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Use the ogive in Exercise 21 to approximate
(a) the cumulative frequency for a weight of 27.5 pounds.
(b) the weight for which the cumulative frequency is 45.
(c) the number of beagles that weigh between 22.5 pounds
and 29.5 pounds.
(d) the number of beagles that weigh more than 30.5 pounds.

Let's analyze the ogive (cumulative frequency graph) provided and answer the questions:

### (a) The cumulative frequency for a weight of 27.5 pounds:
To find this, locate 27.5 pounds on the x-axis, then move up vertically to the point on the ogive curve. From there, move horizontally to the y-axis to determine the cumulative frequency. 

- **Approximation**: At 27.5 pounds, the cumulative frequency is approximately **40**.

### (b) The weight for which the cumulative frequency is 45:
To find this, locate 45 on the y-axis, then move horizontally to intersect the ogive curve. From there, move vertically down to the x-axis to determine the corresponding weight.

- **Approximation**: A cumulative frequency of 45 corresponds to a weight of approximately **30 pounds**.

### (c) The number of beagles that weigh between 22.5 pounds and 29.5 pounds:
To find this, determine the cumulative frequencies at 22.5 pounds and 29.5 pounds, then subtract the smaller from the larger.

- **Cumulative frequency at 22.5 pounds**: Approximately 10.
- **Cumulative frequency at 29.5 pounds**: Approximately 50.
- **Number of beagles**: \( 50 - 10 = 40 \).

So, approximately **40 beagles** weigh between 22.5 and 29.5 pounds.

### (d) The number of beagles that weigh more than 30.5 pounds:
To find this, first determine the cumulative frequency at 30.5 pounds, then subtract this value from the total cumulative frequency.

- **Cumulative frequency at 30.5 pounds**: Approximately 50.
- **Total cumulative frequency**: 55.
- **Number of beagles**: \( 55 - 50 = 5 \).

So, approximately **5 beagles** weigh more than 30.5 pounds.

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Would you like further details or have any other questions?

### Related Questions:
1. How would you interpret the cumulative frequency in a different context?
2. Can you explain the difference between an ogive and a histogram?
3. How can an ogive help in understanding the distribution of data?
4. What are the benefits of using an ogive for cumulative frequency analysis?
5. How would the graph change if the sample size were larger?
6. What other types of graphs can represent the same data?
7. How can you estimate the median weight from the ogive?
8. How would you describe the shape of the distribution based on the ogive?

### Tip:
When reading from an ogive, always make sure to interpolate between points for more accurate results, especially if the graph is not perfectly smooth.

Learn how to approximate cumulative frequencies using an ogive (cumulative frequency graph) with detailed examples. Understand how to find cumulative frequencies for specific weights and ranges. Includes solutions for different scenarios such as finding cumulative frequencies, determining weights, and analyzing distributions. Suitable for high school students studying statistics.

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