In 1990, the average duration of long-distance telephone calls originating in one town was A long-distance telephone company wants to perform a hypothesis test to determine whether the average duration of long-distance phone calls has changed from the 1990 mean of The hypotheses are:

The null hypothesis is

The alternate hypothesis is

Step-by-step explanation:

Mean of X in 1990, test if the mean duration has changed:

At the null hypothesis, we test if the mean duration is still X, that is:

At the alternate hypothesis, we test if the mean duration has the changed, that is, if it is different of X. So

take the logarithm of both sides (the base of the log doesn't matter):

apply the exponent property - this says that :

distribute the log terms on both sides:

group like terms together:

divide through both sides by the coefficient on :

you can do some additional rewriting here using the properties of the logarithm to simplify things if you like: