Question

the accompanying data are the heart rates (in beats per minute) and qt intervals (in milliseconds) for 13 males. the qt interval is a measure of electrical waves of the heart. a lengthened qt interval can indicate heart health problems. find the equation of the regression line. then construct a scatter plot of the data and draw the regression line. then use the regression equation to predict the value of y for each of the given x - values. if meaningful. if the x - value is not meaningful to predict the value of y, explain why not.
(a) ( x = 120 ) bpm
(b) ( x = 68 ) bpm
(c) ( x = 90 ) bpm
(d) ( x = 84 ) bpm
click the icon to view the table of heart rates and qt intervals.
the equation of the regression line is ( hat{y}=square x+square ).
(round to two decimal places as needed.)
heart rates and qt intervals
heart rate, x qt interval, y
61 403
74 363
63 381
70 367
83 342
98 316
67 400
65 384
85 342
78 378
94 328
76 376
88 350
print
done

Explanation:

First, list the paired data (x: heart rate, y: QT interval):
(61,403), (74,363), (63,381), (70,367), (83,342), (98,316), (67,400), (65,384), (85,342), (78,378), (94,328), (76,376), (88,350)

Step1: Calculate sums

$n=13$
$\sum x = 61+74+63+70+83+98+67+65+85+78+94+76+88 = 1002$
$\sum y = 403+363+381+367+342+316+400+384+342+378+328+376+350 = 4630$
$\sum xy = 61×403 + 74×363 + ... +88×350 = 359158$
$\sum x^2 = 61^2+74^2+...+88^2 = 79156$

Step2: Compute slope $b_1$

$b_1 = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2} = \frac{13×359158 - 1002×4630}{13×79156 - 1002^2} ≈ -1.23$

Step3: Compute intercept $b_0$

$\bar{x} = \frac{1002}{13} ≈77.08$, $\bar{y} = \frac{4630}{13}≈356.15$
$b_0 = \bar{y} - b_1\bar{x} ≈356.15 - (-1.23)×77.08≈449.06$

Step4: Predict for valid x

(a) x=120 is outside data range (61-98): invalid
(b) x=68 is within range: $\hat{y}=-1.23×68+449.06≈367.02$
(c) x=90 is within range: $\hat{y}=-1.23×90+449.06≈338.36$
(d) x=84 is within range: $\hat{y}=-1.23×84+449.06≈341.42$

Answer:

Regression equation: $\hat{y}=-1.23x + 449.06$
(a) Not meaningful (x outside data range)
(b) 338.36
(c) 367.02
(d) 341.42

(Note: Exact values may vary slightly with rounding during calculations.)