Answer :
Based on the data collected and the statistical analysis performed, we cannot conclude with 95% confidence that smokers take longer to fall asleep.
To determine if the results support the hypothesis that smokers take longer to fall asleep, we need to compare the sleep times of smokers and non-smokers. First, let's calculate the mean and standard deviation of the sleep times for each group.
For the non-smokers:
Mean = (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) / 15 = 30.2133 minutes
Standard Deviation = √[((27.4 - 30.2133)² + (25.9 - 30.2133)² + ... + (37.4 - 30.2133)² + (12.9 - 30.2133)²) / 14] ≈ 7.8246 minutes
For the smokers:
Mean = (52.1 + 22.1 + 47.6 + 53.1 + 45.0 + 51.7 + 36 + 61.1 + 43.8 + 23.7 + 14.8) / 11 = 41.3 minutes
Standard Deviation = √[((52.1 - 41.3)² + (22.1 - 41.3)² + ... + (23.7 - 41.3)² + (14.8 - 41.3)²) / 10] ≈ 14.6188 minutes
Now, let's investigate the variances of the two groups. To do this, we'll use the F-test with a significance level of 2%. The F-value is calculated by dividing the larger variance by the smaller variance.
F-value = (14.6188)² / (7.8246)² ≈ 3.0006
Next, we compare the F-value to the critical F-value at a significance level of 2%. Since the degrees of freedom for the numerator is 10 (number of smokers - 1) and the degrees of freedom for the denominator is 14 (number of non-smokers - 1), we find that the critical F-value is approximately 3.20.
Since the calculated F-value (3.0006) is less than the critical F-value (3.20), we do not have enough evidence to support the hypothesis that smokers take longer to fall asleep. The results do not suggest a significant difference in the sleep times between smokers and non-smokers.
In conclusion, we cannot say with 95% certainty that smokers take longer to fall asleep based on the data gathered and statistical analysis carried out.
To know more about confidence refer here:
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