A sleep specialist wants to know if meditation before you go to sleep can help you fall asleep. She conducted a study where people would record how many minutes they meditated before getting into bed and how many minutes it took them to fall asleep. Meditation Time Time to fall asleep 39 28 46 24 49 17 52 4 53 12 calculate the following: r= y= x + If someone was to meditate for 20 minutes, what would the regression line predict for the amount of time it would take for them to fall asleep?
To calculate the correlation coefficient (r) and the regression equation (y = mx + b) for the given data, we will use the formulas:
n = 5 (number of data points)
Σx = 239 (sum of meditation times)
Σy = 85 (sum of time to fall asleep)
Σxy = 887.5 (sum of the product of meditation time and time to fall asleep)
Σx^2 = 11307 (sum of squared meditation times)
Σy^2 = 2449 (sum of squared times to fall asleep)
r = (nΣxy - ΣxΣy) / √[(nΣx^2 - (Σx)^2)(nΣy^2 - (Σy)^2)]
r = (5 * 887.5 - 239 * 85) / √[(5 * 11307 - 239^2)(5 * 2449 - 85^2)]
r = (4437.5 - 20315) / √[(56535 - 57121)(12245 - 7225)]
r = -15877.5 / √[(-586)(5020)]
r = -15877.5 / √(-2942920)
r = -15877.5 / -1716.192
r = 9.256
Now, we can calculate the regression equation (y = mx + b):
m = r * (Σy/Σx)
m = 9.256 * (85/239)
m = 3.291
b = (Σy - mΣx) / n
b = (85 - 3.291 * 239) / 5
b = (85 - 786.9) / 5
b = -140.98
Therefore, the regression equation is y = 3.291x - 140.98.
To predict the time it would take for someone to fall asleep if they meditated for 20 minutes, we substitute x = 20 into the regression equation:
y = 3.291(20) - 140.98
y = 65.82 - 140.98
y = -75.16
The regression line predicts that if someone meditates for 20 minutes, they would likely take approximately -75.16 minutes to fall asleep. However, this prediction is not practical because time to fall asleep cannot be negative. In this case, it would indicate that the person would fall asleep around 1 hour and 15 minutes before they started meditating, which is illogical.