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Chapter 5: Problem 20

The relationship between hospital patient-to-nurse ratio and various characteristics of job satisfaction and patient care has been the focus of a number of research studies. Suppose \(x=\) patient-to-nurse ratio is the predictor variable. For each of the following potential dependent variables, indicate whether you expect the slope of the least-squares line to be positive or negative and give a brief explanation for your choice. a. \(y=\) a measure of nurse's job satisfaction (higher values indicate higher satisfaction) b. \(y=\) a measure of patient satisfaction with hospital care (higher values indicate higher satisfaction) c. \(y=\) a measure of patient quality of care.

Short Answer

Expert verified
a. The slope for nurse's job satisfaction is expected to be negative. \b. The slope for patient satisfaction with hospital care is expected to be negative.\c. The slope for patient's quality of care is also predicted to be negative.

Step by step solution

01

Understand the relationship between variables

Identify and understand the concept of the patient-to-nurse ratio being the predictor or independent variable. Also grasp the fact that the measures of nurse's job satisfaction, patient satisfaction with hospital care, and patient's quality of care are the dependent variables. Recognise that the slope of a least-squares line indicates the kind of relationship between the independent and dependent variables. A positive slope suggests that as the independent variable increases, the dependent variable also increases. Meanwhile, a negative slope indicates that as the independent variable rises, the dependent variable decreases.
02

Predict the direction of the slope for each dependent variable

a. For nurses' job satisfaction: It is reasonable to predict a negative slope because as the patient-to-nurse ratio increases (i.e., more patients per nurse), this might imply more workload for each nurse which might lead to reduced job satisfaction.\b. For patient satisfaction with hospital care: Again, it seems reasonable to predict a negative slope because a high patient-to-nurse ratio could lead to inadequate attention being given to each patient, hence lower patient satisfaction.\c. For patient's quality of care: It would be logical to expect a negative slope because high patient-to-nurse ratios could imply less available time per patient, which could result in reduced quality of care.

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Most popular questions from this chapter

A study was carried out to investigate the relationship between the hardness of molded plastic \((y\), in Brinell units) and the amount of time elapsed since termination of the molding process \((x\), in hours). Summary quantities include \(n=15\), SSResid \(=1235.470\), and SSTo = \(25,321.368\). Calculate and interpret the coefficient of determination.

The accompanying data resulted from an experiment in which weld diameter \(x\) and shear strength \(y\) (in pounds) were determined for five different spot welds on steel. A scatterplot shows a pronounced linear pattern. With \(\sum(x-\bar{x})=1000\) and \(\sum(x-\bar{x})(y-\bar{y})=8577\), the least-squares line is \(\hat{y}=-936.22+8.577 x\). \(\begin{array}{llllll}x & 200.1 & 210.1 & 220.1 & 230.1 & 240.0\end{array}\) \(\begin{array}{llllll}y & 813.7 & 785.3 & 960.4 & 1118.0 & 1076.2\end{array}\) a. Because \(1 \mathrm{lb}=0.4536 \mathrm{~kg}\), strength observations can be re-expressed in kilograms through multiplication by this conversion factor: new \(y=0.4536(\) old \(y) .\) What is the equation of the least-squares line when \(y\) is expressed in kilograms? b. More generally, suppose that each \(y\) value in a data set consisting of \(n(x, y)\) pairs is multiplied by a conversion factor \(c\) (which changes the units of measurement for \(y\) ). What effect does this have on the slope \(b\) (i.e., how does the new value of \(b\) compare to the value before conversion), on the intercept \(a\), and on the equation of the least-squares line? Verify your conjectures by using the given formulas for \(b\) and \(a\). (Hint: Replace \(y\) with \(c y\), and see what happens \- and remember, this conversion will affect \(\bar{y} .\) )

The paper "Effects of Canine Parvovirus (CPV) on Gray Wolves in Minnesota" (Journal of Wildlife Management \([1995]: 565-570\) ) summarized a regression of \(y=\) percentage of pups in a capture on \(x=\) percentage of \(\mathrm{CPV}\) prevalence among adults and pups. The equation of the least-squares line, based on \(n=10\) observations, was \(\hat{y}=62.9476-0.54975 x\), with \(r^{2}=.57\) a. One observation was \((25,70)\). What is the corresponding residual? b. What is the value of the sample correlation coefficient? c. Suppose that \(\mathrm{SSTo}=2520.0\) (this value was not given in the paper). What is the value of \(s_{e} ?\)

Explain why it can be dangerous to use the leastsquares line to obtain predictions for \(x\) values that are substantially larger or smaller than those contained in the sample.

The article "Examined Life: What Stanley H. Kaplan Taught Us About the SAT" (The New Yorker [December 17,2001\(]: 86-92\) ) included a summary of findings regarding the use of SAT I scores, SAT II scores, and high school grade point average (GPA) to predict first-year college GPA. The article states that "among these, SAT II scores are the best predictor, explaining 16 percent of the variance in first-year college grades. GPA was second at \(15.4\) percent, and SAT I was last at \(13.3\) percent." a. If the data from this study were used to fit a leastsquares line with \(y=\) first-year college GPA and \(x=\) high school GPA, what would the value of \(r^{2}\) have been? b. The article stated that SAT II was the best predictor of first-year college grades. Do you think that predictions based on a least-squares line with \(y=\) first-year college GPA and \(x=\) SAT II score would have been very accurate? Explain why or why not.
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