Final answer:
To determine if the average time spent on homework has increased, we set up a null hypothesis stating the mean is 2.5 hours and an alternative hypothesis stating it is greater. We then calculate a test statistic and compare it with a critical value; if larger, the null hypothesis is rejected, indicating an increase.
Step-by-step explanation:
Hypothesis Testing in Statistics
To conduct a hypothesis test on whether the mean time students spend on homework has increased, we begin by setting up our null and alternative hypotheses. The null hypothesis (H0) represents no change or the status quo. In this case, it would be that the mean time spent on homework is still 2.5 hours: H0: μ = 2.5 hours. The alternative hypothesis (H1) suggests that there is an increase, hence: H1: μ > 2.5 hours.
After setting the hypotheses, the next steps involve calculating the test statistic, deciding upon a significance level (α), referring to a statistical distribution (like the z-distribution if the population standard deviation is known), and then making a decision on whether to reject H0 based on whether the test statistic falls in the rejection region defined by α.
With a sample mean of 3 hours, a known population standard deviation of 1.5, and a sample size of 26, we can calculate the test statistic using the formula for the z-test since the population standard deviation is known and the distribution is assumed to be normal. If the calculated z is greater than the z-value corresponding to the chosen level of significance, H0 would be rejected, indicating that there is enough evidence to conclude the average time spent on homework has increased.