High School

A study of the effects of smoking on sleep patterns is conducted. The measure observed is the time, in minutes, that it takes to fall asleep. These data are obtained:

Smokers: 69.3, 56.0, 22.1, 47.6, 53.2, 48.1, 52.7, 34.4, 60.2, 43.8, 23.2, 13.8, 28.4

Nonsmokers: 28.6, 25.1, 26.4, 34.9, 29.8, 31.8, 41.6, 21.1, 36.0, 37.9, 13.9, 38.5, 30.2, 30.6

What is the standard deviation of both groups?

A. \(s(\text{smokers}) = 18.93\), \(s(\text{nonsmokers}) = 7.13\)

B. \(s(\text{smokers}) = 16.93\), \(s(\text{nonsmokers}) = 9.13\)

C. \(s(\text{smokers}) = 7.13\), \(s(\text{nonsmokers}) = 16.93\)

D. \(s(\text{smokers}) = 16.93\), \(s(\text{nonsmokers}) = 7.13\)

Answer :

Final answer:

The standard deviation of the smokers group is approximately 16.57 and the standard deviation of the nonsmokers group is approximately 6.89.

Explanation:

To calculate the standard deviation for both groups, we need to follow the steps mentioned earlier:

  1. Calculate the mean of the data set for each group.
  2. Subtract the mean from each data value and square the result.
  3. Calculate the mean of the squared differences for each group.
  4. Take the square root of the mean of the squared differences for each group.

Let's calculate the standard deviation for both groups:

Smokers:

Mean = (69.3 + 56.0 + 22.1 + 47.6 + 53.2 + 48.1 + 52.7 + 34.4 + 60.2 + 43.8 + 23.2 + 13.8 + 28.4) / 13 = 43.6

Subtract the mean from each data value and square the result:

(69.3 - 43.6)^2 = 669.64

(56.0 - 43.6)^2 = 154.24

(22.1 - 43.6)^2 = 459.21

(47.6 - 43.6)^2 = 16.00

(53.2 - 43.6)^2 = 92.16

(48.1 - 43.6)^2 = 20.25

(52.7 - 43.6)^2 = 82.81

(34.4 - 43.6)^2 = 84.64

(60.2 - 43.6)^2 = 275.84

(43.8 - 43.6)^2 = 0.04

(23.2 - 43.6)^2 = 420.64

(13.8 - 43.6)^2 = 912.04

(28.4 - 43.6)^2 = 230.40

Calculate the mean of the squared differences:

(669.64 + 154.24 + 459.21 + 16.00 + 92.16 + 20.25 + 82.81 + 84.64 + 275.84 + 0.04 + 420.64 + 912.04 + 230.40) / 13 = 274.42

Take the square root of the mean of the squared differences:

sqrt(274.42) ≈ 16.57

Nonsmokers:

Mean = (28.6 + 25.1 + 26.4 + 34.9 + 29.8 + 31.8 + 41.6 + 21.1 + 36.0 + 37.9 + 13.9 + 38.5 + 30.2 + 30.6) / 14 = 30.79

Subtract the mean from each data value and square the result:

(28.6 - 30.79)^2 = 4.79

(25.1 - 30.79)^2 = 32.49

(26.4 - 30.79)^2 = 18.09

(34.9 - 30.79)^2 = 17.16

(29.8 - 30.79)^2 = 0.98

(31.8 - 30.79)^2 = 1.04

(41.6 - 30.79)^2 = 116.64

(21.1 - 30.79)^2 = 93.21

(36.0 - 30.79)^2 = 27.04

(37.9 - 30.79)^2 = 50.41

(13.9 - 30.79)^2 = 287.04

(38.5 - 30.79)^2 = 59.84

(30.2 - 30.79)^2 = 0.35

(30.6 - 30.79)^2 = 0.03

Calculate the mean of the squared differences:

(4.79 + 32.49 + 18.09 + 17.16 + 0.98 + 1.04 + 116.64 + 93.21 + 27.04 + 50.41 + 287.04 + 59.84 + 0.35 + 0.03) / 14 = 47.47

Take the square root of the mean of the squared differences:

sqrt(47.47) ≈ 6.89

Therefore, the standard deviation of the smokers group is approximately 16.57 and the standard deviation of the nonsmokers group is approximately 6.89.

Learn more about standard deviation here:

https://brainly.com/question/28198896

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