Dance Attendance Estimate and Raffle Probability
The student council wants to estimate how many of the 420 seventh graders will attend the spring dance. They survey 25 students during an honor-roll study hall; 20 say they will attend. A council member uses 20/25 = 80% to predict 336 attendees. The council also has attendance data from the last five spring dances: 180, 200, 210, 220, and 400 students. At the dance, each attendee will receive one raffle ticket, and 6 prizes will be awarded. There will also be a game with a spinner divided into 8 equal sections; 3 sections win a small prize. The advisor questions the survey method, the prediction, and the typical attendance figure. The council must choose a reasonable attendance estimate for planning and compute the probability of winning a raffle prize and the expected number of spinner winners if 240 students play once each.
1. Identify the sampling problem and explain why 80% may not represent all 420 seventh graders. Is 336 a reliable prediction?
Read the worked answer
The sample is taken only from an honor-roll study hall, so it is not a random sample of all seventh graders. Students in that group may differ from the full grade in dance interest, schedules, or other factors. The 80% result applies to the 25 surveyed students, not necessarily to all 420. Predicting 336 attendees assumes the sample is representative, which is not justified. A better estimate would come from a random sample that includes students from different classes and achievement levels.
2. Compare the mean and median of the five attendance values. Which better represents typical attendance, and why?
Read the worked answer
The mean is (180 + 200 + 210 + 220 + 400) ÷ 5 = 1,210 ÷ 5 = 242. The median is the middle value when the data are ordered: 180, 200, 210, 220, 400, so the median is 210. The value 400 is much larger than the other four values and pulls the mean upward. The median, 210, is a better description of typical attendance because it is not affected by the outlier. For planning, the council should use about 210 rather than 242 or 336.
3. If 240 students attend and each gets one raffle ticket, what is the probability a ticket wins one of 6 prizes? If 240 students each play the spinner once, how many small prizes should be expected?
Read the worked answer
For the raffle, there are 6 winning tickets out of 240 total tickets, so P(win) = 6/240 = 1/40 = 0.025 = 2.5%. For the spinner, 3 of 8 equal sections win, so P(win) = 3/8. If 240 students play once each, the expected number of winners is 240 × 3/8 = 90. This is an expected value, not a guarantee; actual results may vary.