.5, the mean of \(x\) values. This is because the variance, or the squared deviation from the mean is minimized at the mean value.

Step by step solution

01

Calculate \(s_{a+b(2,0)}\)

Begin by calculating the estimated standard deviation of the statistic \(a+b(2.0)\) using the formula \(s_{a+b x} = s_e \sqrt{1/n + (x - \overline{x})^2/S_{xx}}\). However instead of \(x\), use \(2.0\). This will give you \(s_{a+b(2,0)}\).
02

Calculate \(s_{d+k 3,0}\) without further calculations

As \(a+bX\) and \(d+kX\) are linear transformations of each other, their standard deviations will also be the same regardless of the constants and coefficients you add. So, we conclude that \(s_{d+k 3,0}\) will be the same as \(s_{a+b(2,0)}\). This is due to the principle that the standard deviation stays the same under linear transformations.
03

Calculate the estimated standard deviation of the statistic \(a+b(2.8)\)

Again use the formula, \(s_{a+b x} = s_e \sqrt{1/n + (x - \overline{x})^2/S_{xx}}\). However replace \(x\) with \(2.8\) instead of \(2.0\). Calculate this to find \(s_{a+b(2.8)}\).
04

Identify the value of \(x^{*}\) that results in the smallest standard deviation

In order to minimize the standard deviation of \(a+b x^{*}\), the value \(x^{*}\) should be chosen as \(\overline{x}\), the mean of the \(x\) values. This is because the component \((x - \overline{x})^2\) in the formula for \(s_{a+b x}\) represents the variance from the mean. Variance is squared deviation from the mean, so it'll be zero when \(x\) is exactly the mean, which in this case is \(\overline{x} = 2.5\).

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Key Concepts

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

Standard Deviation
Standard deviation is a statistical measure that shows how much variation or dispersion exists in a set of values. It's a crucial concept in regression analysis as it helps in understanding the spread of data points around the mean. A low standard deviation indicates that the values are close to the mean, whereas a high standard deviation shows that the values are more spread out.
  • Standard deviation is often represented as \(s\) or \(\sigma\).
  • It's calculated as the square root of the variance.
  • In the context of regression, standard deviation is used to measure the accuracy of predictions.
The formula for standard deviation in regression models helps us understand how well the fit represents the data and informs adjustments needed for calculations like estimated standard deviation.
Linear Transformation
In mathematics and statistics, a linear transformation is a function between two vector spaces that preserves the operations of addition and scalar multiplication. When discussing linear transformations in relation to standard deviation, the concept is that the scale and units of measurement can change while maintaining relationships between data points.
  • Linear transformations can be expressed in the form \(y = ax + b\).
  • Despite changes in scale or units, the relative distribution (standard deviation) remains the same.
  • This property is utilized in parts of the regression analysis process to simplify calculations.
For example, say you have a transformation \(a + bX\) and another \(d + kX\) -- their standard deviations are equivalent due to the linear nature of transformations, which preserves variance.
Estimated Standard Deviation
Estimated standard deviation is a modification of standard deviation when applied to a sample instead of a whole population, providing an estimate of deviation within data. It鈥檚 important in statistical estimations to gauge the predicted error or uncertainty.
  • In regression, the estimated standard deviation of the statistic \(a + bX\) is especially crucial.
  • It provides an insight into the reliability of regression predictions.
  • The formula \(s_{a+b x} = s_e \sqrt{1/n + (x - \overline{x})^2/S_{xx}}\) is used to find estimated standard deviations where \(s_e\) is the standard error.
This formula helps calculate how the deviation changes with respect to different predictor values \(x\), allowing for assessment of precision and reliability in predictions.
Mean
The mean is one of the most fundamental concepts in statistics, often referred to as the average. It is calculated as the sum of all values divided by the number of values.
  • The mean is represented by \(\overline{x}\) for a sample and \(\mu\) for a population.
  • In the context of regression, it serves as a baseline to measure other data points' deviation.
  • Using the mean minimizes the component \((x - \overline{x})^2\) in deviation formulas, which is crucial for reducing standard deviation.
Understanding the role of mean helps explain why optimal calculations, like minimizing estimated standard deviation, are typically around the mean value. In regression estimates, choosing \(x = \overline{x}\) minimizes deviations, highlighting the stability and balance the mean provides.

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Most popular questions from this chapter

Explain the difference between a confidence interval and a prediction interval. How can a prediction level of \(95 \%\) be interpreted?

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