AP Statistics 2026 FRQ #3 FULL WALKTHROUGH | How to Solve Every Part of the Free Response Question

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[00:00:00]What's up my sad stars? In this video, we're going to tackle the third free response question from the 2026 AP Statistics exam. Now, this question dealt with the length of song and well, if certain people sing a certain song, well, then it could take a little bit longer, a little bit shorter. So, the problem started out using the normal distribution, then it turned into the binomial distribution, and then it even turned into the geometric distribution.

[00:00:20]So, it actually ended up being a really good probability based question. So, let's start looking at the answers right now.

[00:00:26]So, question three deals with a certain sports team that has been that has a team song performed by different musicians at the beginning of each of its games. The time it takes for the team song to be performed by different musicians can be modeled using a normal distribution with a mean of 109 seconds and a standard deviation of 16 seconds.

[00:00:44]All performances of the team song are independent. So, basically, a team has a team song that they play before every game, but they allow different musicians to play it. And well, that incorporates variability because some people might take a little bit longer or a little bit shorter to sing that song. Well, we know the mean is 109 seconds and the standard deviation is 16. All right, let's talk about part A. Part A says, "What is the probability that a randomly selected performance of the team song will take longer than 120 seconds?"

[00:01:12]Now, all we have to do is remember this is a normal distribution. That's going to save us. So, all we have to do is figure out exactly the probability that in a normal distribution, we are more than 120 seconds. All right, so what I did was first, I said, "All right, we're going to find the probability that the length of the song, I just call that S, is greater than 120 seconds." What I did was I went ahead and got its Z score.

[00:01:34]So, I took 120 seconds, subtracted the mean of 109, divided by the standard deviation of 16, I got a Z score of 0.6875.

[00:01:42]So, asking about a song being greater than 120 seconds is the same thing as a Z score being greater than 0.6875 in a normal distribution. Now, you could use Desmos, you could use NumWorks, you could use a TI-84 calculator to get the answer, but the answer is or you could actually even use an old-school table, but the answer is.24588, or if we want to round that, it's approximately.246.

[00:02:05]About 24.6% chance that a performance of the song is over 120 seconds. Hopefully, that wasn't too bad. A lot of kids are going to ask, "What work do I have to show?" Well, honestly, this, right? You do not have to show that you are, you know, using normal CDF or anything like that. All you really need to do is make sure that you identify that it's a normal distribution with a mean of 109 and a standard deviation of 16, and then when you show this notation right here, and then the Z score being greater than.6875, you're allowed to just get your answer pretty simply without having to show, you know, calculator talk.

[00:02:40]All right, the next question is a little bit of a binomial one. 10 performances.

[00:02:45]Now, we have a set number of trials, right? 10 performances of the team song will be randomly selected. Let the random variable X represent the number of games in which the performance is longer than 120 seconds. Now, this is important because what we're going to need is that value that we just found in the previous question, and that the probability that we are over 120 seconds is.246. You have to be able to connect that we need that number from part A to do part B.

[00:03:12]Now, the question is, what is the probability that X, the number of songs in 10 performances is greater than 120 seconds, is three or greater? Greater than or equal to three. So, the first thing you should do here is identify that this follows a binomial distribution with 10 as the number of trials, and the probability of success is.246.

[00:03:33]Now, if this is the case, you could actually just jump to NumWorks ti-84 calculator or Desmos to get your answer.

[00:03:39]Makes it really, really simple. The probability that the X, the number of performances out of 10 that are over 120 seconds, is greater than or equal to three is.462. But, as long as you've identified that it's a binomial distribution, showing your N and your P, you do not have to use any calculator speak. If you use Desmos, you could literally just set it up, binomial distribution, N equals 10, P equals 0.246, and then just type in three or greater, and you're going to get the answer really, really simple. Same thing with the NumWorks, really, really easy to use and really, really easy to get that simple solution. But, some teachers and some people stress wanting to show work. So, let's talk about the work.

[00:04:18]Now, three or greater would be three, four, five, six, seven, eight, nine, or 10 songs out of the 10 performances lasting over 120 seconds. I do not want to show all that work, so I'm going to go the route by doing what I don't want. If I want three or greater, I don't want zero, I don't want one, I don't want two. So, here is me showing that work for zero out of 10, one out of 10, and two out of 10. Then I can add those three probabilities together, but then I'm going to have to subtract that away from one, because remember, zero, one, and two are what I do not want. I want three or greater. So, here's me showing all the work for zero, one, and two, we get 0.538, and then doing one minus that, we get 0.462, which is our final answer. But, I cannot stress enough, you actually don't have to show all that work. You could use your Desmos or your NumWorks or your TI-84 to get that answer for you, but you do have to identify that it's a binomial distribution with N equal to 10 and P equals 0.246.

[00:05:17]All right, now let's get to part C, which a lot of kids were very unsure of this year. Ben will attend all of the games for this team.

[00:05:25]Let the random variable Y represent the number of games Ben will attend until a performance of the team song lasts longer than 120 seconds. Bingo, right there. There's no set number of games.

[00:05:37]He's going to attend games until the song is over 120 seconds. This is a geometric distribution where we want to think about that first success. As soon as Ben gets that first success, a song being over 120 seconds, he is done. Now, that's how we know what we're going to do here. So, we're going to follow a, you know, where Y is following a geometric distribution with a P value or not, excuse me, not a P value, a probability of success of.246.

[00:06:06]All right. Now, where a lot of kids struggle was they said, "Well, I don't know how to find the mean and standard deviation." The formulas were on the formula sheet on your AP Statistics formula sheet, there is the geometric distribution formula, and right here is the mean, and right here is the standard deviation. Now, interesting enough, this topic of geometric is going away after 2026. So, if you are studying for the 2027, 2028, 2029, yada yada yada, AP Statistics exam, this type of question will not show up because the geometric distribution is no longer in the curriculum after 2026. All right, but the key thing is the formulas are given to you. All you have to do is go and use them. So, I'm going to do the mean, 1 divided by the probability of success, which was.246. I got 4.0650.

[00:06:53]And then the standard deviation is the square root of 1 minus.246 divided by.246, and then we get 3.530. But, make sure there's that square root on the top, though.

[00:07:03]So, that's all you had to do was calculate the mean and the standard deviation for these particular values.

[00:07:08]Now, what are the units on these? Games, right? This is the number of games he would have to attend until he gets that first song performance being over 120 seconds.

[00:07:20]And now for part D, interpret the standard deviation calculated in part C II in context. So, they want us to interpret the standard deviation that we just calculated, but the funny thing is it's really hard to interpret standard deviation if you don't also incorporate the mean.

[00:07:35]So, here's what I said. The number of games Ben needs to attend until the first time the team song is over 120 seconds, typically differs from the mean by about 3.53 games. So, we expect it to take an average of 4.065 games until the first time the song is over 120 seconds, but that number of games could vary by 3.53 games. So, the idea is the mean is the number of games we expect it to take until we get a song over 120 seconds, and that's 4.065, but that number can certainly vary, and that's where the standard deviation comes in. The number of games that that number could vary by is 3.53 games. All right, hopefully that made sense. Definitely a tough question that I bet a lot of kids are not going to get right this year, but no worries.

[00:08:19]If you're studying for 2027 or further, you're actually never going to see a question like this come up, so good for you. All right, see you in the next video.