Question

A clinical trial was conducted to test whether a drug for treating insomnia is effective. Researchers administered the drug and measured the time it took 23 randomly selected subjects to fall asleep. The subjects had a mean wake time of 96.6 min and a sample standard deviation of 24.5 min. Assume that the population of wake times is normally distributed.

(a) Construct a 95% confidence interval. You may use your calculator. Round to 1 decimals.



(b) Are the assumptions met? Explain.




(c) Interpret the interval.

Suppose unmedicated insomniacs typically have an average wake time of 110 minutes. Run a hypothesis test, at the 0.05 level of significance, to determine whether this drug treatment was able to decrease the amount of time it takes subjects to fall asleep.

Hypotheses: ___________________________ ___________________________

p-value (round to 4 decimals) = _______________

State your conclusion in terms of the problem

Answer 1

a)
Required confidence interval = (96.6-2.0739(5.1086), 96.6+2.0739(5.1086))
Required confidence interval = (96.6 - 10.5947, 96.6 + 10.5947)
Required confidence interval = (86.0, 107.2)
Interpretion: We are 95.0% confident that the true mean of the population lie between the interval 86.0 and 107.2.

b) Yes, the assumptions are met.

Randomization Condition: The data must be sampled randomly.

Independence Assumption: The sample values must be independent of each other. This means that the occurrence of one event has no influence on the next event. Usually, if we know that people or items were selected randomly we can assume that the independence assumption is met.

10% Condition: When the sample is drawn without replacement (usually the case), the sample size, n, should be no more than 10% of the population.

Homogeneity: The population should be normally distributed and the two populations have the same variance.

c)


The test hypothesis is

This is a left-tailed test because the alternative hypothesis is formulated to detect claim if mean is less than 110.0.
Now, the value of test static can be found out by following formula:

OR

Conclusion : There is sufficient evidence to conclude that unmedicated insomniacs typically have an average wake time of 110 minutes