91Ó°ÊÓ

A study conducted at. VPI\&SU to determine if certain static arm-strength measures have an influence on the "dynamic lift" characteristics of an individual. Twenty-five individuals were subjected to strength tests and then were asked to perform a weight-lifting test in which weight was dynamically lifted overhead. The data are given here. (a) Estimate \(\alpha\) and 0 for the linear regression curve \(\mu_{Y \mid x}=a+0 x\) (b) Find a point estimate of \(\mu_{Y \mid 30}\). (c) Plot the residuals versus the \(X\) s (arm strength). Comment. $$ \begin{array}{c|c|c} & \text { Arm } & \text { Dynamic } \\ \text { Individual } & \text { Strength, } \mathrm{x} & \text { Lift, } y \\ \hline 1 & 17.3 & 71.7 \\ 2 & 19.3 & 48.3 \\ 3 & 19.5 & 88.3 \\ 4 & 19.7 & 75.0 \\ 5 & 22.9 & 91.7 \\ 6 & 23.1 & 100.0 \\ 7 & 26.4 & 73.3 \\ 8 & 26.8 & 65.0 \\ 9 & 27.6 & 75.0 \\ 10 & 28.1 & 88.3 \\ 11 & 28.2 & 68.3 \\ 12 & 28.7 & 96.7 \end{array} $$ $$ \begin{array}{c|c|c} & \text { Arm } & \text { Dynamic } \\ \text { Individual } & \text { Strength, } \boldsymbol{x} & \text { Lift, } \boldsymbol{y} \\ \hline 13 & 29.0 & 76.7 \\ 14 & 29.6 & 78.3 \\ 15 & 29.9 & 60.0 \\ 16 & 29.9 & 71.7 \\ \mathbf{1 7} & \mathbf{3 0 . 3} & 85.0 \\ 18 & 31.3 & 85.0 \\ \mathbf{1 9} & 36.0 & 88.3 \\ 20 & 39.5 & 100.0 \\ 21 & 40.4 & 100.0 \\ 22 & 44.3 & 100.0 \\ 23 & 44.6 & 91.7 \\ 24 & 50.4 & 100.0 \\ 25 & 55.9 & 71.7 \end{array} $$

A study was made on the amount of converted sugar in a certain process at various temperatures. The data were coded and recorded as follows: $$ \begin{array}{cc} \text { Temperature, } \boldsymbol{x} & \text { Converted Sugar, } \boldsymbol{v} \\ \hline 1.0 & 8.1 \\ 1.1 & 7.8 \\ 1.2 & 8.5 \\ 1.3 & 9.8 \\ 1.4 & 9.5 \\ 1.5 & 8.9 \\ 1.6 & 8.6 \\ 1.7 & 10.2 \\ 1.8 & 9.3 \\ 1.9 & 9.2 \\ 2.0 & 10.5 \end{array} $$ (a) Estimate the linear regression line. (b) Estimate the mean amount of converted sugar produced when the coded temperature is \(1.75 .\) (c) Plot the residuals versus temperature. Comment.

A mathematics placement test is given to all entering freshmen at a small college. A student who receives a grade below 35 is denied admission to the regular mathematics course and placed in a remedial class. The placement test scores and the final grades for 20 students who took the regular course were recorded as follows: $$ \begin{array}{cc} \text { Placement Test } & \text { Course Grade } \\ \hline 50 & 53 \\ 35 & 41 \\ 35 & 61 \\ 40 & 56 \\ 55 & 68 \\ 65 & 36 \\ 35 & 11 \\ 60 & 70 \\ 90 & 79 \\ 35 & 59 \\ 90 & 54 \\ 80 & 91 \\ 60 & 48 \\ 60 & 71 \\ 60 & 71 \\ 40 & 47 \\ 55 & 53 \\ 50 & 68 \\ 65 & 57 \\ 50 & 79 \end{array} $$ (a) Plot a scatter diagram. (b) Find the equation of the regression line to predict course grades from placement test scores. (c) Graph the line on the scatter diagram. (d) If 60 is the minimum passing grade, below which placement test score should students in the future be denied admission to this course?

A study was made by a retail merchant to determine the relation between weekly advertising expenditures and sales. The following data were recorded: $$ \begin{array}{cc} \text { Advertising Costs (\$) } & \text { Sales (\$) } \\ \hline 40 & 385 \\ 20 & 400 \\ 25 & 395 \\ 20 & 365 \\ 30 & 475 \\ 50 & 440 \\ 40 & 490 \\ 20 & 420 \\ 50 & 560 \\ 40 & 525 \\ 25 & 480 \\ 50 & 510 \end{array} $$ (a) Plot a scatter diagram. (b) Find the equation of the regression line to predict weekly sales from advertising expenditures. (c) Estimate the weekly sales when advertising costs are \(\$ 35 .\) (d) Plot the residuals versus advertising costs. Comment.

The following data were collected to determine the relationship between pressure and the corresponding scale reading for the purpose of calibration. $$ \begin{array}{cc} \text { Pressure, } x \text { (lb/sqin.) } & \text { Scale Reading, } y \\ \hline 10 & 13 \\ 10 & 18 \\ \text { to } & 16 \\ 10 & 15 \\ 10 & 20 \\ 50 & 86 \\ 50 & 90 \\ 50 & 88 \\ 50 & 88 \\ 50 & 92 \end{array} $$ (a) Find the equation of the regression line. (b) The purpose of calibration in this application is to estimate pressure from an observed scale reading-Estimate the pressure for a scale reading of 54 using \(\dot{x}=(54-a) / b\)-

A study of the amount of rainfall and the quantity of air pollution removed produced the following data: $$ \begin{array}{cc} \text { Daily Rainfall, } & \text { Particulate Removed, } \\ x(0.01 \mathrm{~cm}) & y\left(\mu \mathrm{g} / \mathrm{m}^{3}\right) \\ \hline 4.3 & 126 \\ 4.5 & 121 \\ 5.9 & 116 \\ 5.6 & 118 \\ 6.1 & 114 \\ 5.2 & 118 \\ 3.8 & 132 \\ 2.5 & 141 \\ 7.5 & 108 \end{array} $$ (a) Find the equation of the regression line to predict the particulate removed from the amount of daily rainfall. (b) Estimate the amount of particulate rernoved when the daily rainfall is \(x=4.8\) units.

11.12 A study was done to study the effect of ambient temperature \(x\) on the electric power consumed by a chemical plant, \(y .\) Other factors were held constant and the data were collected from an experimental pilot plant. (a) Plot the data. (b) Estimate the slope and intercept in a simple linear regression model. (c) Predict power consumption for an ambient temperature of \(65^{\circ} \mathrm{F}\). $$ \begin{array}{cc|cc} y \text { (BTU) } & x\left({ }^{\circ} \mathrm{F}\right) & y \text { (BTU) } & x\left({ }^{\circ} \mathrm{F}\right) \\ \hline 250 & 27 & 265 & 31 \\ 285 & 45 & 298 & 60 \\ 320 & 72 & 267 & 31 \\ 295 & 58 & 321 & 74 \end{array} $$

Given the data set $$ \begin{array}{cccc} y & x & y & x \\ \hline 7 & 2 & 40 & 10 \\ 50 & 15 & 70 & 20 \\ 100 & 30 & & \end{array} $$ (a) Plot the data. (b) Fit a regression line "through the origin." (c) Plot the regression line on the graph with the data. (d) Give a general formula for (in terms of the \(y_{i}\) and the slope \(b\) ) the estimator of \(\sigma^{2}\). (e) Give a formula for \(\operatorname{Var}\left(\hat{y}_{i}\right) ; i=1,2, \ldots, n\) for this case. (I) Plot \(95 \%\) confidence limits Tor the mean response on the graph around the regression line.

(a) Find the least squares estimate for the parameter \(\beta\) in the linear equation \(\mu_{Y \mid x}=3 x\). (b) Estimate the regression line passing through the origin for the following data: $$ \begin{array}{c|cccccc} x & 0.5 & 1.5 & 3.2 & 4.2 & 5.1 & 6.5 \\ \hline y & 1.3 & 3.4 & 6.7 & 8.0 & 10.0 & 13.2 \end{array} $$

The following data were obtained in a study of the relationship between the weight and chest size of infants at birth: $$ \begin{array}{cc} \text { Weight (kg) } & \text { Chest Size (cm) } \\ \hline 2.75 & 29.5 \\ 2.15 & 26.3 \\ 4.41 & 32.2 \\ 5.52 & 36.5 \\ 3.21 & 27.2 \\ 4.32 & 27.7 \\ 2.31 & 28.3 \\ 4.30 & 30.3 \\ 3.71 & 28.7 \end{array} $$ (a) Calculate \(r\). (b) Test the null hypothesis that \(p=0\) against the alternative that \(p>0\) at the 0.0 i level of significance. (c) What percentage of the variation in the infant chest sizes is explained by difference in weight?