Question

the five point summary is as follows: min = 1.9, q₁ = 2.6, median = 3, q₃ = 3.2, max = 4.4. the interquartile range is iqr = 0.6. give the outlier fences below.
table with lower fence (1.7) and upper fence (4.1)
list any outliers below. if there is more than 1, separate them by commas.
4.4
part 3 of 3
hint 7 hint 8
the only outlier in the data set is 4.4.
slide the elements of the boxplot below into their correct positions.
boxplot with x-axis: 1.5, 2, 2.5, 3, 3.5, 4, 4.5; points: 1.5 (marked with *), 2.4, 3.2, 4.1, 4.9; blue box between 2.4 and 4.1, line at 3.2, whiskers to 1.5 and 4.9, outlier at 1.4? (note: ocr may have minor errors, but key elements are boxplot and numerical data)

Answer:

To determine the correct positions for the boxplot elements, we use the five - number summary and the outlier fences:

Step 1: Identify the minimum, \(Q_1\), median, \(Q_3\), and maximum (non - outlier)

• The minimum value (excluding outliers) is \(1.9\). But we also have an outlier at \(4.4\), and the upper fence is \(4.1\). So the maximum non - outlier value is the largest value less than or equal to the upper fence. From the given data, the values we have are related to the five - number summary: \(\text{min}=1.9\), \(Q_1 = 2.6\), \(\text{median}=3\), \(Q_3=3.2\), \(\text{max}=4.4\) (outlier), and upper fence \(= 4.1\).
• The left end of the box (lower quartile, \(Q_1\)) should be at \(2.6\) (but in the given numbers for sliding, we have \(2.4\) which seems incorrect, maybe a typo, but following the correct five - number summary logic:
• The left whisker should go to the minimum non - outlier value. The lower fence is \(1.7\), and the minimum data value is \(1.9\) which is greater than the lower fence (\(1.9>1.7\)), so the left end of the whisker is at \(1.9\). But in the given sliding elements, we have \(1.5\) (maybe a different scale, but let's use the given numbers).
• The left side of the box ( \(Q_1\)): In a boxplot, the box starts at \(Q_1\). From the five - number summary, \(Q_1 = 2.6\), but in the sliding elements, we have \(2.4\) (maybe a mistake, but let's proceed with the given sliding numbers and the correct positions based on the five - number summary and fences).
• The median line inside the box: The median is \(3\), but in the sliding elements, we have \(3.2\) (maybe a mistake).
• The right side of the box ( \(Q_3\)): \(Q_3 = 3.2\)
• The right end of the whisker (non - outlier maximum): The upper fence is \(4.1\), so the right end of the whisker is at \(4.1\) (since \(4.4\) is an outlier, it is plotted as a separate point).
• The outlier: \(4.4\) (or \(4.9\) in the sliding elements? Maybe a typo, but based on the earlier calculation, the outlier is \(4.4\))

Correct Positions (assuming the sliding elements are to be matched to the correct boxplot parts):

• The leftmost point (outlier or minimum? But we know the outlier is \(4.4\), and the minimum is \(1.9\). Wait, in the given boxplot, there is a star at \(1.4\) or \(1.5\)? Maybe the lower fence is \(1.7\), and if there was a value less than \(1.7\), it would be an outlier, but our minimum is \(1.9>1.7\), so no lower outlier. The upper outlier is \(4.4\) (or \(4.9\) in the sliding elements, maybe a typo).
• The left end of the box ( \(Q_1\)): Should be at \(2.6\), but in the sliding elements, we have \(2.4\) (maybe a mistake, but if we follow the given sliding numbers and the correct boxplot structure:
• The box starts at \(Q_1\) (let's say \(2.4\) is a misprint for \(2.6\)), the median line at \(3\) (misprint for \(3.2\)? No, median is \(3\)), the box ends at \(Q_3 = 3.2\), the right whisker ends at upper fence \(4.1\), and the outlier is at \(4.4\) (or \(4.9\)).

But based on the given five - number summary and the outlier calculation:

• Lower Fence: \(Q_1-1.5\times\text{IQR}=2.6 - 1.5\times0.6=2.6 - 0.9 = 1.7\)
• Upper Fence: \(Q_3 + 1.5\times\text{IQR}=3.2+1.5\times0.6=3.2 + 0.9 = 4.1\)

So the non - outlier maximum is the largest value \(\leq4.1\). Our original max is \(4.4>4.1\), so \(4.4\) is an outlier.

For the boxplot:

• The left whisker goes from the minimum (\(1.9\)) to \(Q_1(2.6)\)
• The box is from \(Q_1(2.6)\) to \(Q_3(3.2)\) with the median (\(3\)) inside.
• The right whisker goes from \(Q_3(3.2)\) to the upper fence (\(4.1\))
• The outlier is at \(4.4\)

If we are to slide the elements:

• The leftmost non - outlier point (start of left whisker): \(1.9\) (but in the given sliding elements, we have \(1.5\), maybe a different dataset scale, but based on the given numbers in the problem:
• The left end of the box ( \(Q_1\)): \(2.4\) (maybe incorrect, but if we use the given sliding numbers and the correct boxplot logic, the box should be between \(Q_1\) and \(Q_3\), median in the middle, whiskers to non - outlier min and max, and outlier as a separate point.

Assuming the sliding elements are:

• Left outlier (if any): But our lower fence is \(1.7\), min is \(1.9\), so no left outlier. The star at \(1.4\) or \(1.5\) is maybe a distractor.
• Left end of box: \(2.4\) (should be \(2.6\))
• Median line: \(3.2\) (should be \(3\))
• Right end of box: \(3.2\) (correct, \(Q_3\))
• Right end of whisker: \(4.1\) (correct, upper fence)
• Outlier: \(4.4\) (or \(4.9\) in the sliding elements, maybe a typo)

But if we strictly follow the calculation for the boxplot elements:

• Minimum (non - outlier): \(1.9\)
• \(Q_1\): \(2.6\)
• Median: \(3\)
• \(Q_3\): \(3.2\)
• Maximum (non - outlier): \(4.1\)
• Outlier: \(4.4\)

If we have to use the given sliding numbers ( \(1.5\), \(2.4\), \(3.2\), \(4.1\), \(4.9\)):

• The left whisker should be at the minimum non - outlier. Since our min is \(1.9\), but the closest in the sliding numbers is \(1.5\) (maybe a different dataset).
• The left end of the box ( \(Q_1\)): \(2.4\)
• The median line: \(3.2\) (incorrect, but maybe a mistake)
• The right end of the box ( \(Q_3\)): \(3.2\) (correct)
• The right end of the whisker: \(4.1\) (correct)
• The outlier: \(4.9\) (maybe a misprint for \(4.4\))

However, based on the initial five - number summary and outlier calculation, the correct outlier is \(4.4\), and the boxplot elements should be positioned as:

• Left whisker: from \(1.9\) to \(2.6\)
• Box: from \(2.6\) to \(3.2\) with median at \(3\)
• Right whisker: from \(3.2\) to \(4.1\)
• Outlier: at \(4.4\)

If we consider the sliding elements given ( \(1.5\), \(2.4\), \(3.2\), \(4.1\), \(4.9\)) and match them:

• The leftmost point (maybe a lower outlier, but we don't have one, so maybe it's a mistake, but if we assume):
• Left outlier: \(1.5\) (incorrect, but for the sake of sliding)
• Left end of box: \(2.4\)
• Median: \(3.2\) (incorrect)
• Right end of box: \(3.2\) (correct)
• Right end of whisker: \(4.1\) (correct)
• Outlier: \(4.9\) (incorrect, should be \(4.4\))

But the correct outlier is \(4.4\), and the boxplot should be constructed with the five - number summary values.

Final Answer (for the outlier part, which is already given as correct in the problem, and for the boxplot positions, the correct outlier is \(4.4\), and the boxplot elements should be placed as per the five - number summary and fences. If we have to list the outlier as per the question, the outlier is \(\boldsymbol{4.4}\)