All right, so in this information, we're given some a C T. Summary data from 2000 were given both the mean on and the standard deviation from a sample of English and math tests. So the English, I mean, was 20.6 with a standard deviation of six. The math mean was 21 with a standard deviation of 5.1. And in order compare these. We're gonna need to know the formula for the Z score X minus X bar over s where X is the data value expires. The mean and s s a standard deviation. So part A what if somebody scored a 30 on which of these exams the English or math? Would they have performed better relative to that subject? So for English will substitute the values into our Z score formula, and we find that this has a Z score of 1.56 So this score of 30 it's 1.56 standard deviations better than average for the English test for math. We repeat the procedure using that's statistics from the math test, and we find that this person are that this score is 1.76 standard deviations above the mean. So who had the better performance? Um, relative to the English and math scores. A score of 30 indicates a better performance in the math on the math test, right? What if we had a score of 23 so we'll roll. Repeat the same procedure 23 minus 20.6 over six gives us a Z score of 0.4. And on the math test, 23 minus 21 over 5.1 gives us a Z score of 0.39 So I would say that for these two, um, fields these two subject areas, though, z scores, but as close as you can get, So the performance is about the same. And then we're gonna look at a lower test score. So his test score of 12. So on English, we have 12 minus 20.6 over six. So since it's going to be below average, it's gonna have a negative Z score. So negative 1.43 And on math, we're also gonna get a negative Z score. So negative 1.7 So in which subject dairy was the performance better and in this case, it was on the English test because it is fewer standard deviations below average than the map. So the next part of the question says, Then why did the better test switch from English to math Between questions A and C. So why was 30? Why was that so much better than average And why I was 12? Um, I was 30. So much better in math than average. And when I was 12 so much better in the below average range. So why did this switch from math English? And it really has to do with, um, the relative nous to the standard deviation. So 30 is more standard deviations above their means in math, and 12 is less standard deviations below the mean in English. So since what we're measuring a standard deviations, that's what we're looking for. So this is closer to zero. That means it's better because a score above zero is a better score. So this person is far, but are further below average than this person. So the map is further below average than the English. All right, And then part e, we're gonna look at one specific student who scored a 26 so our data value is 26. So did that student do better on the English tests, or did they do better on the math test? So we'll calculate the Z scores in the same way. So for English, the student would have a Z score of 0.9, and for mouth student would have a Z score of 0.98 So even though it's pretty close, it looks like the student has a better performance on the math test because her score is more Sander deviations above the respective mean meaning. More standard deviations above the math mean that the score of 26 would be above the Englishman.