Question

finish times (to the nearest hour) for 10 dogsled teams are shown below. make a frequency table showing class limits, class boundaries, midpoints, frequency, relative frequencies, and cumulative frequencies. use three classes. (round your answer for relative frequency to the nearest hundredth and for midpoint to the nearest tenth.) 283 265 236 310 275 301 287 258 299 280

Explanation:

Step1: Find the range

First, we find the range of the data. The minimum value is 236 and the maximum value is 310. The range is \( 310 - 236 = 74 \). We need to divide this range into 3 classes, so the class width is \( \frac{74}{3}\approx24.67 \), we can round it up to 25 for simplicity.

Step2: Determine class limits

- First class: Lower limit = 236, Upper limit = \( 236 + 25 - 1 = 260 \) (since we want to include 236 and have a width of 25, the upper limit is \( 236+25 - 1 \) to make the next class start at 261)
- Second class: Lower limit = 261, Upper limit = \( 261+25 - 1 = 285 \)
- Third class: Lower limit = 286, Upper limit = \( 286 + 25 - 1=310 \) (which includes the maximum value 310)

Step3: Find class boundaries

- For the first class: Lower boundary = \( 236 - 0.5 = 235.5 \), Upper boundary = \( 260 + 0.5 = 260.5 \)
- For the second class: Lower boundary = \( 261 - 0.5 = 260.5 \), Upper boundary = \( 285 + 0.5 = 285.5 \)
- For the third class: Lower boundary = \( 286 - 0.5 = 285.5 \), Upper boundary = \( 310 + 0.5 = 310.5 \)

Step4: Calculate midpoints

- Midpoint of first class: \( \frac{236 + 260}{2}=\frac{496}{2} = 248.0 \)
- Midpoint of second class: \( \frac{261+285}{2}=\frac{546}{2}=273.0 \)
- Midpoint of third class: \( \frac{286 + 310}{2}=\frac{596}{2}=298.0 \)

Step5: Count frequencies

Now we count how many data points fall into each class:
- First class (236 - 260): Data points are 236, 258, 265? Wait, no. Wait 236, 258, 265? Wait 236 is in, 258 is in, 265: 265 is greater than 260? Wait our first class upper limit is 260, so 265 is not in first class. Wait let's list the data: 283, 265, 236, 310, 275, 301, 287, 258, 299, 280.
So data in [236, 260]: 236, 258. Wait 265 is 265 > 260, so first class: 236, 258. Wait 236, 258: that's 2 data points? Wait no, 236, 258, and 265? Wait 265 is 265, our first class is 236 - 260, so 265 is outside. Wait 236, 258: two? Wait 236, 258, and 265? No, 265 is 265, which is more than 260. Wait 236, 258: two? Wait 236, 258, 265? Wait 265 is 265, so 236 (236), 258 (258), 265: 265 is 265, which is greater than 260, so first class: 236, 258. Wait 236, 258: that's 2? Wait no, 236, 258, 265? Wait 265 is 265, so 236, 258, 265: 265 is 265, which is 265 ≤ 260? No, 265 > 260. So first class: 236, 258 (2 data points). Second class: 261 - 285. Data points: 265, 275, 280. Let's check: 265 (265 is between 261 - 285), 275 (yes), 280 (yes). So 265, 275, 280: 3 data points? Wait 283: 283 is greater than 285? 283 is 283, 285 is the upper limit of second class, so 283 is in second class? Wait 261 - 285: 283 is 283 ≤ 285, so 283 is in second class. Oh, I made a mistake earlier. Let's re - list the data: 283, 265, 236, 310, 275, 301, 287, 258, 299, 280.

Now, first class (236 - 260): 236, 258. That's 2.

Second class (261 - 285): 265, 275, 280, 283. Let's check: 265 (261 ≤ 265 ≤ 285), 275 (yes), 280 (yes), 283 (yes). So 4 data points.

Third class (286 - 310): 310, 301, 287, 299. Let's check: 310 (286 ≤ 310 ≤ 310), 301 (yes), 287 (yes), 299 (yes). So 4 data points. Wait but 2 + 4+4 = 10, which is the total number of data points. Good.

Step6: Calculate relative frequencies

Relative frequency = \( \frac{\text{frequency}}{\text{total number of data points}} \)
- First class: \( \frac{2}{10}=0.20 \)
- Second class: \( \frac{4}{10} = 0.40 \)
- Third class: \( \frac{4}{10}=0.40 \)

Step7: Calculate cumulative frequencies

- Cumulative frequency for first class: 2 (since it's the first class, cumulative frequency is its own frequency)
- Cumulative frequency for second class: \( 2 + 4=6 \)
- Cumulative frequency for third class: \( 6+4 = 10 \)

Now we can construct the frequency table:

| Class Limits | Class Boundaries | Midpoint | Frequency | Relative Frequency | Cumulative Frequency |
|--------------|------------------|----------|-----------|--------------------|----------------------|
| 236 - 260 | 235.5 - 260.5 | 248.0 | 2 | 0.20 | 2 |
| 261 - 285 | 260.5 - 285.5 | 273.0 | 4 | 0.40 | 6 |
| 286 - 310 | 285.5 - 310.5 | 298.0 | 4 | 0.40 | 10 |

Answer:

The frequency table is as shown above:

Class Limits	Class Boundaries	Midpoint	Frequency	Relative Frequency	Cumulative Frequency
261 - 285	260.5 - 285.5	273.0	4	0.40	6
286 - 310	285.5 - 310.5	298.0	4	0.40	10