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Answer :
Based on the analysis of the confidence interval, the drug treatment seems to be effective in reducing the mean wake time compared to the mean wake time before the treatment.
To construct a 95% confidence interval estimate of the mean wake time for a population with the treatment, we can use the formula for the confidence interval for a population mean:
[tex]\[ \text{Confidence Interval} = \text{Sample Mean} \pm \text{Margin of Error} \][/tex]
Where the margin of error is calculated as:
[tex]\[ \text{Margin of Error} = \text{Critical Value} \times \frac{\text{Standard Deviation}}{\sqrt{\text{Sample Size}}} \][/tex]
Since the sample size is 16 and the critical value for a 95% confidence interval (assuming a normal distribution) is approximately 1.96, we can calculate the margin of error:
[tex]\[ \text{Margin of Error} = 1.96 \times \frac{24.9}{\sqrt{16}} = 1.96 \times \frac{24.9}{4} = 12.09 \][/tex]
the confidence interval:
[tex]\[ \text{Lower Limit} = \text{Sample Mean} - \text{Margin of Error} = 97.4 - 12.09 = 85.31 \]\[ \text{Upper Limit} = \text{Sample Mean} + \text{Margin of Error} = 97.4 + 12.09 = 109.49 \][/tex]
So, the 95% confidence interval estimate of the mean wake time for a population with the treatment is [tex]\(85.31\) min < μ < \(109.49\) min.[/tex]
the results in relation to the mean wake time of 104.0 min before the treatment:
- The confidence interval does not include the value 104.0 min within its range. This suggests that the mean wake time before the treatment is not a likely value given the observed data with the treatment.
- Since the entire confidence interval lies below 104.0 min, it indicates that the drug treatment appears to be effective in reducing the mean wake time. The lower limit of the confidence interval (85.31 min) is significantly lower than the initial mean wake time of 104.0 min.
In conclusion, based on the analysis of the confidence interval, the drug treatment seems to be effective in reducing the mean wake time compared to the mean wake time before the treatment.
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