Final answer:
In AP Statistics, the null hypothesis for testing if the mean time spent on homework has increased would be H0: μ = 2.5 hours, while the alternative hypothesis would be H1: μ > 2.5 hours. A Z-test would be employed since the population standard deviation is known.
Step-by-step explanation:
The question pertains to conducting hypothesis tests within the context of AP Statistics, specifically testing whether the mean time spent on homework each week has increased from a previously known average. When formulating a hypothesis test, we establish a null hypothesis (H0) and an alternative hypothesis (H1). The null hypothesis is typically a statement of no effect or no change, and the alternative hypothesis is what you want to test for; in this case, an increase in the mean time spent on homework.
For the scenario provided: the null hypothesis would be H0: μ = 2.5 hours (the mean time students spend on homework per week is 2.5 hours). The alternative hypothesis would be H1: μ > 2.5 hours (the mean time students spend on homework per week has increased).
The distribution of the population is assumed to be normal, which is necessary to apply the Z-test for hypothesis testing. As the population standard deviation is known (1.5 hours), you would use the Z-test to compare the sample mean (3 hours) against the population mean (2.5 hours), using the standard deviation and sample size (26 students) given.