A sample of Alzheimer's patients is tested to assess the amount of time in stage IV sleep. It has been hypothesized that individuals suffering from Alzheimer's Disease may spend less time per night in the deeper stages of sleep. Number of minutes spent is Stage IV sleep is recorded for sixty-one patients. The sample produced a mean of 48 minutes (S=14 minutes) of stage IV sleep over a 24 hour period of time. Compute a 95 percent confidence interval for this data. What does this information tell you about a particular individual's (an Alzheimer's patient) stage IV sleep?
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A 95% confidence interval can be calculated using the formula:
CI = mean ± (Z * (standard deviation / √n))
Where:
- Z is the Z-score, which for a 95% confidence interval is 1.96
- n is the sample size
So, the 95% confidence interval would be:
CI = 48 ± (1.96 * (14 / √61))
CI = 48 ± (1.96 * (14 / 7.81))
CI = 48 ± (1.96 * 1.79)
CI = 48 ± 3.51
So, the 95% confidence interval is (44.49, 51.51) minutes.
This means that we are 95% confident that the true mean time spent in stage IV sleep by Alzheimer's patients is between 44.49 and 51.51 minutes. However, this does not provide information about a particular individual's stage IV sleep as confidence intervals deal with population parameters, not individual data points.
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