Bhisham C. Gupta - Statistics and Probability with Applications for Engineers and Scientists Using MINITAB, R and JMP

Figure 2.9.1MINITAB printout of scatter plot for the data in Table 2.9.1.

The correlation coefficient is a dimensionless measure that can attain any value in the interval . As the strength of the association between the two variables grows, the absolute value of r approaches 1. Thus, when there is a perfect association between the two variables, or , depending on whether the association is positive or negative. In other words, , if the two variables are moving in the same direction, and , if the two variables are moving in the opposite direction.

Perfect association means that if we know the value of one variable, then the value of the other variable can be determined without any error. The other special case is when , which does not mean that there is no association between the two variables, but rather that there is no linear association between the two variables. As a general rule, the linear association is weak, moderate, or strong when the absolute value of is less than 0.3, between 0.3 and 0.7, or greater than 0.7, respectively. For instance, if ( 2.9.1) is computed for the data in Table 2.9.1, then . Hence, we can conclude that the association between the two variables X and Y is strong.

1 Enter the pairs of data in columns C1 and C2. Label the columns X and Y.

2 From the Menu bar select Graph Scatterplot. This prompts a dialog box to appear on the screen. In this dialog box, select scatterplot With Regression and click OK. This prompts the following dialog box to appear:In this dialog box, under the X and Y variables, enter the columns in which you have placed the data. Use the desired options and click OK. The Scatter plot shown in Figure 2.9.1appears in the Session window.

3 For calculating the correlation coefficient, select from the Menu bar Stat Basic Statistics Correlation. Then, enter the variables C1 and C2 in the dialog box.

USING R

We can use a built in ‘plot()’ function in R to generate scatter plots. Extra arguments such as ‘pch’ and ‘cex’ can be used to specify the plotting symbol and size of the symbol, respectively. Finally, the function ‘abline()’ can be used to embed the trend line to the scatter plot as follows. The function ‘cor()’ can be used to calculate the Pearson correlation coefficient. The whole task can be completed by running the following R code in the R Console window.

x = c(195,180,220,160,200,220,200,183,139,155) y = c(130,128,138,122,140,148,142,127,116,123) #To plot the data in a scatter plot plot(x, y, pch = 20, cex = 2, main = ‘Scatterplot for Cholesterol Level and Systolic Blood Pressure Data’, xlab = ‘Cholesterol Level’, ylab = ‘Systolic Blood Pressure’) #To add a trend line abline(lm(y x), col = ‘red’) #To calculate the Pearson correlation coefficient cor(x, y) 0.9242063

The resulting R scatter plot for the data in Table 2.9.1looks exactly the same as in the MINTAB printout in Figure 2.9.1.

1 The following data give the heights (cm) and weights (lb) of 10 male undergraduate students:Heights170167172171165170168172175172Weights182172179172174179188168185169Draw a scatter plot for these data. By observing this scatter plot, do you expect the correlation between heights and weights to be positive or negative?Determine the correlation coefficient between the heights and weights.

2 The following data give the final exam scores in biology and chemistry of eight science majors:Biology scores8588789289837995Chemistry scores9084869594898487Draw a scatter plot for these data. By observing this scatter plot, do you expect the correlation between biology and chemistry scores to be approximately 1, 1, or 0?Determine the correlation coefficient between the biology and chemistry scores.

3 The following data show the experience (in years) and yearly salaries (in thousands of dollars) of 10 engineers:Experience101281561114161512Salaries98959711088102120128105104Construct a scatter plot for these data. By observing this scatter plot, do you expect the correlation between experience and salaries to be positive or negative?Determine the correlation coefficient between the experience and salaries. Is the value of correlation coefficient consistent with what you concluded in part (a)?

4 The following scores give two managers' assessments of ten applicants for the position of a senior engineer:Manager 17897989796Manager 28699879878Construct a scatter plot for these data. By observing this scatter plot, do you expect the correlation between assessments of managers 1 and 2 to be positive or negative?Determine the correlation coefficient between the assessment managers 1 and 2. Is the value of correlation coefficient consistent with what you concluded in part (a)?

Case Study 1( St. Luke's Hospital data ) 1In the fast‐paced world of health care, small fluctuations in the quarterly patient (mean) satisfaction scores may not raise any red flags. But that quickly changed for St. Luke's Hospital in Cedar Rapids, Iowa, during a leadership retreat in spring 2004. Here, managers received a shocking surprise, the effects of which are still driving improvement efforts today. The hospital's inpatient satisfaction measures, which appeared flat, had actually dipped to the 64th percentile when compared to other hospitals in the country.

People were shocked because they thought St. Luke's should be in the 80th or 90th percentiles, explains Kent Jackson, director of children's specialty services, and the leader of the hospital's patient and family experience team. “What became more significant was that in the second quarter of 2004, the hospital dropped to the 49th percentile [for inpatient satisfaction]. So, about the time that people were shocked, it was about to get worse,” Jackson recalls.

St. Luke's Hospital is a fully accredited 560‐bed hospital with more than 2600 associates and 381 physicians who provide 24‐hour coverage in east central Iowa. Founded in 1884 as the community's first hospital, today St. Luke's is a member of the Iowa Health System. The hospital offers a comprehensive array of inpatient services including medical/surgical, cardiology, oncology, neurology, inpatient and outpatient behavioral health, and neonatal intensive care; it delivers a broad range of diagnostic and outpatient services, and provides a 24‐hour trauma center and an air ambulance service. St. Luke's five‐point strategy to gain patient satisfaction focused on: