A medical researcher wishes to determine how the dosage (in mg) of a drug affects the heart rate of the patient. Find the correlation coefficient & interpret it. Find & interpret the slope. Find & interpret the y-intercept. Give the least squares regression line. Dosage Heart rate 0.125 95 0.2 90 0.25 93 0.3 92 0.35 88 0.4 80 0.5 82

A medical researcher wishes to determine how the dosage (in mg) of a drug affects the heart rate of the patient. Find the correlation coefficient & interpret it. 𝒓= −𝟓.𝟏𝟕𝟐𝟐𝟗𝟕 𝟔 =−𝟎.𝟖𝟔 Find & interpret the slope. 𝒃=𝒓 𝒔 𝒚 𝒔 𝒙 = −𝟎.𝟖𝟔 𝟓.𝟔𝟓 𝟎.𝟏𝟐𝟔 =−𝟑𝟖.𝟓𝟔 For every additional mg the heart rate decreases by 38.56 bpm. Find & interpret the y-intercept. 𝒂= 𝒚 −𝒃 𝒙 𝒂=𝟖𝟖.𝟓𝟕− −𝟑𝟖.𝟓𝟔 .𝟑𝟎𝟒 𝒂=𝟏𝟎𝟎.𝟐𝟗 Write LSRL: 𝑯𝒆𝒂𝒓𝒕 𝑹𝒂𝒕𝒆 =𝟏𝟎𝟎.𝟐𝟗−𝟑𝟖.𝟓𝟔 𝑫𝒐𝒔𝒂𝒈𝒆 Dosage Heart rate 0.125 95 0.2 90 0.25 93 0.3 92 0.35 88 0.4 80 0.5 82

Residuals Section 3.2B

Residuals Variation in the y values can be effectively explained when the residuals are small – close to the line. Remember a residual = observed – exp. 𝒓𝒆𝒔𝒊𝒅𝒖𝒂𝒍=𝒚− 𝒚

The equation to explain the relationship between drug dosage and heart rate is shown below. 𝑯𝒆𝒂𝒓𝒕 𝑹𝒂𝒕𝒆 =𝟏𝟎𝟎.𝟐𝟗−𝟑𝟖.𝟓𝟔 𝑫𝒐𝒔𝒂𝒈𝒆 Dosage Heart rate 0.125 95 0.2 90 0.25 93 0.3 92 0.35 88 0.4 80 0.5 82 Find the predicted value for a dosage of 0.4 mg. 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. 𝑯𝒆𝒂𝒓𝒕 𝑹𝒂𝒕𝒆 =𝟏𝟎𝟎.𝟐𝟗−𝟑𝟖.𝟓𝟔 𝑫𝒐𝒔𝒂𝒈𝒆 Dosage Heart rate 𝑦 𝑦− 𝑦 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!

Residual Plot It is a scatterplot of the residuals vs the explanatory variable. They help us to assess how well a regression line fits the data. The residual plot should show no obvious pattern The residuals should be relatively small.

The equation to explain the relationship between drug dosage and heart rate is shown below. Graph the residual Plot. Is the linear model appropriate? 𝑯𝒆𝒂𝒓𝒕 𝑹𝒂𝒕𝒆 =𝟏𝟎𝟎.𝟐𝟗−𝟑𝟖.𝟓𝟔 𝑫𝒐𝒔𝒂𝒈𝒆 Dosage Heart rate 𝑦 𝑦− 𝑦 0.125 95 88.723 6.2775 0.2 90 88.658 1.342 0.25 93 88.615 4.385 0.3 92 88.572 3.428 0.35 88 88.529 -0.529 0.4 80 88.486 -8.486 0.5 82 88.4 -6.4 * The sum of the residuals is always zero!

Residual Plot

Height vs Shoe size – residual plot Good residual plot – show relatively no pattern.

Good or Bad

Standard Deviation of the Residuals It represents the approximate size of a “typical” or “average prediction error (residual). Formula: 𝑠 𝑒 = 𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙𝑠 2 𝑛−2 𝑠 𝑒 = 𝑦− 𝑦 2 𝑛−2

The equation to explain the relationship between drug dosage and heart rate is shown below. Find the standard deviation of the residuals. 𝑯𝒆𝒂𝒓𝒕 𝑹𝒂𝒕𝒆 =𝟏𝟎𝟎.𝟐𝟗−𝟑𝟖.𝟓𝟔 𝑫𝒐𝒔𝒂𝒈𝒆 Dosage Heart rate 𝑦 𝑦− 𝑦 𝑦− 𝑦 2 0.125 95 88.723 6.2775 0.2 90 88.658 1.342 0.25 93 88.615 4.385 0.3 92 88.572 3.428 0.35 88 88.529 -0.529 0.4 80 88.486 -8.486 0.5 82 88.4 -6.4

Homework *Page 191 (43, 45, 55, 60, 62)