A sleep specialist wants to know if meditating 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
r =
y= x+
If someone was to mediate for 20 minutes, what would the regression line predict for the amount of time it would take for them to fall asleep?
To find the regression line equation and use it to predict the time it would take for someone to fall asleep after meditating for 20 minutes, we can use the data given in the study:
Meditation Time (x): 39, 46, 49, 52, 53
Time to fall asleep (y): 28, 24, 17, 4, 12
First, we need to calculate the mean of x and y:
Mean of x = (39+46+49+52+53)/5 = 47.8
Mean of y = (28+24+17+4+12)/5 = 17
Next, we need to calculate the sums of squares:
Σ(xy) = (39*28) + (46*24) + (49*17) + (52*4) + (53*12) = 6567
Σ(x^2) = 39^2 + 46^2 + 49^2 + 52^2 + 53^2 = 12059
Σ(y^2) = 28^2 + 24^2 + 17^2 + 4^2 + 12^2 = 2681
Now, we can calculate the slope (b) and the y-intercept (a) of the regression line equation:
b = (5*6567 - 39*17*47.8) / (5*12059 - 39^2) ≈ -0.58
a = 17 - (-0.58*47.8) ≈ 44.07
Therefore, the regression line equation is:
y = -0.58x + 44.07
To predict the time it would take for someone to fall asleep after meditating for 20 minutes, we substitute x = 20 into the regression line equation:
y = -0.58*20 + 44.07
y ≈ 32.87
Therefore, the regression line predicts that it would take approximately 32.87 minutes for someone to fall asleep after meditating for 20 minutes.