Question

the following are the temperatures, in degrees fahrenheit, of 20 us cities on april 3rd. the boxplot represents the data. 42, 47, 48, 50, 50, 50, 51, 51, 52, 52, 54, 55, 55, 55, 56, 57, 58, 59, 59, 60. april 3 temperatures boxplot image with axis 46, 48, 50, 52, 54, 56, 58, 60. why is the boxplot incorrect? the minimum temperature is 42°f, but the boxplot shows 46°f. the maximum temperature is 60°f, but the boxplot shows 66°f. the median temperature is 53°f, but the boxplot shows 52°f. the first quartile temperature is 50°f, but the boxplot shows 53°f.

Explanation:

Step1: Analyze each option

- **Option 1**: The minimum temperature in the data is \(42^\circ\text{F}\), but the boxplot's left whisker starts at \(46^\circ\text{F}\). Wait, no, let's check the maximum first. Wait, the data's maximum is \(60^\circ\text{F}\). The boxplot's right whisker end is at \(66^\circ\text{F}\)? Wait, no, the x - axis labels are 46,48,50,52,54,56,58,60. Wait, the right end of the whisker (the maximum) in the boxplot: the data's maximum is 60. Let's check each option:
- **Option 1**: Minimum data value is 42. The boxplot's left whisker starts at 46? Wait, the leftmost part of the whisker (the minimum) in the boxplot: the x - axis has 46 as the first label. But the data's minimum is 42. Wait, but let's check the maximum. The data's maximum is 60. The boxplot's right whisker (the maximum) – looking at the boxplot, the right end of the whisker (the maximum) is shown? Wait, the options:
- Option 2: "The maximum temperature is \(60^\circ\text{F}\), but the boxplot shows \(66^\circ\text{F}\)". Wait, the x - axis labels are up to 60. Wait, maybe the right end of the whisker (the maximum) in the boxplot is at 66? Wait, no, the x - axis is labeled 46,48,50,52,54,56,58,60. Wait, maybe the boxplot's maximum (the end of the right whisker) is at 66? But the data's maximum is 60. Wait, let's check the other options.
- Option 3: Median of 20 data points. For \(n = 20\) (even), median is the average of the 10th and 11th terms. The data: 42,47,48,50,50,50,51,51,52,52,54,55,55,55,56,57,58,59,59,60. 10th term is 52, 11th term is 54. Median \(=\frac{52 + 54}{2}=53\). The boxplot's median line: the box is split, and the x - axis has 52 as a label. Wait, the boxplot's median – if the box is between, say, 50 and 56, the median line? Wait, the option says median is 53, boxplot shows 52. But let's check the maximum.
- Option 4: First quartile (Q1) for \(n = 20\). The first quartile is the median of the first 10 data points. First 10 data points: 42,47,48,50,50,50,51,51,52,52. Median of these 10 (even number) is average of 5th and 6th terms: \(\frac{50+50}{2}=50\). The boxplot's Q1 (left end of the box) – the box starts at, looking at the x - axis, 53? Wait, no, the x - axis labels are 46,48,50,52,54,56,58,60. The box starts at, maybe 50? No, the option says Q1 is 50, boxplot shows 53. But let's check the maximum again. The data's maximum is 60. The boxplot's right whisker (the maximum) – if the boxplot's right end (maximum) is at 66, which is not in the data. Wait, the options: Option 2 says "The maximum temperature is \(60^\circ\text{F}\), but the boxplot shows \(66^\circ\text{F}\)". Wait, maybe the boxplot's right whisker (the maximum) is drawn at 66, but the data's maximum is 60. Let's confirm the data: the last value is 60. So the maximum is 60. If the boxplot's maximum (the end of the right whisker) is 66, that's incorrect. Let's check other options:
- Option 1: Minimum is 42, boxplot shows 46. But maybe the left whisker is from 42 to 46? No, the left end of the whisker (the minimum) should be 42. But the x - axis starts at 46. Wait, but the option says "the boxplot shows 46°F" as minimum. But maybe the left whisker is from 42 (but the x - axis label is 46). Wait, no, the key is which option is correct. Let's re - evaluate:
- Data: 20 values.
- Minimum: 42.
- Maximum: 60.
- Median: (52 + 54)/2 = 53.
- Q1: median of first 10: (50 + 50)/2 = 50.
- Q3: median of last 10: data from 11th to 20th: 54,55,55,55,56,57,58,59,59,60. Median of these 10: (56 + 57)/2 = 56.5.
- Now check the options:
- Option 1: Minimum is 42, boxplot shows 46. But maybe the left whisker is from 42 to 46? No, the minimum should be the left end of the whisker. But the x - axis label is 46. Wait, maybe the boxplot's minimum (left whisker end) is 46, but data's minimum is 42. But let's check Option 2: Maximum is 60, boxplot shows 66. If the boxplot's right whisker end is 66, that's wrong. Option 3: Median is 53, boxplot shows 52. The box's median line – if the box is between Q1 and Q3, the median is inside. Option 4: Q1 is 50, boxplot shows 53. The left end of the box (Q1) should be 50, if boxplot shows 53, that's wrong. But which is correct? Wait, the boxplot's x - axis has labels 46,48,50,52,54,56,58,60. The rightmost label is 60, but maybe the whisker goes beyond to 66? The data's maximum is 60, so the boxplot's maximum (whisker end) should be 60. So Option 2 says "The maximum temperature is \(60^\circ\text{F}\), but the boxplot shows \(66^\circ\text{F}\)" – this is correct. Wait, but let's check the other options again. Wait, the first option: minimum is 42, boxplot shows 46. But the left whisker in the boxplot – the left end of the whisker (the minimum) – if the x - axis starts at 46, but the data's minimum is 42, then the left whisker should extend to 42, but the boxplot's left whisker starts at 46? No, maybe the boxplot's minimum (the left end of the whisker) is 46, but data's minimum is 42. But the option says "the boxplot shows 46°F" as minimum. But let's check the maximum. The data's maximum is 60, the boxplot's maximum (right end of whisker) – if the boxplot's right end is 66, which is not in the data, then Option 2 is correct.

Step2: Confirm the correct option

After analyzing each option, we see that the maximum temperature in the data set is \(60^\circ\text{F}\), but the boxplot (from the x - axis and the whisker) appears to show a maximum of \(66^\circ\text{F}\) (or the right whisker is drawn beyond 60 to 66), which is incorrect. The other options:
- Option 1: The minimum is 42, but the boxplot's left whisker starts at 46? But maybe the whisker is from 42 to 46, but the x - axis label is 46. However, the key is that the maximum in the data is 60, and the boxplot's maximum is 66, which is a clear error.
- Option 3: Median is 53, boxplot shows 52. But the median calculation: 10th term is 52, 11th is 54, median is 53. But maybe the boxplot's median line is at 52? But this is less clear than the maximum error.
- Option 4: Q1 is 50, boxplot shows 53. Q1 is 50, if the box starts at 53, that's wrong, but the maximum error is more obvious.

Answer:

B. The maximum temperature is \(60^\circ\text{F}\), but the boxplot shows \(66^\circ\text{F}\).