Question

finish times (to the nearest hour) for 59 dogsled teams are shown below. draw a ogive. use five classes. 261 276 236 244 279 296 284 299 290 290 248 256 338 360 341 333 261 267 287 296 313 311 306 306 299 303 277 283 304 305 287 290 287 288 297 299 332 330 309 327 306 327 285 291 295 298 306 315 310 318 318 320 333 321 323 324 327 302 319

Explanation:

Step1: Find the range

First, find the minimum and maximum values. The minimum value ($\text{min}$) is 236, and the maximum value ($\text{max}$) is 360. The range is $\text{max}-\text{min}=360 - 236 = 124$.

Step2: Determine class width

We need 5 classes, so the class width ($w$) is calculated as $\lceil\frac{\text{range}}{\text{number of classes}}
ceil=\lceil\frac{124}{5}
ceil = 25$ (we round up to ensure all data is covered).

Step3: Define the classes

The first class starts at the minimum value, 236. The classes will be:
- Class 1: $236 - 260$
- Class 2: $261 - 285$
- Class 3: $286 - 310$
- Class 4: $311 - 335$
- Class 5: $336 - 360$

Step4: Tally the frequencies

Count the number of data points in each class:
- Class 1 ($236 - 260$): 236, 244, 248, 256. So frequency ($f_1$) = 4.
- Class 2 ($261 - 285$): 261, 261, 267, 276, 277, 279, 283, 284, 285. So $f_2 = 9$.
- Class 3 ($286 - 310$): 287, 287, 287, 288, 290, 290, 290, 291, 295, 296, 296, 297, 298, 299, 299, 299, 302, 303, 304, 305, 306, 306, 306, 309, 310. Wait, no, let's recount carefully. Wait, 286 - 310: numbers from 286 up to 310. Let's list the data in this range: 287, 287, 287, 288, 290, 290, 290, 291, 295, 296, 296, 297, 298, 299, 299, 299, 302, 303, 304, 305, 306, 306, 306, 309, 310. Wait, actually, let's do it properly. The data points:

From the list:

261, 276, 236, 244, 279, 296, 284, 299, 290, 290, 248, 256,

338, 360, 341, 333, 261, 267, 287, 296, 313, 311, 306, 306,

299, 303, 277, 283, 304, 305, 287, 290, 287, 288, 297, 299,

332, 330, 309, 327, 306, 327, 285, 291, 295, 298, 306, 315,

310, 318, 318, 320, 333, 321, 323, 324, 327, 302, 319.

Now, class 286 - 310:

287 (3 times), 288, 290 (3 times), 291, 295, 296 (2 times), 297, 298, 299 (3 times), 302, 303, 304, 305, 306 (3 times), 309, 310. Let's count: 3 (287) +1 (288)+3 (290)+1 (291)+1 (295)+2 (296)+1 (297)+1 (298)+3 (299)+1 (302)+1 (303)+1 (304)+1 (305)+3 (306)+1 (309)+1 (310) = 3+1=4, +3=7, +1=8, +1=9, +2=11, +1=12, +1=13, +3=16, +1=17, +1=18, +1=19, +1=20, +3=23, +1=24, +1=25. Wait, that can't be. Wait, maybe I made a mistake. Let's use a better approach. Let's list all data points and count:

Data points:

236, 244, 248, 256,

261, 261, 267, 276, 277, 279, 283, 284, 285,

287, 287, 287, 288, 290, 290, 290, 291, 295, 296, 296, 297, 298, 299, 299, 299, 302, 303, 304, 305, 306, 306, 306, 309, 310,

311, 313, 315, 318, 318, 319, 320, 321, 323, 324, 327, 327, 327, 330, 332, 333, 333,

338, 341, 360.

Now, class 1: 236 - 260: 236,244,248,256 → 4

Class 2: 261 - 285: 261,261,267,276,277,279,283,284,285 → 9

Class 3: 286 - 310: 287,287,287,288,290,290,290,291,295,296,296,297,298,299,299,299,302,303,304,305,306,306,306,309,310 → Let's count: 3 (287) +1 (288) +3 (290) +1 (291) +1 (295) +2 (296) +1 (297) +1 (298) +3 (299) +1 (302) +1 (303) +1 (304) +1 (305) +3 (306) +1 (309) +1 (310) = 3+1=4, +3=7, +1=8, +1=9, +2=11, +1=12, +1=13, +3=16, +1=17, +1=18, +1=19, +1=20, +3=23, +1=24, +1=25. Wait, but total data points: 4 (class1) +9 (class2) +25 (class3) +? Let's check total: 59. 4+9=13, 13+25=38. Then class4: 311 - 335: 311,313,315,318,318,319,320,321,323,324,327,327,327,330,332,333,333 → Let's count: 311,313,315,318,318,319,320,321,323,324,327,327,327,330,332,333,333. That's 17. Then class5: 336 - 360: 338,341,360 → 3. Now total: 4+9+25+17+3=58. Wait, missing one. Oh, maybe I miscounted class3. Let's check again. Wait, the data point 285 is in class2 (261 - 285), correct. Then class3 starts at 286. So 287 is first in class3. Let's list class3 data points:

287,287,287,288,290,290,290,291,295,296,296,297,298,299,299,299,302,303,304,305,306,306,306,309,310. Wait, that's 25. Then class4: 311,313,315,318,318,319,320,321,323,324,327,327,327,330,332,333,333. Wait, 311 (1), 313 (1), 315 (1), 318 (2), 319 (1), 320 (1), 321 (1), 323 (1), 324 (1), 327 (3), 330 (1), 332 (1), 333 (2). Let's count: 1+1=2, +1=3, +2=5, +1=6, +1=7, +1=8, +1=9, +1=10, +3=13, +1=14, +1=15, +2=17. Then class5: 338,341,360 → 3. Total: 4+9+25+17+3=58. Wait, there are 59 data points. Oh, I see, I missed one. Let's check the original data:

The last row: 310, 318, 318, 320, 333, 321, 323, 324, 327, 302, 319. Wait, that's 11 data points? Wait, original data:

First row: 12 numbers (261,276,236,244,279,296,284,299,290,290,248,256)

Second row: 12 (338,360,341,333,261,267,287,296,313,311,306,306)

Third row: 12 (299,303,277,283,304,305,287,290,287,288,297,299)

Fourth row: 12 (332,330,309,327,306,327,285,291,295,298,306,315)

Fifth row: 11 (310,318,318,320,333,321,323,324,327,302,319)

Total: 12+12+12+12+11=59. Ah, so in the fifth row, there are 11 numbers. So I missed one in class3. Let's check the fifth row: 302 is in class3 (286 - 310), correct. 310 is in class3. Wait, maybe the data point 285 is in class2, correct. Wait, let's recount class3:

From all rows:

Row1: 296,284,299,290,290 (284 is class2, 296,299,290,290 are class3)

Row2: 287,296,306,306 (287,296,306,306 are class3)

Row3: 299,303,283,304,305,287,290,287,288,297,299 (283 is class2, others class3: 299,303,304,305,287,290,287,288,297,299)

Row4: 309,327,306,327,285,291,295,298,306,315 (285 class2, 327,327,315 class4; others class3: 309,306,291,295,298,306)

Row5: 310,318,318,320,333,321,323,324,327,302,319 (302 class3, 310 class3; others class4 or 5)

Now, let's count class3:

Row1: 296,299,290,290 → 4

Row2: 287,296,306,306 → 4 (total 8)

Row3: 299,303,304,305,287,290,287,288,297,299 → 10 (total 18)

Row4: 309,306,291,295,298,306 → 6 (total 24)

Row5: 302,310 → 2 (total 26)

Ah, there we go. I missed the two in row5. So class3: 26. Then class4:

Row2: 313,311 → 2

Row4: 327,327,315 → 3 (total 5)

Row5: 318,318,320,333,321,323,324,327,319 → 9 (total 14)

Wait, no, let's do it properly. Let's list all data points and assign to classes:

Class 1: 236,244,248,256 → 4

Class 2: 261,261,267,276,277,279,283,284,285 → 9 (total 13)

Class 3: 287,287,287,288,290,290,290,291,295,296,296,297,298,299,299,299,302,303,304,305,30

Answer:

Step1: Find the range

First, find the minimum and maximum values. The minimum value ($\text{min}$) is 236, and the maximum value ($\text{max}$) is 360. The range is $\text{max}-\text{min}=360 - 236 = 124$.

Step2: Determine class width

We need 5 classes, so the class width ($w$) is calculated as $\lceil\frac{\text{range}}{\text{number of classes}}
ceil=\lceil\frac{124}{5}
ceil = 25$ (we round up to ensure all data is covered).

Step3: Define the classes

The first class starts at the minimum value, 236. The classes will be:

• Class 1: $236 - 260$
• Class 2: $261 - 285$
• Class 3: $286 - 310$
• Class 4: $311 - 335$
• Class 5: $336 - 360$

Step4: Tally the frequencies

Count the number of data points in each class:

• Class 1 ($236 - 260$): 236, 244, 248, 256. So frequency ($f_1$) = 4.
• Class 2 ($261 - 285$): 261, 261, 267, 276, 277, 279, 283, 284, 285. So $f_2 = 9$.
• Class 3 ($286 - 310$): 287, 287, 287, 288, 290, 290, 290, 291, 295, 296, 296, 297, 298, 299, 299, 299, 302, 303, 304, 305, 306, 306, 306, 309, 310. Wait, no, let's recount carefully. Wait, 286 - 310: numbers from 286 up to 310. Let's list the data in this range: 287, 287, 287, 288, 290, 290, 290, 291, 295, 296, 296, 297, 298, 299, 299, 299, 302, 303, 304, 305, 306, 306, 306, 309, 310. Wait, actually, let's do it properly. The data points:

From the list:

261, 276, 236, 244, 279, 296, 284, 299, 290, 290, 248, 256,

338, 360, 341, 333, 261, 267, 287, 296, 313, 311, 306, 306,

299, 303, 277, 283, 304, 305, 287, 290, 287, 288, 297, 299,

332, 330, 309, 327, 306, 327, 285, 291, 295, 298, 306, 315,

310, 318, 318, 320, 333, 321, 323, 324, 327, 302, 319.

Now, class 286 - 310:

287 (3 times), 288, 290 (3 times), 291, 295, 296 (2 times), 297, 298, 299 (3 times), 302, 303, 304, 305, 306 (3 times), 309, 310. Let's count: 3 (287) +1 (288)+3 (290)+1 (291)+1 (295)+2 (296)+1 (297)+1 (298)+3 (299)+1 (302)+1 (303)+1 (304)+1 (305)+3 (306)+1 (309)+1 (310) = 3+1=4, +3=7, +1=8, +1=9, +2=11, +1=12, +1=13, +3=16, +1=17, +1=18, +1=19, +1=20, +3=23, +1=24, +1=25. Wait, that can't be. Wait, maybe I made a mistake. Let's use a better approach. Let's list all data points and count:

Data points:

236, 244, 248, 256,

261, 261, 267, 276, 277, 279, 283, 284, 285,

287, 287, 287, 288, 290, 290, 290, 291, 295, 296, 296, 297, 298, 299, 299, 299, 302, 303, 304, 305, 306, 306, 306, 309, 310,

311, 313, 315, 318, 318, 319, 320, 321, 323, 324, 327, 327, 327, 330, 332, 333, 333,

338, 341, 360.

Now, class 1: 236 - 260: 236,244,248,256 → 4

Class 2: 261 - 285: 261,261,267,276,277,279,283,284,285 → 9

Class 3: 286 - 310: 287,287,287,288,290,290,290,291,295,296,296,297,298,299,299,299,302,303,304,305,306,306,306,309,310 → Let's count: 3 (287) +1 (288) +3 (290) +1 (291) +1 (295) +2 (296) +1 (297) +1 (298) +3 (299) +1 (302) +1 (303) +1 (304) +1 (305) +3 (306) +1 (309) +1 (310) = 3+1=4, +3=7, +1=8, +1=9, +2=11, +1=12, +1=13, +3=16, +1=17, +1=18, +1=19, +1=20, +3=23, +1=24, +1=25. Wait, but total data points: 4 (class1) +9 (class2) +25 (class3) +? Let's check total: 59. 4+9=13, 13+25=38. Then class4: 311 - 335: 311,313,315,318,318,319,320,321,323,324,327,327,327,330,332,333,333 → Let's count: 311,313,315,318,318,319,320,321,323,324,327,327,327,330,332,333,333. That's 17. Then class5: 336 - 360: 338,341,360 → 3. Now total: 4+9+25+17+3=58. Wait, missing one. Oh, maybe I miscounted class3. Let's check again. Wait, the data point 285 is in class2 (261 - 285), correct. Then class3 starts at 286. So 287 is first in class3. Let's list class3 data points:

287,287,287,288,290,290,290,291,295,296,296,297,298,299,299,299,302,303,304,305,306,306,306,309,310. Wait, that's 25. Then class4: 311,313,315,318,318,319,320,321,323,324,327,327,327,330,332,333,333. Wait, 311 (1), 313 (1), 315 (1), 318 (2), 319 (1), 320 (1), 321 (1), 323 (1), 324 (1), 327 (3), 330 (1), 332 (1), 333 (2). Let's count: 1+1=2, +1=3, +2=5, +1=6, +1=7, +1=8, +1=9, +1=10, +3=13, +1=14, +1=15, +2=17. Then class5: 338,341,360 → 3. Total: 4+9+25+17+3=58. Wait, there are 59 data points. Oh, I see, I missed one. Let's check the original data:

The last row: 310, 318, 318, 320, 333, 321, 323, 324, 327, 302, 319. Wait, that's 11 data points? Wait, original data:

First row: 12 numbers (261,276,236,244,279,296,284,299,290,290,248,256)

Second row: 12 (338,360,341,333,261,267,287,296,313,311,306,306)

Third row: 12 (299,303,277,283,304,305,287,290,287,288,297,299)

Fourth row: 12 (332,330,309,327,306,327,285,291,295,298,306,315)

Fifth row: 11 (310,318,318,320,333,321,323,324,327,302,319)

Total: 12+12+12+12+11=59. Ah, so in the fifth row, there are 11 numbers. So I missed one in class3. Let's check the fifth row: 302 is in class3 (286 - 310), correct. 310 is in class3. Wait, maybe the data point 285 is in class2, correct. Wait, let's recount class3:

From all rows:

Row1: 296,284,299,290,290 (284 is class2, 296,299,290,290 are class3)

Row2: 287,296,306,306 (287,296,306,306 are class3)

Row3: 299,303,283,304,305,287,290,287,288,297,299 (283 is class2, others class3: 299,303,304,305,287,290,287,288,297,299)

Row4: 309,327,306,327,285,291,295,298,306,315 (285 class2, 327,327,315 class4; others class3: 309,306,291,295,298,306)

Row5: 310,318,318,320,333,321,323,324,327,302,319 (302 class3, 310 class3; others class4 or 5)

Now, let's count class3:

Row1: 296,299,290,290 → 4

Row2: 287,296,306,306 → 4 (total 8)

Row3: 299,303,304,305,287,290,287,288,297,299 → 10 (total 18)

Row4: 309,306,291,295,298,306 → 6 (total 24)

Row5: 302,310 → 2 (total 26)

Ah, there we go. I missed the two in row5. So class3: 26. Then class4:

Row2: 313,311 → 2

Row4: 327,327,315 → 3 (total 5)

Row5: 318,318,320,333,321,323,324,327,319 → 9 (total 14)

Wait, no, let's do it properly. Let's list all data points and assign to classes:

Class 1: 236,244,248,256 → 4

Class 2: 261,261,267,276,277,279,283,284,285 → 9 (total 13)

Class 3: 287,287,287,288,290,290,290,291,295,296,296,297,298,299,299,299,302,303,304,305,30