Question

consider histogram (iii), which has a sample size of 100. (iii) sample of size 100 in this histogram, the first midpoint label is at 27 and the final midpoint label is at 31. so, the lowest value, represented by the bar to the left of the 27, is . the highest value represented by the bar to the right of 31, is .

Explanation:

Step1: Determine class width

The midpoints are at 27, 28, 29, 30, 31. The difference between consecutive midpoints is \(28 - 27 = 1\), so the class width is 1. But actually, for a histogram, the class width is the distance between the lower (or upper) bounds of consecutive classes. Since midpoints are spaced by 1, the class width (range of each bar) is 1? Wait, no, midpoint of a class is \(\frac{\text{lower bound}+\text{upper bound}}{2}\). Let's assume the classes are symmetric around the midpoints. So for midpoint 27, the class is from \(27 - 0.5 = 26.5\) to \(27 + 0.5 = 27.5\)? Wait, no, if the midpoints are 27,28,29,30,31, the class width (the interval between the start of one class and the start of the next) should be 1? Wait, maybe the class width is 1, so the first class (left of 27 midpoint) would have midpoint less than 27. Wait, the first midpoint is 27, so the class before (left of 27) would have midpoint \(27 - 1 = 26\)? Wait, no, the distance between midpoints is the class width. Wait, let's think again. The midpoints are 27,28,29,30,31. So the class width (the width of each bar, i.e., the range of values each bar represents) is \(28 - 27 = 1\) (since midpoints are 1 unit apart). Therefore, the lower bound of the class with midpoint 27 is \(27 - 0.5 = 26.5\), and upper bound is \(27 + 0.5 = 27.5\). Wait, but the problem says "the bar to the left of the 27" (the first midpoint label is 27, so the bar left of 27 is the previous class). So its midpoint would be \(27 - 1 = 26\) (since class width is 1, midpoints are 1 apart). Then the lower bound of that class is \(26 - 0.5 = 25.5\)? No, maybe the class width is 1, so each class is 1 unit wide. So midpoint 27: class is 26.5 - 27.5, midpoint 28: 27.5 - 28.5, etc. Wait, no, that would make class width 1 (27.5 - 26.5 = 1). So the bar to the left of 27 (midpoint 27) would be the class before 26.5 - 27.5, which is 25.5 - 26.5, so its midpoint is 26. But the problem is about the lowest value represented by that bar. The lowest value in a class is the lower bound. Wait, maybe the class width is 1, so the first midpoint is 27, so the class for midpoint 27 is from \(27 - 0.5 = 26.5\) to \(27 + 0.5 = 27.5\). Then the bar to the left of 27 (the midpoint 27 bar) would be the class before, which is from \(26.5 - 1 = 25.5\) to \(26.5\)? No, that doesn't make sense. Wait, maybe the midpoints are at 27,28,29,30,31, and the class width is 1, so each class is 1 unit, so the first class (left of 27) has midpoint 26, so its range is 25.5 - 26.5, so the lowest value is 25.5? Wait, no, maybe the class width is 1, so the midpoint is the center, so the class is midpoint - 0.5 to midpoint + 0.5. So for midpoint 27: 26.5 - 27.5, midpoint 28: 27.5 - 28.5, etc. Then the bar to the left of 27 (the 26.5 - 27.5 bar) would be the bar before, which is 25.5 - 26.5, so its midpoint is 26. Then the lowest value in that bar is 25.5? Wait, but maybe the class width is 1, so the distance between midpoints is 1, so the class width is 1. So the first midpoint is 27, so the class is 27 - 0.5 to 27 + 0.5 = 26.5 - 27.5. Then the bar to the left (lower values) would be 26.5 - 1 = 25.5 to 26.5, so midpoint 26. Then the lowest value in that bar is 25.5? Wait, but the problem says "the lowest value, represented by the bar to the left of the 27, is...". Similarly, the bar to the right of 31 (midpoint 31) would be 31.5 - 32.5? Wait, no, midpoint 31 is 30.5 - 31.5, so the bar to the right is 31.5 - 32.5, so the highest value is 32.5? Wait, maybe I made a mistake. Wait, let's check the midpoints: 27,28,29,30,31. The difference between midpoints is 1, so the class width (the width of each bar) is 1. So each class is 1 unit wide. So the midpoint is the center, so the class is (midpoint - 0.5) to (midpoint + 0.5). So for midpoint 27: 26.5 - 27.5, midpoint 28: 27.5 - 28.5, midpoint 29: 28.5 - 29.5, midpoint 30: 29.5 - 30.5, midpoint 31: 30.5 - 31.5. Then the bar to the left of the 27 midpoint bar (which is 26.5 - 27.5) would be the bar before, which is 25.5 - 26.5 (midpoint 26), so the lowest value in that bar is 25.5. The bar to the right of the 31 midpoint bar (30.5 - 31.5) would be 31.5 - 32.5 (midpoint 32), so the highest value in that bar is 32.5. Wait, but maybe the class width is 1, so the midpoints are at 27,28,29,30,31, so the classes are 27-28, 28-29, etc.? No, that would make midpoint 27.5, 28.5, etc. So maybe the midpoints are at the integers, so the class is from (midpoint - 0.5) to (midpoint + 0.5). So that's the correct way. So the first midpoint is 27, so class is 26.5 - 27.5. The bar to the left (lower) is 25.5 - 26.5, so lowest value is 25.5. The bar to the right of 31 (class 30.5 - 31.5) is 31.5 - 32.5, so highest value is 32.5. Wait, but maybe the problem is simpler. The midpoints are 27,28,29,30,31, so the class width is 1 (since 28-27=1). So the first class (left of 27) has midpoint 26, so its range is 26 - 0.5 = 25.5 to 26 + 0.5 = 26.5? No, that's the same as before. Alternatively, maybe the class width is 1, so the classes are 27-28, 28-29, etc., with midpoints at 27.5, 28.5, etc. But the problem says the first midpoint label is at 27, so that can't be. So maybe the class width is 1, and the midpoints are at the integers, so each class is from (midpoint - 0.5) to (midpoint + 0.5). So that's the key. So for midpoint 27: 26.5 - 27.5, midpoint 28: 27.5 - 28.5, etc. Then the bar to the left of 27 (the 26.5 - 27.5 bar) is the bar before, which is 25.5 - 26.5, so the lowest value in that bar is 25.5. The bar to the right of 31 (the 30.5 - 31.5 bar) is 31.5 - 32.5, so the highest value in that bar is 32.5. Wait, but maybe the problem is that the midpoints are 27,28,29,30,31, so the distance between midpoints is 1, so the class width is 1. Therefore, the lower bound of the first class (left of 27) is \(27 - 1 = 26\)? No, that's not right. Wait, let's think of the midpoint formula: midpoint = (lower + upper)/2. If the class width is \(w\), then upper = lower + \(w\). So midpoint = (lower + lower + \(w\))/2 = lower + \(w/2\). So lower = midpoint - \(w/2\), upper = midpoint + \(w/2\). Now, the midpoints are 27,28,29,30,31. The difference between consecutive midpoints is 1, so \(w/2 = 1\)? No, the difference between midpoints is \(w/2 + w/2 = w\)? Wait, no. If two consecutive classes have midpoints \(m_1\) and \(m_2\), then the distance between \(m_1\) and \(m_2\) is \(w/2 + w/2 = w\), because the upper bound of the first class is \(m_1 + w/2\), and the lower bound of the second class is \(m_2 - w/2\), and since they are consecutive, \(m_1 + w/2 = m_2 - w/2\), so \(m_2 - m_1 = w\). Ah! So the class width \(w\) is equal to the difference between consecutive midpoints. So here, midpoints are 27,28,29,30,31, so \(w = 28 - 27 = 1\). Therefore, for midpoint 27: lower = 27 - 1/2 = 26.5, upper = 27 + 1/2 = 27.5. For midpoint 28: lower = 28 - 0.5 = 27.5, upper = 28 + 0.5 = 28.5, etc. So the bar to the left of the 27 midpoint bar (which is 26.5 - 27.5) is the bar before, which has midpoint \(27 - 1 = 26\) (since \(w = 1\), so midpoints are 1 apart). So for midpoint 26: lower = 26 - 0.5 = 25.5, upper = 26 + 0.5 = 26.5. Therefore, the lowest value in that bar is 25.5. Similarly, the bar to the right of the 31 midpoint bar (30.5 - 31.5) has midpoint \(31 + 1 = 32\), so lower = 32 - 0.5 = 31.5, upper = 32 + 0.5 = 32.5. Therefore, the highest value in that bar is 32.5.

Step2: Confirm lowest and highest values

So the lowest value (left of 27 midpoint bar) is 25.5, and the highest value (right of 31 midpoint bar) is 32.5. Wait, but maybe the problem considers the class width as 1, so the midpoints are at the integers, so the classes are 27-28, 28-29, etc., with midpoints at 27.5, 28.5, etc. But the problem says the first midpoint label is at 27, so that's conflicting. Wait, maybe the class width is 1, and the midpoints are at the integers, so each class is from (midpoint - 0.5) to (midpoint + 0.5). So that's the only way the midpoint is at the integer. So yes, that makes sense. So the first midpoint is 27, so class is 26.5 - 27.5, midpoint 28: 27.5 - 28.5, etc. Then the bar to the left of 27 (the 26.5 - 27.5 bar) is the bar before, which is 25.5 - 26.5, so lowest value is 25.5. The bar to the right of 31 (the 30.5 - 31.5 bar) is 31.5 - 32.5, so highest value is 32.5.

Answer:

The lowest value is \(\boldsymbol{25.5}\) and the highest value is \(\boldsymbol{32.5}\).