Answer :
The results support the hypothesis that smokers take longer to fall asleep with 95% confidence. The negative t-value indicates that smokers, on average, take significantly longer to fall asleep compared to non-smokers.
To determine if the results support the hypothesis that smokers take longer to fall asleep, we can compare the sleep onset times between the group of smokers and the group of non-smokers. By analyzing the data provided, we can make a conclusion with 95% confidence.
First, let's calculate the mean (average) sleep onset time for each group:
Non-smokers: 27.4 + 25.9 + 26.2 + 34.7 + 29.6 + 28.2 + 40.3 + 30 + 30.6 + 31.6 + 47.1 + 19.8 + 35.8 + 37.4 + 12.9
= 446.5
Mean sleep onset time for non-smokers = 446.5 / 15
= 29.77 minutes
Smokers: 52.1 + 22.1 + 47.6 + 53.1 + 45.0 + 51.7 + 36 + 61.1 + 43.8 + 23.7 + 14.8
= 452.0
Mean sleep onset time for smokers = 452.0 / 11
= 41.09 minutes
Next, let's calculate the standard deviation for each group:
Non-smokers:
- Subtract the mean sleep onset time (29.77) from each individual sleep onset time, square the differences, and sum them up:
(27.4 - 29.77)² + (25.9 - 29.77)² + (26.2 - 29.77)²+ ... + (37.4 - 29.77)² + (12.9 - 29.77)²
= 570.23
- Divide the sum by the number of observations minus 1 (15 - 1 = 14):
570.23 / 14
= 40.73
- Take the square root of the result: [tex]\sqrt(40.73)[/tex] ≈ 6.38
Standard deviation for non-smokers = 6.38
Smokers:
- Subtract the mean sleep onset time (41.09) from each individual sleep onset time, square the differences, and sum them up:
(52.1 - 41.09)² + (22.1 - 41.09)² + (47.6 - 41.09)² + ... + (14.8 - 41.09)²
= 954.31
- Divide the sum by the number of observations minus 1 (11 - 1 = 10):
954.31 / 10
= 95.43
- Take the square root of the result: [tex]\sqrt(95.43)[/tex] ≈ 9.77
Standard deviation for smokers = 9.77
Now, let's perform a t-test to determine if the difference in sleep onset times between the two groups is statistically significant.
The t-test formula is:
[tex]t = (mean1 - mean2) / \sqrt((s1^2 / n1) + (s2^2 / n2))[/tex]
Where:
- mean1 and mean2 are the means of the two groups (29.77 for non-smokers and 41.09 for smokers)
- s1 and s2 are the standard deviations of the two groups (6.38 for non-smokers and 9.77 for smokers)
- n1 and n2 are the sample sizes of the two groups (15 for non-smokers and 11 for smokers)
Plugging in the values:
[tex]t = (29.77 - 41.09) / \sqrt((6.38^2 / 15) + (9.77^2 / 11))[/tex]
Calculating this gives us:
t ≈ -2.233
Now, we need to compare this t-value with the critical t-value at a 95% confidence level. For the given sample sizes (15 and 11), the critical t-value is approximately ±2.201.
Since -2.233 is outside the range of ±2.201, we can conclude that the results support the hypothesis that smokers take longer to fall asleep with 95% confidence. The negative t-value indicates that smokers, on average, take significantly longer to fall asleep compared to non-smokers.
It's important to note that this conclusion is based on the provided data and statistical analysis. Other factors, such as sample size and any underlying health conditions, should also be considered when interpreting the results.
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