2 A chemical company wishing to study the effect of extraction time on the

# 2 a chemical company wishing to study the effect of

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2. A chemical company, wishing to study the effect of extraction time on the efficiency of an extraction operation, obtained the data shown in the following table. Calculate r Extraction time X Extraction efficiency (%) y 27 57 45 64 41 80 19 46 35 62 39 72 19 52 49 77 15 57 31 68
174 Correlation coefficient Command: This command is used to find the correlation coefficient between two variables. R=correcoef(A,B) Where A is a vector of variance of first variable and B is the vector of values of second variable. This command returns the correlation coefficient between two variables A and B as a matrix. (i) (1,1) Elements is the correlation coefficient between A and A (ii) (1,2) Elements is the correlation coefficient between A and B (iii) (2,1) Elements is the correlation coefficient between B and A (iv) (2,2) Elements is the correlation coefficient between B and B Example: >> a=[ 70 92 80 74 65 83]; >>b=[ 74 84 63 87 78 90]; >> r=corrcoef(a,b) r= 1.0000 0.6474 0.6474 1.0000 Tutorial problems: 1. Show that for the Bivariate normal distribution Independence implies zero correlation. 2. The following table gives the data on corn yield x and peanut yield (y) (mt/ha) for eight different types of soil X: 2.4 3.4 4.6 3.7 2.2 3.3 4.0 2.1 Y: 1.33 2.12 1.80 1.65 2.00 1.76 2.11 1.63 Find the correlation between corn yields x a peanut yield and interpret the value.
175 3. The following table gives the concentrations of two different pollutants ozone concentration x (ppm) and secondary carbon concentration y (ug/m 3 ). Find the correlation coefficient between ozone concentration and secondary carbon concentration and interpret the values X: 0.666 0.088 0.120 0.050 0.162 0.186 0.057 0.100 Y: 4.6 11.6 9.5 6.3 13.8 15.4 2.5 11.8 4. The following table gives the data on x=% light absorption at 5800A and y=peak photo voltage X: 4.0 8.7 12.7 19.1 21.4 24.6 28.9 29.8 30.5 Y: 0.12 0.28 0.55 0.68 0.85 1.02 1.15 1.34 1.29 a) Construct a scatter plot of this data. What does it suggest? b) Assume that the simple linear regression model is appropriate; obtain the equation of the estimated regression line. c) Predict photo voltage when % absorption is 19% and compute the value of the corresponding residual.
176 5. Compute and interpret the correlation coefficient for the following grades of 6 students selected at random. Mathematics grade: 70 92 80 74 65 83 English grade: 74 84 63 87 78 90 6. The following data represent the chemistry grades for a random sample of 12 freshmen at a certain college along with their scores on an intelligence test administered while they were still senior in high school:
177 Student: 1 2 3 4 5 6 7 8 9 10 11 12 Test score x: 65 50 55 65 55 70 65 70 55 70 50 55 Chemistry y: 85 74 76 90 85 87 94 98 81 91 76 74 Compute and interpret the sample correlation coefficient. 7. Heavy metals can inhibit the biological treatment of waste in municipal treatment plants. Monthly measurements were made at a state of the art treatment plant of the amount of chromium 9ug/l) in both the influent and effluent. Influent: 250 290 270 100 300 410 110 130 1100 Effluent: 19 10 17 11 70 60 18 30 180 a) Make a scatter plot b) Make a scatter plot after taking the natural logarithm of both variables c) Calculate the correlation coefficient, r, in part (a) and Part (b) d) Comment on the appropriateness of r in each case.
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