Final answer:
A student with a percentile rank of 65 has performed better than 65% of the class. Percentiles rank the position of an individual within a group, and higher percentiles generally indicate better performance or faster speed, except in contexts where lower is better, such as waiting times.
Step-by-step explanation:
When a student in a class of 50 has a percentile rank of 65, it implies that the student's performance is better than 65% of their peers. Percentiles are particularly useful when dealing with large populations, but even in smaller groups like a classroom, they help to compare individual scores. For instance, if 90 percent of the test scores are less than your score, you are at or beyond the 90th percentile. This is significant even in a small dataset since each score represents 2% of the population.
In a statistical context, if we reject \( \mu = 65 \), we're saying that we believe the average score is higher than 65, based on sample data at a certain level of significance. For individual scores, a student's percentile can give a clear idea of where they stand relative to others, and it is particularly important for standardized testing criteria, such as the SAT, where colleges might set a minimum percentile for admission.
For example, if a student is in the 70th percentile for a 20-question math test with 16 correct answers, it indicates they performed better than 70% of the students. Similarly, being in the 80th percentile in a race implies running faster than 80% of the participants. Interpretation of percentiles can vary with context, as higher percentiles are usually more desirable for test scores and speed, but for a wait time at the DMV, a lower percentile would be preferable.