2026 AP Statistics Free Response #6

Ajouté : 2026-05-16

[00:00:00]All right. Number six, the investigative task for the AP Statistics 2026 version J.

[00:00:07]Okay, when a baseball game, baseball need to score runs. Score runs, player needs to get on base. Primary way to get on base is hitting the ball.

[00:00:13]Okay, cool. Explanation of baseball.

[00:00:14]Figure one shows the relationship between the number of hits and the number of runs for 30 randomly selected professional baseball teams. Consider the scatter plot in figure one. Describe the relationship between the number of hits and the number of runs on the baseball teams in context. Okay, um there is a I don't know. There is a strong strong positive correlation. Basically, it's a positive correlation between the number of hits, number of hits of what is it? Number of um hits and number of runs.

[00:01:04]You saying thing more than that? I'm trying to remember if there's anything else we want to add into that. Um describe the relationship. Um It's no apparent outliers.

[00:01:18]I don't know. I would say moderately strong.

[00:01:21]Maybe.

[00:01:22]I'm trying to remember all the things I want to say.

[00:01:24]Uh with There are no There are They don't want to analyze the data.

[00:01:28]They just want to know the relation.

[00:01:30]There are no significant outliers.

[00:01:38]Okay.

[00:01:39]Based on the model from the sample of 30 teams, equation of least squares regression for predicting the number of runs from the number of hits is given by that. Okay, cool. They did it for you.

[00:01:47]Using the given regression equation, calculate the point estimate for the predicted number of runs a team would score if they achieve their goal of 1250 hits. Show your work. So, the predicted number of teams would be would be -372.2 +0.823 times the number of hits, which would be 1250.

[00:02:05]That's uh -372.2 +0.823 * 1250.

[00:02:13]That's going to be 656.55 runs.

[00:02:20]Okay, I don't hm.

[00:02:22]Each team has a total salary equal to the sum of the salaries of all the players on the team. The team median total salary for the 30 teams is 160 million. Figure two, points with dots represent teams with total salary greater than the median.

[00:02:33]And the uh square teams represent total salary less than the median.

[00:02:37]Okay.

[00:02:38]Compare the team represented by the point that's circled in black uh uh circled with A with the other teams that have the same total total total salary classification.

[00:02:48]So, the other salary classification the other ones with the light squares, right? This one definitely has the most number of hits and the most number of runs. So, it is they have the most number of hits and runs in their category.

[00:03:13]Or their salary classification.

[00:03:16]Cuz it's furthest to the right and furthest up.

[00:03:20]I don't think I would say anything else than that.

[00:03:23]Feels It's all they want you to say. For each salary classification consider the linear relationship between the number of hits and the number of runs. For teams with total salary greater than the median, is the strength of the linear relationship stronger or weaker than similar teams are similar to the strength of the linear relationship for teams with total salary less than the median, explain. So, if you look at kind of like turn your head a little bit, look at the black dots, look at that that that correlation. The white ones, look at that correlation. I would say there's not really other than this point, this is like a high leverage point. If you ignored that one, this would be almost like there's no correlation, whereas the black one seemed to have a correlation. So, for the black ones, for the ones greater than the median, there definitely appears to be just a trend. So, um um the strength of the linear relation linear relationship is stronger because they uh are closer to a line with positive slope.

[00:04:37]And the other one is not.

[00:04:42]The other the the the less than median is not especially if removing this kind of strong outlier espe- removing the strong outlier. Strong leverage or high leverage outlier.

[00:05:15]It's high leverage because this guy makes the squared outs look like there's a linear trend, but if you eliminate him, then it doesn't appear there's to be much of a trend anyway. That's what makes it a kind of a high leverage affects the slope of that line of best fit. Okay.

[00:05:27]Part C, different types of intervals can be used when working with linear regression models. One type of interval is a confidence interval with this for the slope of the least squares regression line. Two other types of intervals in the context of problems are as follows. Confidence interval estimates the mean number of runs for all teams with a specific number of hits.

[00:05:41]Prediction interval that predicts the number of runs for a single team with a specific number of hits. All three intervals use the same formula. Point estimate plus or minus T star times standard error. Where T star is a critical value comes from a T distribution and minus two degrees of freedom. Same critical values when finding each of these intervals at the same level of confidence. Okay, so we're going to use T star for all of them.

[00:06:01]It's just the point estimate and the standard error is going to vary based on what we're going to do. Confidence interval for the mean number of runs or the predicted number of runs with a specific number of hits. Okay.

[00:06:12]Consider a point estimate for the predicted number of runs found in part A for the team who scores to achieve 1250 hits next year. Recall this value is calculated using a linear regression model based on the data from the sample of 30 teams. Okay, so there I have recalled that. What is the critical value for 95% confidence interval that will be used for the confidence interval for the mean number of as well for the prediction interval for the number of runs. Indicate the answer to two decimal places. Okay. So confidence intervals are about T distributions.

[00:06:38]So your degrees of freedom is always going to be the N minus two in this case. So it's going to be 28.

[00:06:45]And um we want to contain 95% of the data in here. So I want the 95% in here for a T distribution. Degrees of freedom eight. Um so we want the 95% area to be in there. And so using our calculator, how do we go about doing that in our calculation here? Is we are going to do um we're going to do a statistical like invT.

[00:07:07]And I just have to remember when I use the the invT, what is it going to tell us? We're going to use invT.

[00:07:14]Now invT always on my T-84 will always just give you area to the left. So you got to do a little bit. You can't just put 0.95 in there. It's always the area to the left of the point. Or you could use this point if you want the negative value cuz it's symmetric. But this is 95%. Each of these is 0.025 cuz they're 0.025. So that means the area to the left of here. Actually, I'll just put 0.025. That's probably easiest. 0.025.

[00:07:37]That'll give you this this value right over here.

[00:07:40]Degrees of freedom is 28.

[00:07:43]And that's going to give you a value there of -2.05.

[00:07:48]And this value here would be positive 2.05. So, our T crit here is going to be 2.05.

[00:07:55]Okay, for that. Cool.

[00:07:58]Standard error for the confidence interval mean number of seven is 17.48.

[00:08:02]Um assuming conditions for inference are met, calculate the 95% confidence interval for the mean number of hits for all teams with 1250.

[00:08:09]Uh confidence interval. So, we have the standard error. We have our T star. We just want to know what the point estimate is.

[00:08:18]And the point estimate is our our value we calculated here, like what we assume the center to be, 656.55.

[00:08:24]So, it's going to be 656.55 plus or minus the 2.05.

[00:08:32]And the standard error they gave us was 17.48.

[00:08:35]So, we're just going to do that calculation. I'm going to do that on my hand calculator instead of typing it out because I think that's 2.05 * 17.48. That's going to be an interval from 620.72 or I'll just do 716. I don't know.

[00:08:52]16 And then we'll just do the same value with a plus sign.

[00:08:59]And that's going to be 692.384.

[00:09:04]Okay.

[00:09:05]Standard error for prediction interval for the number of runs is 56.78.

[00:09:12]Assuming the conditions for calculate 95% interval for the with the 1250 hits.

[00:09:16]So, here we're going to um I want to just double-check if I'm if I should be using this point estimate for both of them.

[00:09:30]Yeah, I think it's the still the same.

[00:09:32]Um for part three, you're just going to do 656.55 plus or minus the 56.78 times the 2.05. So, exact same thing.

[00:09:51]It's going to be 540.

[00:09:53]0.151 Probably the difference is how we're going to interpret what that means because they're just two different ways we can calculate. One's for a single team and one is for So, I wonder if that's what the next part of the question's going to be.

[00:10:05]I have not pre-read the questions.

[00:10:07]Probably should have.

[00:10:08]>> [sighs] >> Would distribution of sample means be expected to have more variability or less variable distribution individual observations? Um Um sample means to be expected to have less variable distribution individual observations. Um it'll be less >> [snorts] >> because uh sample means have less variation.

[00:10:35]tion as n increases by central limit theorem, right?

[00:10:43]The standard error using the confidence interval that estimates the mean number of runs for the team X can be found using this formula.

[00:10:49]The standard error with the prediction interval predicts the number of runs can be found using this formula.

[00:10:55]Please don't ask me to do that. Um in both standard formulas, s is standard deviation of the residuals.

[00:11:00]n is the sample size and x bar is the sample mean of the number of hits.

[00:11:05]Okay, so sample mean. Based on the answer from part DI and the standard error formulas for the confidence interval and the prediction interval, explain why the prediction interval calculated in part three is wider than the confidence interval in part this part here.

[00:11:17]Um >> [snorts] >> Yeah, so I mean they're both proportional to s. You can pull out the s squared, right? They're all going to be proportional to s.

[00:11:32]You can factor out an S squared.

[00:11:34]So, um both calculations are proportional to the sample standard deviation.

[00:11:49]Sample standard deviation.

[00:11:57]Um but the individual team observations have a higher standard deviation.

[00:12:19]More variability.

[00:12:21]That matches what we said here. More variability.

[00:12:27]And that implies a larger standard error.

[00:12:35]Right? So, that's it's just sort of like it seemed like a lot, but they just wanted to just you analyze the formula and understand the connection between that sample standard deviation and how it affects that. So, we're saying the individual team one, which is the second one here, mean number is uh for the second one, prediction for a single team, versus the estimate mean number of runs for all teams, right? Like this one's going to have less standard uh deviations in the data cuz it's a sample mean. And so, because that's smaller, you expect the variation to be less, which is why like this number ended up being so much bigger than this number over here.

[00:13:10]Okay?