2026 AP Statistics Free Response #5

Added: 2026-05-10

[00:00:00]All right, uh 2026 AP Statistics FRQ number five. So, a researcher is investigating whether there is an association Oh, this is version J, by the way. Among professional athletes in all types of sports plays, the age group and sports plays, and a recent year given two-way table. Consider the two-way table. What is the probability a randomly selected professional athlete is a football player? So, these are all professional athletes. We want the probability that they are football players. So, we're going to take this divided by this. So, it's going to be 2461 divided by 4193.

[00:00:32]Um I don't know notationally. I'm not I mean, okay. Okay. I'm being lazy. So, let's do the probability that they're football is equal to this. So, 24 Oops. 24 61 divided by 4193. So, then we need you to show the calculations. So, we're going to get 0.5869.

[00:00:58]Okay? So, hopefully that one was pretty straightforward.

[00:01:01]Uh sorry, it's cutting off on the screen. What's the probability a randomly selected professional athlete Let me make it a little bit smaller and just fit it underneath here.

[00:01:10]>> [snorts] >> Randomly selected professional athletes in the age group 25 to 30, given that they are football players. So, the probability that they are 25 less than or equal to age less than 30, given that they are football player is um So, we're going to just look at this column. We're going to exclude all of the other ones, and we're going to say like, "Oh, if they're between here and here, it's the 1326 divided by the 2461." Right? It's the probability It's It's Yeah, that that would be that ratio there. You're going to take how many are in there out of all the football players, so we're going to do 1326 divided by 2461.

[00:01:51]That's 0.5388.

[00:01:55]Cool. All right, a mosaic plot was constructed with the information on a two-way table.

[00:01:59]Uh [snorts] using the mosaic plot, answer the following questions. The B and H displayed represent the probability calculated in part A.

[00:02:06]Uh Which probability does B correspond to?

[00:02:11]B is the width.

[00:02:13]Relative frequency, and this is just the football player. So, this is the football players.

[00:02:20]And uh The width is just the fraction of the football players, right? So, it's going to be It's going to be the 0.58 um Oh, B B is the width here. So, it corresponds to the 0.

[00:02:37]5869.

[00:02:40]It is the the football player proportion.

[00:02:50]What probability does X represent in the in context?

[00:02:55]Um so, X is going to be this area, which is the combination. So, X is the product. It's the area here.

[00:03:02]Right? So, it's the area. So, it's the intersection that they're both football players in there. And so, it is the probability that a randomly cuz the area is the probability from the randomly selected athlete follows both of these things is that they're in this age group and that is a football player player aged between 25 and 30.

[00:03:40]Okay.

[00:03:42]Part C, use the information to answer the following questions. Uh are the events baseball and 35 listening mutually exclusive events? So, let's look at baseball.

[00:03:53]And um less than or equal to 35 age are mutually exclusive. No, there's some people in there. There's a dark region right here of baseball players, so they're not mutually exclusive. No, because there are some people less than or equal to 35 who play baseball.

[00:04:17]And we probably have a specific number of that. There are how many of them that do that? Uh less than or equal to 35 Oh, greater than equal to 35, sorry. Uh there's 61.

[00:04:29]61 people greater than or equal to 35 who play baseball. Okay. Are the events baseball and are they independent events?

[00:04:36]So, here we have to know whether or not if the probability of baseball What's the probability of baseball?

[00:04:44]Um we want to know Okay, what we ultimately want to know for independence is whether baseball given you know age greater than or equal to 35 whether or not those are equal. So, this this one over here, probability that they're playing baseball Um baseball is 259 out of 4193. Uh 259 out of What is that number? 4,193?

[00:05:16]So, let's just Oh.

[00:05:20]Let's 259 / 41 93.

[00:05:28]There's a lot of ways to show independence, but this is the this is the definition of it.

[00:05:34]Probability of baseball given that their age is greater than or equal to 35 would be looking at this row here. That would be 61 out of 121. So, this divided by the total cuz we're going to say assuming we're given that their age is greater than 3 And like I said, this is not the only way you could have done it.

[00:05:52]Um that's just the one I chose.

[00:05:55]>> [sighs] >> 61 out of 121. Yeah, I got that right.

[00:05:57]>> [snorts] >> 61 / 121 um point This is equal to 0.5041.

[00:06:06]These are not equal, so they are not independent.

[00:06:11]Because those probabilities, if they're independent, then those probabilities ought to be the same.

[00:06:17]Consider the data for all professional athletes from the 2-way table. Determine if it's appropriate to carry out chi-squared test for independence to investigate whether there's association age group and sport played using this table. Explain your answer. Okay, so chi-squared um checks we got to do is each square needs to have at least five values in it.

[00:06:38]So, we have to do the expected values of them or have to be greater than equal to five. So, first let's do are they independent?

[00:06:46]So, they tell us that they were independent um independent uh selected people. So, research association um No reason to think that they're not independent. They just did say do an observation. I mean, depends on how they selected those people, but we'll say that that's fine.

[00:07:07]Probably the biggest one they want you to do here is just to do the um the um the expected value. So, we want to do the expected value.

[00:07:19]have to be greater than equal to five.

[00:07:21]And that's what we'll that's probably what they're wanting you to verify in this question is just to make sure you understand how to do the expected value calculation here.

[00:07:30]So, the simplest way to do it is for each of them if we're going to make an expected table, you're going to do the row row total you're going to do the row total times the column total divided by the total total.

[00:07:44]Okay, now I want to look at like which one might be like the smallest one maybe. So, I'm going to look at like you know, cuz any of the other ones are going to be bigger. So, I'm just going to look for the smallest one probably cuz I'm just a little bit lazy.

[00:07:57]So, let's look at this one. This one is the the smallest one. 121 * 516 / 4193.

[00:08:04]These are the smallest row and column totals. So, honestly, if this one is greater than five, then all the other ones will be greater than five and so I can make a logical argument and appeal that I don't need to actually calculate it for the entire table.

[00:08:17]Um I don't know if they would accept that on a test if I'm being too thinking, but this is equal to 14.890.

[00:08:27]So, what I would do, just because I am lazy, I'm saying the smallest expected value entry is for which one is that? Is is for um that the basketball people over the age of 35 is for age greater than equal to 35.

[00:08:53]Um and basketball is 14.890, which is greater than five. So, yes, can use chi-squared test for independence.

[00:09:14]Okay.

[00:09:15]I don't know if there's anything else to check. That's all that comes to mind from my part. I'll double check. You guys can correct me if I'm wrong there, but that's all I can think of is all I would be checking for.