Final answer:
In the mathematics problem, the mean difference in time between relaxed sitting and jumping for eight girls at Rachel's birthday is tested statistically, likely using a t-test. For the soccer scenario, the displacement over time for a ball kicked to the goalie and back would create a V-shaped graph, resembling the hockey puck's displacement graph from a previous question.
Step-by-step explanation:
Hypothesis Testing in Mathematics
When considering the problem from Rachel's eleventh birthday party, where eight girls measured the time difference between sitting relaxed and jumping, a hypothesis test could be used to determine if the mean difference is statistically significant from zero. These eight trials form a data set from which the mean, standard deviation, and standard error can be calculated. A paired sample t-test or a similar procedure would compare the mean difference to zero, which is the null hypothesis in this case. If the calculated p-value is below a predetermined significance level (often 0.05), the hypothesis that there is no difference can be rejected. This demonstrates an application of statistical methods including mean, standard deviation, p-value, and hypothesis testing in analyzing a data set from a real-world situation.
In the soccer scenario, the displacement over time diagram for a ball being kicked to the goalie and then back would appear as a V-shaped graph. This is because the ball starts from a certain point, moves towards the goalie (increasing displacement), stops (displacement remains constant for a moment), and is then kicked back to the player, reducing the displacement over time back to the original point. The graph would look similar to one showing the displacement of a hockey puck moving back and forth over the same distance.