Answer:
(a) [tex]\bar x_1 = 43.7[/tex] Â Â [tex]\bar x_2 = 30.25[/tex]
(b) [tex]\sigma_1 = 16.93[/tex] Â Â [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.
