Question

part 1 of 2
draw the percentile graph using class boundaries for the x - axis and the percentages for the y - axis. round the answer to nearest percent.
cumulative percentages
(graph with class boundaries: 365.5, 386.5, 407.5, 428.5, 449.5, 470.5, 491.5, 512.5, 533.5 and cumulative percentages points)
part: 1 / 2
part 2 of 2
find the approximate values that corresponds to the given percentiles by constructing a percentile graph. round your answers to the nearest whole number.
percentiles speeds (mph)
8th
21st
27th
52nd
97th

Explanation:

To find the speed corresponding to each percentile, we use the percentile graph (ogive) by locating the percentile on the y - axis (cumulative percentage), drawing a horizontal line to the graph, then a vertical line down to the x - axis (class boundaries) to get the speed.

Step 1: 8th Percentile

• Locate 8% on the y - axis. Draw a horizontal line to the ogive curve, then a vertical line down to the x - axis. From the graph, the x - value (class boundary) corresponding to 8% is approximately 386 (since at 386.5, the percentage is 14.8, and we interpolate or directly read from the graph's trend. Looking at the points, at 365.5 it's 0, at 386.5 it's 14.8. We can do a linear approximation: Let \(y = mx + b\), where \(x\) is class boundary, \(y\) is percentage. At \(x = 365.5,y = 0\); at \(x = 386.5,y = 14.8\). The slope \(m=\frac{14.8 - 0}{386.5 - 365.5}=\frac{14.8}{21}\approx0.7048\). We want \(y = 8\), so \(8=0.7048(x - 365.5)+0\), \(x=\frac{8}{0.7048}+365.5\approx11.35 + 365.5\approx376.85\approx377\)? Wait, maybe a better way is to look at the graph. Wait, the first point after 0 is at 386.5 with 14.8. So 8% is between 365.5 and 386.5. But maybe the graph is such that at 8th percentile, the speed is around 380? Wait, no, let's re - examine. Wait the x - axis labels are 365.5, 386.5, 407.5, 428.5, 449.5, 470.5, 491.5, 512.5, 533.5. The y - axis at 365.5 is 0, at 386.5 is 14.8. So to find 8th percentile, we can use proportionality. The distance from 0 to 14.8 on y - axis is 14.8 units, corresponding to x - distance from 365.5 to 386.5 (21 units). We need to find x when y = 8. So \(\frac{x - 365.5}{21}=\frac{8}{14.8}\), \(x=365.5+\frac{8\times21}{14.8}\approx365.5 + 11.35\approx376.85\approx377\). But maybe the graph is designed to have the 8th percentile around 380? Wait, maybe I made a mistake. Alternatively, looking at the graph, the first point is (365.5, 0), next (386.5, 14.8). So 8 is about half of 14.8, so x is about 365.5+\( \frac{8}{14.8}\times21\approx365.5 + 11.35 = 376.85\approx377\). But maybe the answer is 380? Wait, perhaps the graph is more straightforward. Let's check the other percentiles.

Step 2: 21st Percentile

• Locate 21% on the y - axis. The point at 407.5 has 24.9. So 21% is between 386.5 (14.8) and 407.5 (24.9). The difference in y: \(24.9 - 14.8 = 10.1\), difference in x: \(407.5 - 386.5 = 21\). We need \(y = 21\), so \(y - 14.8=6.2\). So \(x=386.5+\frac{6.2}{10.1}\times21\approx386.5+12.91\approx399.41\approx399\).

Step 3: 27th Percentile

• Locate 27% on the y - axis. The point at 407.5 is 24.9, next at 428.5 is 33.6. Difference in y: \(33.6 - 24.9 = 8.7\), difference in x: \(428.5 - 407.5 = 21\). We need \(y = 27\), so \(y - 24.9 = 2.1\). So \(x=407.5+\frac{2.1}{8.7}\times21\approx407.5 + 5.06\approx412.56\approx413\).

Step 4: 52nd Percentile

• Locate 52% on the y - axis. The point at 470.5 is 48.0, next at 491.5 is 57.9. Difference in y: \(57.9 - 48.0 = 9.9\), difference in x: \(491.5 - 470.5 = 21\). We need \(y = 52\), so \(y - 48.0 = 4.0\). So \(x=470.5+\frac{4.0}{9.9}\times21\approx470.5+8.48\approx478.98\approx479\).

Step 5: 97th Percentile

• Locate 97% on the y - axis. The last point is (533.5, 100.0). The point at 512.5 is 71.5. So 97% is close to 100%, so we can see that from 512.5 (71.5) to 533.5 (100.0), difference in y: \(100 - 71.5 = 28.5\), difference in x: \(21\). We need \(y = 97\), so \(y - 71.5 = 25.5\). So \(x=512.5+\frac{25.5}{28.5}\times21\approx512.5 + 18.77\approx531.27\approx531\). But since the last point is 533.5 at 100%, 97% is very close to 533. So approximately…

Answer:

• 8th: \(\boldsymbol{377}\) (or 380 depending on graph interpretation)
• 21st: \(\boldsymbol{400}\) (or 399)
• 27th: \(\boldsymbol{413}\)
• 52nd: \(\boldsymbol{479}\)
• 97th: \(\boldsymbol{533}\)

(Note: The answers may vary slightly depending on the precision of reading the percentile graph. The above is based on linear interpolation between the given points on the ogive.)