Question 1169806: Q−1: [4+6+8+2 Marks] A study of the effects of smoking on sleep pattern is conducted. The measure observed in the time, in minutes that it takes to fall asleep. These data are:
Smokers 48.1 52.7 53.2 56 60.2 69.3 13.8 22.1 23.2 34.4 34.8 47.6
Nonsmokers 30.2 30.6 31.8 34.9 36 37.9 38.5 41.6 13.9 21.1 25.1 26.4 28.4 28.6 29.8
a) Calculate the mean, median for each group.
b) Find the inter-quartile range for each group.
c) Calculate the standard deviation for each group.
d) Comment on what kind of impact smoking appears to have on the time required to fall asleep.
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Okay, let's analyze the sleep pattern data for smokers and non-smokers.
**a) Calculate the Mean and Median for Each Group**
**Smokers:**
1. **Data:** 48.1, 52.7, 53.2, 56, 60.2, 69.3, 13.8, 22.1, 23.2, 34.4, 34.8, 47.6
2. **Number of data points (n):** 12
3. **Sum of data points:** 48.1 + 52.7 + 53.2 + 56 + 60.2 + 69.3 + 13.8 + 22.1 + 23.2 + 34.4 + 34.8 + 47.6 = 545.4
4. **Mean:** 545.4 / 12 = 45.45 minutes
5. **Ordered data:** 13.8, 22.1, 23.2, 34.4, 34.8, 47.6, 48.1, 52.7, 53.2, 56, 60.2, 69.3
6. **Median:** Since there are 12 data points (even number), the median is the average of the 6th and 7th values: (47.6 + 48.1) / 2 = 47.85 minutes
**Non-smokers:**
1. **Data:** 30.2, 30.6, 31.8, 34.9, 36, 37.9, 38.5, 41.6, 13.9, 21.1, 25.1, 26.4, 28.4, 28.6, 29.8
2. **Number of data points (n):** 15
3. **Sum of data points:** 30.2 + 30.6 + 31.8 + 34.9 + 36 + 37.9 + 38.5 + 41.6 + 13.9 + 21.1 + 25.1 + 26.4 + 28.4 + 28.6 + 29.8 = 464.8
4. **Mean:** 464.8 / 15 = 30.99 minutes (approximately 31.0 minutes)
5. **Ordered data:** 13.9, 21.1, 25.1, 26.4, 28.4, 28.6, 29.8, 30.2, 30.6, 31.8, 34.9, 36, 37.9, 38.5, 41.6
6. **Median:** Since there are 15 data points (odd number), the median is the 8th value: 30.2 minutes
**b) Find the Inter-Quartile Range (IQR) for Each Group**
**Smokers:**
1. **Ordered data:** 13.8, 22.1, 23.2, 34.4, 34.8, 47.6, 48.1, 52.7, 53.2, 56, 60.2, 69.3
2. **Q1 (25th percentile):** Since n = 12, Q1 is between the 3rd and 4th values: (23.2 + 34.4) / 2 = 28.8 minutes
3. **Q3 (75th percentile):** Q3 is between the 9th and 10th values: (53.2 + 56) / 2 = 54.6 minutes
4. **IQR:** Q3 - Q1 = 54.6 - 28.8 = 25.8 minutes
**Non-smokers:**
1. **Ordered data:** 13.9, 21.1, 25.1, 26.4, 28.4, 28.6, 29.8, 30.2, 30.6, 31.8, 34.9, 36, 37.9, 38.5, 41.6
2. **Q1 (25th percentile):** Q1 is the 4th value: 26.4 minutes
3. **Q3 (75th percentile):** Q3 is the 12th value: 36 minutes
4. **IQR:** Q3 - Q1 = 36 - 26.4 = 9.6 minutes
**c) Calculate the Standard Deviation for Each Group**
**Smokers:**
1. **Data:** 48.1, 52.7, 53.2, 56, 60.2, 69.3, 13.8, 22.1, 23.2, 34.4, 34.8, 47.6
2. **Mean:** 45.45 minutes
3. **Calculate the variance:**
* (48.1 - 45.45)² + (52.7 - 45.45)² + (53.2 - 45.45)² + (56 - 45.45)² + (60.2 - 45.45)² + (69.3 - 45.45)² + (13.8 - 45.45)² + (22.1 - 45.45)² + (23.2 - 45.45)² + (34.4 - 45.45)² + (34.8 - 45.45)² + (47.6 - 45.45)² = 4786.67
* Variance = 4786.67 / (12 - 1) = 4786.67 / 11 = 435.15
4. **Standard deviation:** √435.15 = 20.86 minutes (approximately)
**Non-smokers:**
1. **Data:** 30.2, 30.6, 31.8, 34.9, 36, 37.9, 38.5, 41.6, 13.9, 21.1, 25.1, 26.4, 28.4, 28.6, 29.8
2. **Mean:** 30.99 minutes
3. **Calculate the variance:**
* (30.2 - 30.99)² + (30.6 - 30.99)² + (31.8 - 30.99)² + (34.9 - 30.99)² + (36 - 30.99)² + (37.9 - 30.99)² + (38.5 - 30.99)² + (41.6 - 30.99)² + (13.9 - 30.99)² + (21.1 - 30.99)² + (25.1 - 30.99)² + (26.4 - 30.99)² + (28.4 - 30.99)² + (28.6 - 30.99)² + (29.8 - 30.99)² = 566.97
* Variance = 566.97 / (15 - 1) = 566.97 / 14 = 40.49
4. **Standard deviation:** √40.49 = 6.36 minutes (approximately)
**d) Comment on the Impact of Smoking on Time to Fall Asleep**
Based on the calculations:
* **Mean:** Smokers (45.45 minutes) have a considerably higher mean time to fall asleep compared to non-smokers (30.99 minutes).
* **Median:** The median time for smokers (47.85 minutes) is also higher than for non-smokers (30.2 minutes).
* **IQR:** The IQR for smokers (25.8 minutes) is much larger than for non-smokers (9.6 minutes), indicating greater variability in the time it takes for smokers to fall asleep.
* **Standard Deviation:** The standard deviation for smokers (20.86 minutes) is much higher than for non-smokers (6.36 minutes), further supporting the higher variability in smokers' sleep patterns.
**Conclusion:**
The data suggests that smoking appears to be associated with:
* A longer average time to fall asleep.
* A wider range of times it takes to fall asleep, with some smokers taking a significantly longer or shorter time.
* More inconsistent sleep patterns.
In simpler terms, smokers, on average, take longer to fall asleep, and the time it takes for them to fall asleep is less predictable compared to non-smokers.
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