0 smokers were questioned about the number of hours they sleep each day you want to test the hypothesis that the smokers need less sleep than the general public which needs an average 7.7 hours of sle?

Answer 1

Disclaimer: This is not an easy answer because a lot depends on the sample size, distribution of population, etc. But for the sake of this I will assume that we are talking about a normally-distributed population and a sufficiently large sample size.

You want to test the hypothesis that smokers need less sleep than the general public. You are told that the general public needs an average of 7.7 hours of sleep. So:

Avgpop = 7.7

The hypothesis that you are testing (H1) is that smokers need less sleep:

Avgsmoker < Avgpopulation or Avgsmoker < 7.7 hours

Your null hypothesis (Ho) would be that there is no difference between the two groups OR that smokers need MORE sleep than the general population:

Avgpopulation >= Avgsmoker or Avgsmoker <= 7.7 hours

You test that hypothesis by taking a sample of X smokers and comparing their average number of sleep to that of your population. You then test to see whether any differences you observe would likely have been found due to chance or because the sample is different from the population. The more smokers that you randomly sample, the more likely it will be that a given difference will be due to smokers need a different amount of sleep than the general public.

Lets say that you sampled 200 smokers and their average night of sleep was 7.55 hours with a standard deviation of 1.0. Now we can calculate the test statistic. Because we are assuming a normal population and the sample size is greater than 30, we can use a one-sample z-test (it is one sample because we are taking one sample and comparing it to a known mean).

Z = (Sample Mean - Population Mean)/(Standard Deviation of Sample/Square Root of Sample Size)

Z = (7.55 - 7.7)/(1.0/Sqrt(200))

Z = -.15/.0707

Z = -2.12

We now compare our test statistic to a critical value table. (one for the z-statistic can be found here: http://math2.org/math/stat/distributions/z-dist.htm

That Z-statistic corresponds to a p-value of .017. This means that the probability of randomly drawing a sample with this observed average assuming the null hypothesis is true is .017 or 1.7%. At a confidence level of 98.3% and below you would reject the null hypothesis. THIS DOES NOT MEAN WITH CERTAINTY THAT SMOKERS NEED LESS SLEEP THAN THE GENERAL PUBLIC, however the data would suggest that the hypothesis that smokers need less sleep is true (in this case).