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A medical researcher wishes to determine how the dosage (in mg) of a drug affects the heart rate of the patient. Dosage Heart rate 0.125 95 0.2 90 0.25 93 0.3 92 0.35 88 0.4 80 0.5 82 1. Find the correlation coefficient & interpret it. 2. Find & interpret the slope. 3. Find & interpret the y-intercept. 4. Give the least squares regression line. A medical researcher wishes to determine how the dosage (in mg) of a drug affects the heart rate of the patient. Dosage Heart rate 0.125 95 0.2 90 0.25 93 0.3 92 0.35 88 0.4 80 0.5 82 1. Find the correlation coefficient & interpret it. βπ. ππππππ π= = βπ. ππ π 2. Find & interpret the slope. ππ π. ππ π = π = βπ. ππ = βππ. ππ ππ π. πππ For every additional mg the heart rate decreases by 38.56 bpm. 3. Find & interpret the y-intercept. π = π β ππ π = ππ. ππ β βππ. ππ . πππ π = πππ. ππ 4. Write LSRL: π―ππππ πΉπππ = πππ. ππ β ππ. ππ π«πππππ RESIDUALS Section 3.2B Residuals β’ Variation in the y values can be effectively explained when the residuals are small β close to the line. β’ Remember a residual = observed β exp. β’πππππ πππ = π β π The equation to explain the relationship between drug dosage and heart rate is shown below. π―ππππ πΉπππ = πππ. ππ β ππ. ππ π«πππππ Dosage Heart rate 0.125 95 0.2 90 0.25 93 0.3 92 0.35 88 0.4 80 0.5 82 1. Find the predicted value for a dosage of 0.4 mg. 2. Find the residual for (0.4, 80). The equation to explain the relationship between drug dosage and heart rate is shown below. Find the residuals for each value. π―ππππ πΉπππ = πππ. ππ β ππ. ππ π«πππππ Dosage Heart rate 0.125 95 0.2 90 0.25 93 0.3 92 0.35 88 0.4 80 0.5 82 * The sum of the residuals is always zero! π¦ π¦βπ¦ Residual Plot β’ It is a scatterplot of the residuals vs the explanatory variable. β’ They help us to assess how well a regression line fits the data. β’ The residual plot should show no obvious pattern β’ The residuals should be relatively small. The equation to explain the relationship between drug dosage and heart rate is shown below. Find the residuals for each value. π―ππππ πΉπππ = πππ. ππ β ππ. ππ π«πππππ π¦ π¦βπ¦ Dosage Heart rate 0.125 95 95.47 -0.47 0.2 90 92.58 -2.578 0.25 93 90.65 2.35 0.3 92 88.72 3.278 0.35 88 86.79 1.206 0.4 80 84.87 -4.866 0.5 82 81.01 0.99 * The sum of the residuals is always zero! Residual Plot Dosage Residual Plot 4 3 2 1 Residuals 0 0 0.1 0.2 0.3 -1 -2 -3 -4 -5 -6 Dosage 0.4 0.5 0.6 Height vs Shoe size β residual plot Good residual plot β show relatively no pattern. Good or Bad Standard Deviation of the Residuals β’ It represents the approximate size of a βtypicalβ or βaverage prediction error (residual). β’ Formula: π π = π π = πππ πππ’πππ 2 πβ2 π¦βπ¦ 2 πβ2 The equation to explain the relationship between drug dosage and heart rate is shown below. Find the standard deviation of the residuals. π―ππππ πΉπππ = πππ. ππ β ππ. ππ π«πππππ π¦ π¦βπ¦ π¦βπ¦ Dosage Heart rate 0.125 95 95.47 -0.47 0.2209 0.2 90 92.59 -2.578 6.6461 0.25 93 90.65 2.35 5.5225 0.3 92 88.72 3.278 10.745 0.35 88 86.79 1.206 1.4544 0.4 80 84.87 -4.866 23.678 0.5 82 81.01 0.99 0.9801 ο₯ο¨ y ο yΛ ο© ο½ nο2 2 se ο½ 49.24726 ο½ 3.14 5 2 Homework *Page 191 (43, 45, 55, 60, 62)