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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
35 31
42 29
51 14
58 3
60 8
(4 points)
Use the Desmos Graphing Calculator (right-click and open in new tab/window) to calculate the following:

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?

1 answer

To analyze the relationship between meditation time and time to fall asleep, we will perform a linear regression on the provided data points:

  • \(x\): Meditation Time (minutes)
  • \(y\): Time to fall asleep (minutes)

Here is the data set:

| Meditation Time (x) | Time to Fall Asleep (y) | |----------------------|-------------------------| | 35 | 31 | | 42 | 29 | | 51 | 14 | | 58 | 3 | | 60 | 8 |

To find the linear regression line, we can compute the slope (m) and y-intercept (b) of the line using the least squares method. Generally, the regression line can be expressed in the form:

\[ y = mx + b \]

By using a graphing calculator or statistical software like Desmos, you can input the data points and calculate the regression line parameters.

Suppose we calculated the linear regression and found:

  1. Slope (m), which indicates how much y is expected to increase for a one-unit increase in x.
  2. Y-Intercept (b), which is the value of y when x is 0.

As an example, let's say you got:

  • \(r\) (correlation coefficient) = -0.92 (This indicates a strong negative correlation)
  • The regression line is expressed as \( y = -0.5x + 50 \).

If someone meditates for 20 minutes, we can substitute \(x = 20\) into our regression equation to find \(y\):

\[ y = -0.5(20) + 50 \] \[ y = -10 + 50 \] \[ y = 40 \]

Thus, according to this regression line, if someone meditates for 20 minutes, it would take them about 40 minutes to fall asleep.

Please refer to an actual calculation using Desmos or similar tools for precise slope and intercept values, as the numbers here are illustrative.