Question

  A sample of Alzheimer's patients is tested to assess the amount of time in stage IV sleep these patients get in a 24-hour period. Number of minutes spent in Stage IV sleep is recorded for 61 patients. The mean stage IV sleep over a 24 hour period of time for these 61 patients was 48 minutes with a standard deviation of 14 minutes.

(a)     Compute 95% confidence interval for mean stage IV sleep. Interpret this confidence interval.

(b) It has been believed that individuals suffering from Alzheimer's Disease may spend less time per night in the deeper stages of sleep. Test the hypothesis at 5% significance level if the true mean stage IV sleep of Alzheimer patients is less than 50 minutes.

1. Provide the hypothesis statement  
2. Calculate the test statistic value
3. Determine the probability value

(c) Could the confidence interval in part (a) be used to test the hypothesis in part (b)? Why or why not?

Answer 1

(a) The 95% confidence interval for mean stage IV sleep is between 44.41 and 51.59.

We are 95% confident that the true mean stage IV sleep is between 44.41 and 51.59.

(b) The hypothesis being tested is:

H0: µ = 50

Ha: µ < 50

The p-value is 0.1345.

Since the p-value (0.1345) is greater than the significance level (0.05), we cannot reject the null hypothesis.

Therefore, we cannot conclude that the true mean stage IV sleep is less than 50 minutes.

(c) Yes, the confidence interval can be used. Since the confidence interval is above 50 minutes, we cannot conclude that the true mean stage IV sleep is less than 50 minutes.

50.00	hypothesized value
48.00	mean 1
14.00	std. dev.
1.79	std. error
61	n
60	df
-1.116	t
.1345	p-value (one-tailed, lower)
44.41	confidence interval 95.% lower
51.59	confidence interval 95.% upper
3.59	margin of error