Question

Using the empirical rule, 68% of male sleep amounts should be between what two values? Either show work or explain how your answer was calculated. (need to be able to show the work as a typed out problem in Word, so if possible type out the solutions and formulas so that there are easier to read)

7

5

8

8

6

8

8

8

10

7

7

4

9

8

7

8

8

10

Answer 1

The empirical rule states that 68% of data lies within 1 standard deviation of the mean, this can be understood by the image shown below:

Now to find the values we need to calculate the mean and standard deviation of the given data as:

Mean = (7 + 5 + 8 + 8 + 6 + 8 + 8 + 8 + 10 + 7 + 7 + 4 + 9 + 8 + 7 + 8 + 8 + 10)/18
= 136/18
Mean = 7.5556

And the standard deviation is calculated as:

Standard Deviation s = √(1/18 - 1) x ((7 - 7.5556)2 + ( 5 - 7.5556)2 + ( 8 - 7.5556)2 + ( 8 - 7.5556)2 + ( 6 - 7.5556)2 + ( 8 - 7.5556)2 + ( 8 - 7.5556)2 + ( 8 - 7.5556)2 + ( 10 - 7.5556)2 + ( 7 - 7.5556)2 + ( 7 - 7.5556)2 + ( 4 - 7.5556)2 + ( 9 - 7.5556)2 + ( 8 - 7.5556)2 + ( 7 - 7.5556)2 + ( 8 - 7.5556)2 + ( 8 - 7.5556)2 + ( 10 - 7.5556)2)
= √(1/17) x ((-0.5556)2 + (-2.5556)2 + (0.4444)2 + (0.4444)2 + (-1.5556)2 + (0.4444)2 + (0.4444)2 + (0.4444)2 + (2.4444)2 + (-0.5556)2 + (-0.5556)2 + (-3.5556)2 + (1.4444)2 + (0.4444)2 + (-0.5556)2 + (0.4444)2 + (0.4444)2 + (2.4444)2)
= √(0.0588) x ((0.30869136) + (6.53109136) + (0.19749136) + (0.19749136) + (2.41989136) + (0.19749136) + (0.19749136) + (0.19749136) + (5.97509136) + (0.30869136) + (0.30869136) + (12.64229136) + (2.08629136) + (0.19749136) + (0.30869136) + (0.19749136) + (0.19749136) + (5.97509136))
= √(0.0588) x (38.44444448)
= √(2.260533335424)
= 1.5038

Thus the Values at -1 and +1 standard deviation is calculated as;

So, The values within which 68% of male sleep amounts should be

{ 6.0518, 9.0594}

Approsimately between {6 and 9}