Probability in statistics and standard deviation

Question

The number of hours an American adult sleeps each night follows a Normal distribution with an average of 6.9 hours per night with a standard deviation of 0.68 hours. If we take a random sample of 15 American adults, what is the probability they have a sample mean less than 7.5 hours per night?

Answer 1

x = sleep time, which is normally distributed with mean 6.9 and standard deviation 0.68.

We are trying to find P(x > 7.5). To use standard normal probability table (z-scores), we must convert that inequality to a z-score: (x - mean)/(standard deviation/square root (sample size))/

So the inequality becomes P(z > (7.5 - 6.9)/(0.68/square root(15)).

We can then look up that probability from the standard normal probability table. Since probabilities in that table are cumulative less than, you will use the fact that P(z > value) = 1 - P(z < value).