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? (1 mark)

Answer 1

a) At 95% confidence level, the critical value is t* = 2

The 95% confidence interval is

We are 95% confident that the true mean stage IV sleep lies in the above confidence interval.

b)

  

The test statistic is

  

P-value = P(T < -1.12)

= 0.1336

Since the P-value is greater than the significance level, so we we should not reject the null hypothesis.

At alpha = 0.05, the critical value is -t0.05 = -1.671

c) Yes, the confidence interval can be be used to test the hypothesis in part(b).

Since the interval contains the hypothesized value 50, so we should not reject the null hypothesis.