Final answer:
To test if the mean time spent on homework has increased, we set a null hypothesis that it remains at 2.5 hours and an alternative hypothesis that it has increased. We calculate a test statistic using the sample data and population standard deviation. Percentiles like the first quartile and 30th percentile help us understand the distribution of data, such as test completion times and study hours.
Step-by-step explanation:
The question pertains to conducting a hypothesis test to determine if there has been an increase in the mean length of time students spend doing homework each week. The null hypothesis (H0) would be that the mean time spent on homework has not increased, and is still 2.5 hours per week, while the alternative hypothesis (H1) would state that the mean time has increased from 2.5 hours. To conduct the hypothesis test, one would use the given sample mean, population standard deviation, and the sample size to calculate a test statistic and compare it to the critical value from a normal distribution table to reach a conclusion.
In the context of interpreting percentiles, the first quartile represents the time below which 25% of the data falls. If the first quartile for time to finish a math test is 35 minutes, it means that 25% of the students finished the test in 35 minutes or less. Similarly, the 30th percentile for hours of study per week means that 30% of the students studied for 7 hours per week or less.