Accidents on a bike path. Examining the location of accidents on a level, 5-mile

bike path shows that they occur uniformly along the length of the path. Figure 3.4
displays the density curve that describes the distribution of accidents.
(a) Explain why this curve satisfies the two requirements for a density curve.
(b) The proportion of accidents that occur in the first mile of the path is the area
under the density curve between 0 miles and 1 mile. What is this area?
(c) Sue’s property adjoins the bike path between the 0.8 mile mark and the 1.1
mile mark. What proportion of accidents happen in front of Sue’s property?

(a) A density curve satisfies two main requirements: it must be non-negative and the total area under the curve must be equal to 1.

In this case, the density curve describing the distribution of accidents is non-negative because accidents cannot have a negative frequency. Additionally, the area under the curve represents the proportion of accidents, and since it is a probability distribution, the total area under the curve must sum up to 1. Therefore, both requirements are met.

(b) To find the proportion of accidents that occur in the first mile of the path, we need to find the area under the density curve between 0 miles and 1 mile.

Since the density curve is uniform along the length of the path, the curve will form a rectangle from 0 to 1 on the x-axis, with a height equal to the value of the density at any given point within that range.

Since the density curve is uniform, the height of the rectangle is constant throughout the range. Let's call this height "h".

To find the area under the curve, we need to calculate the base of the rectangle, which is 1 mile, and multiply it by the height, which is "h".

Therefore, the area under the curve between 0 miles and 1 mile is given by "1 * h".

(c) To find the proportion of accidents that happen in front of Sue's property, we need to find the area under the density curve between the 0.8 mile mark and the 1.1 mile mark.

Since the density curve is uniform, the height of the rectangle is constant. Again, let's call this height "h".

To find the area under the curve, we calculate the base of the rectangle, which is 1.1 - 0.8 = 0.3 mile, and multiply it by the height, which is "h".

Therefore, the area under the curve between the 0.8 mile mark and the 1.1 mile mark is given by "0.3 * h".

(a) A density curve describes the distribution of a variable and has two main requirements:

1. Non-negativity: The curve must always be non-negative, meaning that the height or value of the curve at any point cannot be negative. In the case of accidents on a bike path, it would not make sense to have a negative number of accidents at any location.

2. Area under the curve equals 1: The total area under the curve must be equal to 1, representing the entire distribution. This ensures that the proportion of accidents occurring at any specific location can be calculated by finding the area under the curve within that range.

The density curve displayed in Figure 3.4 satisfies these requirements because it is always non-negative and the total area under the curve is equal to 1.

(b) To find the proportion of accidents that occur in the first mile of the path, we need to calculate the area under the density curve between 0 miles and 1 mile.

The area under the curve represents the proportion, so we can use basic geometric principles to compute it. Since the density curve is uniformly distributed, the area under the curve between 0 miles and 1 mile will be a rectangle. The length of the rectangle is 1 mile, and the height of the rectangle is given by the density curve.

Therefore, the area is simply the product of the length and the height of the rectangle. In this case, as the accidents are uniformly distributed, the height of the rectangle is constant along the entire length. Thus, the area is equal to the length of the interval, which is 1 mile.

(c) To determine the proportion of accidents that happen in front of Sue's property, we need to calculate the area under the density curve between the 0.8 mile mark and the 1.1 mile mark.

Using a similar approach as in part (b), we can consider the area under the curve between these two points as a rectangle. The length of this rectangle is 0.3 mile (1.1 - 0.8 mile), and the height is given by the density curve.

To calculate the proportion, we need to find the area of this rectangle. Using the same logic as in part (b), the area is equal to the length multiplied by the height. Since the density curve is uniform, the height is constant within this interval.

Therefore, the proportion of accidents that happen in front of Sue's property is equal to the length of the interval (0.3 mile) multiplied by the height of the density curve within that interval.

We have no access to Figure 3.4.

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