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 +
To find the correlation coefficient (r) and the line of best fit (y = mx + b) between the meditation time and time to fall asleep, we first need to calculate the mean values for both sets of data.
Mean meditation time (x):
(39 + 46 + 49 + 52 + 53) / 5 = 239 / 5 = 47.8 minutes
Mean time to fall asleep (y):
(28 + 24 + 17 + 4 + 12) / 5 = 85 / 5 = 17 minutes
Next, we will calculate the values needed for the correlation coefficient (r) formula:
Sum of the products of the deviations from the mean:
Σ((x - mean(x))(y - mean(y))) = (39-47.8)(28-17) + (46-47.8)(24-17) + (49-47.8)(17-17) + (52-47.8)(4-17) + (53-47.8)(12-17)
= (-8.8)(11) + (-1.8)(7) + (1.2)(0) + (4.2)(-13) + (5.2)(-5)
= -96.8 - 12.6 + 0 - 54.6 - 26
= -190.8
Sum of the squares of the deviations from the mean for meditation time:
Σ((x - mean(x))^2) = (39-47.8)^2 + (46-47.8)^2 + (49-47.8)^2 + (52-47.8)^2 + (53-47.8)^2
= (-8.8)^2 + (-1.8)^2 + (1.2)^2 + (4.2)^2 + (5.2)^2
= 77.44 + 3.24 + 1.44 + 17.64 + 27.04
= 126.8
Sum of the squares of the deviations from the mean for time to fall asleep:
Σ((y - mean(y))^2) = (28-17)^2 + (24-17)^2 + (17-17)^2 + (4-17)^2 + (12-17)^2
= (11)^2 + (7)^2 + (0)^2 + (-13)^2 + (-5)^2
= 121 + 49 + 0 + 169 + 25
= 364
Now, we can calculate the correlation coefficient (r):
r = Σ((x - mean(x))(y - mean(y))) / √(Σ((x - mean(x))^2) * Σ((y - mean(y))^2))
= -190.8 / √(126.8 * 364)
= -190.8 / √(46113.92)
= -190.8 / 214.71
r ≈ -0.888
Therefore, the correlation coefficient between meditation time and time to fall asleep is approximately -0.888, indicating a strong negative correlation.
To find the equation of the line of best fit (y = mx + b), we can use the following formulas:
m = r * (SD(y) / SD(x))
b = mean(y) - (m * mean(x))
Where:
r = -0.888
SD(x) = √(Σ((x - mean(x))^2) / (n-1))
SD(y) = √(Σ((y - mean(y))^2) / (n-1))
Calculating SD(x):
SD(x) = √(126.8 / 4)
= √31.7
≈ 5.63
Calculating SD(y):
SD(y) = √(364 / 4)
= √91
≈ 9.54
Now, calculate m:
m = -0.888 * (9.54 / 5.63)
= -0.888 * 1.696
m ≈ -1.507
Now, calculate b:
b = 17 - (-1.507 * 47.8)
= 17 + 71.89
b ≈ 88.89
Therefore, the equation of the line of best fit is:
y = -1.507x + 88.89
This equation can be used to predict the time to fall asleep based on the meditation time.