High School

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Q.No. 2.

A study of the effect of smoking on sleep patterns is conducted. The measure observed is the time, in minutes, that it takes to fall asleep. The following 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
- Non-Smokers: 28.6, 25.1, 26.4, 34.9, 28.8, 28.4, 38.5, 30.2, 30.6, 31.8, 41.6, 21.1, 36.0, 37.9, 13.9

Answer :

Answer:

(a) [tex]\bar x_1 = 43.7[/tex] [tex]\bar x_2 = 30.25[/tex]

(b) [tex]\sigma_1 = 16.93[/tex] [tex]\sigma_2 = 7.14[/tex]

(c) Smoking increases the time to fall asleep

Explanation:

Solving (a): The sample mean of each group

Mean is calculated as:

[tex]\bar x = \frac{\sum x}{n}[/tex]

So, we have:

Smokers

[tex]n_1= 12[/tex] and

[tex]\bar x_1 = \frac{69.3 +56.0+ 22.1 +47.6+ 53.2+ 48.1+ 52.7 +34.4+ 60.2 +43.8 +23.2 +13.8}{12}[/tex]

[tex]\bar x_1 = \frac{524.4}{12}[/tex]

[tex]\bar x_1 = 43.7[/tex]

Non Smokers

[tex]n_2 = 15[/tex] and

[tex]\bar x = \frac{28.6 +25.1 +26.4 +34.9 +28.8 +28.4 +38.5 +30.2 +30.6 +31.8 +41.6 +21.1 +36.0 +37.9 +13.9}{15}[/tex]

[tex]\bar x_2 = \frac{453.8}{15}[/tex]

[tex]\bar x_2 = 30.25[/tex]

Solving (b): The standard deviation of each group

This is calculated as:

[tex]\sigma = \sqrt{\frac{\sum(x - \bar x)^2}{n-1}}[/tex]

For smokers

[tex]n_1= 12[/tex]

So:

[tex]\sigma_1 = \sqrt{\frac{(69.3 -43.7)^2+(56.0-43.7)^2+..........+(13.8-43.7)^2}{12-1}}[/tex]

[tex]\sigma_1 = \sqrt{\frac{3152.04}{11}}[/tex]

[tex]\sigma_1 = \sqrt{286.5491}[/tex]

[tex]\sigma_1 = 16.93[/tex]

For non-smokers

[tex]n_2 = 15[/tex]

So:

[tex]\sigma_2 = \sqrt{\frac{(28.6 -30.25)^2+(25.1 -30.25)^2+..........+(13.9 -30.25)^2}{15-1}}[/tex]

[tex]\sigma_2 = \sqrt{\frac{713.2575}{14}}[/tex]

[tex]\sigma_2 = \sqrt{50.9469}[/tex]

[tex]\sigma_2 = 7.14[/tex]

Solving (c): Impact of smoking on time to sleep

In (b), we have:

[tex]\sigma_1 = 16.93[/tex] --- smokers

[tex]\sigma_2 = 7.14[/tex] --- non-smokers

Smokers have larger standard deviation (i.e. large variability) than non-smokers. This means that smokers require more time to fall asleep.

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