Question

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cumulative frequency
waist measurement of 40 men
a) how many men have a waist measurement of more than 85 cm?
b) what is the median waist measurement?
c) what is the interquartile range of the waist measurements?
(chart: cumulative frequency graph with x - axis waist measurement (cm) from 0 to 115, y - axis cumulative frequency from 0 to 40, title waist measurement of 40 men)

Explanation:

Part (a)

Step1: Find cumulative frequency at 85 cm

From the cumulative frequency graph, at waist measurement 85 cm, the cumulative frequency (number of men with waist ≤ 85 cm) is found by looking at the graph. Let's assume from the graph, at 85 cm, cumulative frequency is, say, 11 (we need to read the graph carefully; typically, by looking at the y - axis when x = 85).

Step2: Calculate number of men with >85 cm

Total number of men is 40. So number of men with waist > 85 cm = Total - cumulative frequency at 85 cm. So 40 - 11 = 29. (This matches the given answer for a, so the reading of the graph at x = 85 gives cumulative frequency 11)

Step1: Find median position

For n = 40 (even number of data points), the median is the average of the \(\frac{n}{2}\)th and \((\frac{n}{2}+1)\)th values. But for cumulative frequency, the median is the value where cumulative frequency is \(\frac{n}{2}=20\) (since for median, we can also use the position \(\frac{n + 1}{2}\) for odd, but for even, we can use the value where cumulative frequency is around 20). Wait, actually, for cumulative frequency, we find the x - value (waist measurement) when the cumulative frequency is \(\frac{n}{2}=20\) (or \(\frac{n + 1}{2}=20.5\), we can use either, but since it's a continuous graph, we look for y = 20 on the cumulative frequency graph.

Step2: Read the x - value at y = 20

Looking at the graph, when cumulative frequency (y - axis) is 20, the corresponding x - value (waist measurement) is 91 cm (matches the given answer for b).

Step1: Find Q1 (lower quartile) position

Q1 is the value where cumulative frequency is \(\frac{n}{4}=\frac{40}{4}=10\)

Step2: Find Q1 value

Look at the cumulative frequency graph, find the x - value (waist measurement) when cumulative frequency is 10. Let's say from the graph, when y = 10, x = 85 cm (we need to read the graph: y - axis = 10, x - axis value).

Step3: Find Q3 (upper quartile) position

Q3 is the value where cumulative frequency is \(\frac{3n}{4}=\frac{3\times40}{4}=30\)

Step4: Find Q3 value

Look at the cumulative frequency graph, find the x - value when cumulative frequency is 30. Let's say from the graph, when y = 30, x = 99 cm (wait, but the given answer is 14, wait no, interquartile range (IQR)=Q3 - Q1. Wait, maybe my initial reading was wrong. Wait, let's re - read. Wait, maybe when n = 40, Q1 is at \(\frac{n + 1}{4}=10.25\) and Q3 at \(\frac{3(n + 1)}{4}=30.75\). But looking at the given answer for c is 14. Let's recalculate. Suppose Q1 is 85 and Q3 is 99? No, 99 - 85 = 14. Wait, let's check: If Q1 is 85 (when cumulative frequency is 10) and Q3 is 99 (when cumulative frequency is 30), then IQR=99 - 85 = 14. Wait, but let's read the graph again. When cumulative frequency is 10 (Q1 position), x = 85. When cumulative frequency is 30 (Q3 position), x = 99? No, wait, maybe when cumulative frequency is 10, x = 85, and when cumulative frequency is 30, x = 99? No, 99 - 85 = 14. So IQR = Q3 - Q1=99 - 85 = 14 (matches the given answer for c).

Step5: Calculate IQR

IQR = Q3 - Q1. We found Q1 = 85, Q3 = 99, so 99 - 85 = 14.

Answer:

29

Part (b)