Question

finish times (to the nearest hour) for 59 dogsled teams are shown below. draw a histogram. use five classes.
261 269 236 244 278 296 284 299 289 289 249 256
338 360 341 333 261 268 287 296 313 311 309 309
299 303 277 283 304 305 288 290 288 289 297 299
332 330 309 327 309 327 285 291 295 298 306 315
310 318 318 320 333 321 323 324 327 239 358

Explanation:

Step1: Find the range

First, we find the minimum and maximum values in the data set. The minimum value ($\text{min}$) is 236 and the maximum value ($\text{max}$) is 360. The range $R$ is calculated as:
$$R=\text{max}-\text{min}=360 - 236=124$$

Step2: Determine the class width

We are using 5 classes. The class width ($w$) is calculated by dividing the range by the number of classes and rounding up to a convenient number.
$$w=\lceil\frac{R}{\text{number of classes}} ceil=\lceil\frac{124}{5} ceil=\lceil24.8 ceil = 25$$

Step3: Define the class intervals

We start from the minimum value and add the class width to define each class interval.
- Class 1: $236 - 260$ (since $236+25 - 1=260$)
- Class 2: $261 - 285$ ( $261+25 - 1 = 285$)
- Class 3: $286 - 310$ ( $286+25 - 1=310$)
- Class 4: $311 - 335$ ( $311+25 - 1 = 335$)
- Class 5: $336 - 360$ ( $336+25 - 1=360$)

Step4: Count the frequency of each class

We go through each data point and count how many fall into each class:
- Class 1 ($236 - 260$): The data points are 236, 244, 249, 256, 239. So frequency $f_1 = 5$
- Class 2 ($261 - 285$): The data points are 261, 269, 278, 261, 268, 277, 283, 284. So frequency $f_2=8$
- Class 3 ($286 - 310$): The data points are 289, 289, 287, 296, 309, 309, 299, 303, 304, 305, 288, 290, 288, 289, 297, 299, 285, 309, 306. Let's count: 289 (2), 287, 296, 309 (3), 303, 304, 305, 288 (2), 290, 289, 297, 299 (2), 285, 306. Total: $2 + 1+1+3+1+1+1+2+1+1+1+2+1+1 = 20$ (Wait, let's do it more carefully: list all in 286 - 310: 287,288,288,289,289,289,285,290,296,297,298,299,299,303,304,305,306,309,309,309. Wait, 285 is 285 which is in 286 - 310? Wait 286 - 310: lower bound 286, upper bound 310. 285 is less than 286, so my mistake earlier. Let's correct:

Data points in 286 - 310: 287,288,288,289,289,289,290,296,297,298,299,299,303,304,305,306,309,309,309. Let's count: 287 (1), 288 (2), 289 (3), 290 (1), 296 (1), 297 (1), 298 (1), 299 (2), 303 (1), 304 (1), 305 (1), 306 (1), 309 (3). Total: $1+2 + 3+1+1+1+1+2+1+1+1+1+3= 19$? Wait, let's list all data points and check:

Original data:

261, 269, 236, 244, 278, 296, 284, 299, 289, 289, 249, 256,

338, 360, 341, 333, 261, 268, 287, 296, 313, 311, 309, 309,

299, 303, 277, 283, 304, 305, 288, 290, 288, 289, 297, 299,

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

310, 318, 318, 320, 333, 321, 323, 324, 327, 239, 358

Now, for 286 - 310:

287 (from second row: 287), 288 (third row: 288, 288), 289 (first row: 289,289; third row: 289), 290 (third row: 290), 296 (first row:296; second row:296), 297 (third row:297), 298 (fourth row:298), 299 (first row:299; third row:299,299), 303 (third row:303), 304 (third row:304), 305 (third row:305), 306 (fourth row:306), 309 (second row:309,309; third row:309; fourth row:309), 310 (fifth row:310). Wait, 310 is included? The class is 286 - 310, so 310 is included (since 310 <= 310). So 310 is in this class. So let's recount:

287:1

288:2

289:3 (first row:2, third row:1)

290:1

296:2 (first and second row)

297:1

298:1

299:3 (first, third (2))

303:1

304:1

305:1

306:1

309:4 (second row:2, third row:1, fourth row:1)

310:1

Now sum: 1+2+3+1+2+1+1+3+1+1+1+1+4+1 = let's calculate step by step:

1+2=3; 3+3=6; 6+1=7; 7+2=9; 9+1=10; 10+1=11; 11+3=14; 14+1=15; 15+1=16; 16+1=17; 17+4=21; 21+1=22. Wait, I think I made a mistake earlier. Let's use a better approach: count each number:

First row (excluding those in other classes): 296,284,299,289,289,249,256 (284 is in 261 - 285? 261 - 285: 261,269,278,261,268,277,283,284. So 284 is in class 2. So first row data in class 3: 296,299,289,289.

Second row: 287,296,313,311,309,309. 287,296,309,309 are in class 3.

Third row:299,303,283,304,305,288,290,288,289,297,299. 299,303,304,305,288,290,288,289,297,299 are in class 3.

Fourth row:309,327,309,327,285,291,295,298,306,315. 309,309,285 (no, 285 < 286), 291,295,298,306 are in class 3.

Fifth row:310,318,318,320,333,321,323,324,327,239,358. 310 is in class 3.

Now let's count:

First row: 296,299,289,289 → 4

Second row:287,296,309,309 →4

Third row:299,303,304,305,288,290,288,289,297,299 →10

Fourth row:309,309,291,295,298,306 →6

Fifth row:310 →1

Total: 4+4+10+6+1 =25? Wait, this is getting confusing. Maybe a better way is to list all data points and assign to classes:

Data points:

236 (class1), 244 (class1), 249 (class1), 256 (class1), 239 (class1) → class1: 5

261 (class2), 269 (class2), 278 (class2), 261 (class2), 268 (class2), 277 (class2), 283 (class2), 284 (class2) → class2: 8

287 (class3), 288 (class3), 288 (class3), 289 (class3), 289 (class3), 289 (class3), 290 (class3), 296 (class3), 296 (class3), 297 (class3), 298 (class3), 299 (class3), 299 (class3), 299 (class3), 303 (class3), 304 (class3), 305 (class3), 306 (class3), 309 (class3), 309 (class3), 309 (class3), 309 (class3), 310 (class3) → let's count these: 287(1), 288(2), 289(3), 290(1), 296(2), 297(1), 298(1), 299(3), 303(1), 304(1), 305(1), 306(1), 309(4), 310(1). Sum:1+2+3+1+2+1+1+3+1+1+1+1+4+1= 23? Wait, maybe I should use a frequency table:

Let's list all data points and their classes:

1. 261 → class2
2. 269 → class2
3. 236 → class1
4. 244 → class1
5. 278 → class2
6. 296 → class3
7. 284 → class2
8. 299 → class3
9. 289 → class3
10. 289 → class3
11. 249 → class1
12. 256 → class1
13. 338 → class5
14. 360 → class5
15. 341 → class5
16. 333 → class4
17. 261 → class2
18. 268 → class2
19. 287 → class3
20. 296 → class3
21. 313 → class4
22. 311 → class4
23. 309 → class3
24. 309 → class3
25. 299 → class3
26. 303 → class3
27. 277 → class2
28. 283 → class2
29. 304 → class3
30. 305 → class3
31. 288 → class3
32. 290 → class3
33. 288 → class3
34. 289 → class3
35. 297 → class3
36. 299 → class3
37. 332 → class4
38. 330 → class4
39. 309 → class3
40. 327 → class4
41. 309 → class3
42. 327 → class4
43. 285 → class3? Wait 285: 286 - 310, 285 < 286, so class2? Wait class2 is 261 - 285. So 285 is in class2? Wait 261 - 285: 261 ≤ x ≤ 285. So 285 is included. So 285 is in class2. Oh! I made a mistake earlier. 285 is in class2 (261 - 285), not class3. So let's correct:

43. 285 → class2

44. 291 → class3

45. 295 → class3

46. 298 → class3

47. 306 → class3

48. 315 → class4

49. 310 → class3

50. 318 → class4

51. 318 → class4

52. 320 → class4

53. 333 → class4

54. 321 → class4

55. 323 → class4

56. 324 → class4

57. 327 → class4

58. 239 → class1

59. 358 → class5

Now let's recount the classes:

Class1 (236 - 260): 236,244,249,256,239 → 5 (correct)

Class2 (261 - 285): 261,269,278,261,268,277,283,284,285 → Wait, 285 is now included. So data points:

261 (1), 269 (2), 278 (3), 261 (4), 268 (5), 277 (6), 283 (7), 284 (8), 285 (9). So class2: 9

Class3 (286 - 310): 287,288,2

Answer:

Step1: Find the range

First, we find the minimum and maximum values in the data set. The minimum value ($\text{min}$) is 236 and the maximum value ($\text{max}$) is 360. The range $R$ is calculated as:
$$R=\text{max}-\text{min}=360 - 236=124$$

Step2: Determine the class width

We are using 5 classes. The class width ($w$) is calculated by dividing the range by the number of classes and rounding up to a convenient number.

$$w=\lceil\frac{R}{\text{number of classes}} ceil=\lceil\frac{124}{5} ceil=\lceil24.8 ceil = 25$$

Step3: Define the class intervals

We start from the minimum value and add the class width to define each class interval.

• Class 1: $236 - 260$ (since $236+25 - 1=260$)
• Class 2: $261 - 285$ ( $261+25 - 1 = 285$)
• Class 3: $286 - 310$ ( $286+25 - 1=310$)
• Class 4: $311 - 335$ ( $311+25 - 1 = 335$)
• Class 5: $336 - 360$ ( $336+25 - 1=360$)

Step4: Count the frequency of each class

We go through each data point and count how many fall into each class:

• Class 1 ($236 - 260$): The data points are 236, 244, 249, 256, 239. So frequency $f_1 = 5$
• Class 2 ($261 - 285$): The data points are 261, 269, 278, 261, 268, 277, 283, 284. So frequency $f_2=8$
• Class 3 ($286 - 310$): The data points are 289, 289, 287, 296, 309, 309, 299, 303, 304, 305, 288, 290, 288, 289, 297, 299, 285, 309, 306. Let's count: 289 (2), 287, 296, 309 (3), 303, 304, 305, 288 (2), 290, 289, 297, 299 (2), 285, 306. Total: $2 + 1+1+3+1+1+1+2+1+1+1+2+1+1 = 20$ (Wait, let's do it more carefully: list all in 286 - 310: 287,288,288,289,289,289,285,290,296,297,298,299,299,303,304,305,306,309,309,309. Wait, 285 is 285 which is in 286 - 310? Wait 286 - 310: lower bound 286, upper bound 310. 285 is less than 286, so my mistake earlier. Let's correct:

Data points in 286 - 310: 287,288,288,289,289,289,290,296,297,298,299,299,303,304,305,306,309,309,309. Let's count: 287 (1), 288 (2), 289 (3), 290 (1), 296 (1), 297 (1), 298 (1), 299 (2), 303 (1), 304 (1), 305 (1), 306 (1), 309 (3). Total: $1+2 + 3+1+1+1+1+2+1+1+1+1+3= 19$? Wait, let's list all data points and check:

Original data:

261, 269, 236, 244, 278, 296, 284, 299, 289, 289, 249, 256,

338, 360, 341, 333, 261, 268, 287, 296, 313, 311, 309, 309,

299, 303, 277, 283, 304, 305, 288, 290, 288, 289, 297, 299,

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

310, 318, 318, 320, 333, 321, 323, 324, 327, 239, 358

Now, for 286 - 310:

287 (from second row: 287), 288 (third row: 288, 288), 289 (first row: 289,289; third row: 289), 290 (third row: 290), 296 (first row:296; second row:296), 297 (third row:297), 298 (fourth row:298), 299 (first row:299; third row:299,299), 303 (third row:303), 304 (third row:304), 305 (third row:305), 306 (fourth row:306), 309 (second row:309,309; third row:309; fourth row:309), 310 (fifth row:310). Wait, 310 is included? The class is 286 - 310, so 310 is included (since 310 <= 310). So 310 is in this class. So let's recount:

287:1

288:2

289:3 (first row:2, third row:1)

290:1

296:2 (first and second row)

297:1

298:1

299:3 (first, third (2))

303:1

304:1

305:1

306:1

309:4 (second row:2, third row:1, fourth row:1)

310:1

Now sum: 1+2+3+1+2+1+1+3+1+1+1+1+4+1 = let's calculate step by step:

1+2=3; 3+3=6; 6+1=7; 7+2=9; 9+1=10; 10+1=11; 11+3=14; 14+1=15; 15+1=16; 16+1=17; 17+4=21; 21+1=22. Wait, I think I made a mistake earlier. Let's use a better approach: count each number:

First row (excluding those in other classes): 296,284,299,289,289,249,256 (284 is in 261 - 285? 261 - 285: 261,269,278,261,268,277,283,284. So 284 is in class 2. So first row data in class 3: 296,299,289,289.

Second row: 287,296,313,311,309,309. 287,296,309,309 are in class 3.

Third row:299,303,283,304,305,288,290,288,289,297,299. 299,303,304,305,288,290,288,289,297,299 are in class 3.

Fourth row:309,327,309,327,285,291,295,298,306,315. 309,309,285 (no, 285 < 286), 291,295,298,306 are in class 3.

Fifth row:310,318,318,320,333,321,323,324,327,239,358. 310 is in class 3.

Now let's count:

First row: 296,299,289,289 → 4

Second row:287,296,309,309 →4

Third row:299,303,304,305,288,290,288,289,297,299 →10

Fourth row:309,309,291,295,298,306 →6

Fifth row:310 →1

Total: 4+4+10+6+1 =25? Wait, this is getting confusing. Maybe a better way is to list all data points and assign to classes:

Data points:

236 (class1), 244 (class1), 249 (class1), 256 (class1), 239 (class1) → class1: 5

261 (class2), 269 (class2), 278 (class2), 261 (class2), 268 (class2), 277 (class2), 283 (class2), 284 (class2) → class2: 8

287 (class3), 288 (class3), 288 (class3), 289 (class3), 289 (class3), 289 (class3), 290 (class3), 296 (class3), 296 (class3), 297 (class3), 298 (class3), 299 (class3), 299 (class3), 299 (class3), 303 (class3), 304 (class3), 305 (class3), 306 (class3), 309 (class3), 309 (class3), 309 (class3), 309 (class3), 310 (class3) → let's count these: 287(1), 288(2), 289(3), 290(1), 296(2), 297(1), 298(1), 299(3), 303(1), 304(1), 305(1), 306(1), 309(4), 310(1). Sum:1+2+3+1+2+1+1+3+1+1+1+1+4+1= 23? Wait, maybe I should use a frequency table:

Let's list all data points and their classes:

1. 261 → class2
2. 269 → class2
3. 236 → class1
4. 244 → class1
5. 278 → class2
6. 296 → class3
7. 284 → class2
8. 299 → class3
9. 289 → class3
10. 289 → class3
11. 249 → class1
12. 256 → class1
13. 338 → class5
14. 360 → class5
15. 341 → class5
16. 333 → class4
17. 261 → class2
18. 268 → class2
19. 287 → class3
20. 296 → class3
21. 313 → class4
22. 311 → class4
23. 309 → class3
24. 309 → class3
25. 299 → class3
26. 303 → class3
27. 277 → class2
28. 283 → class2
29. 304 → class3
30. 305 → class3
31. 288 → class3
32. 290 → class3
33. 288 → class3
34. 289 → class3
35. 297 → class3
36. 299 → class3
37. 332 → class4
38. 330 → class4
39. 309 → class3
40. 327 → class4
41. 309 → class3
42. 327 → class4
43. 285 → class3? Wait 285: 286 - 310, 285 < 286, so class2? Wait class2 is 261 - 285. So 285 is in class2? Wait 261 - 285: 261 ≤ x ≤ 285. So 285 is included. So 285 is in class2. Oh! I made a mistake earlier. 285 is in class2 (261 - 285), not class3. So let's correct:

1. 285 → class2

1. 291 → class3

1. 295 → class3

1. 298 → class3

1. 306 → class3

1. 315 → class4

1. 310 → class3

1. 318 → class4

1. 318 → class4

1. 320 → class4

1. 333 → class4

1. 321 → class4

1. 323 → class4

1. 324 → class4

1. 327 → class4

1. 239 → class1

1. 358 → class5

Now let's recount the classes:

Class1 (236 - 260): 236,244,249,256,239 → 5 (correct)

Class2 (261 - 285): 261,269,278,261,268,277,283,284,285 → Wait, 285 is now included. So data points:

261 (1), 269 (2), 278 (3), 261 (4), 268 (5), 277 (6), 283 (7), 284 (8), 285 (9). So class2: 9

Class3 (286 - 310): 287,288,2