Question

Biking accidents Accidents on a level, $3$-mile bike path occur uniformly along the length of the path. The figure below displays the density curve that

describes the uniform 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 happens in front of Sue’s property?

Explain.

Short answer

Part (a) Because the curve is always on or above the horizontal axis, it generates a density of $1/3$by $3$rectangles: the area beneath the curve is $1$square meter.

Part (b) The first mile of the path is responsible for $1/3$of all accidents.

Part (c) The percentage of accidents that occur in front of Sue's house is $0.1$

Step-by-step solution

Part (a) Step 1. Given

Part (a) Step 2. Concept

These inflection points are always one standard deviation distant from the mean on a normal density curve.

Part (a) Step 3. Explanation

Because the curve is always on or above the horizontal axis and generates a density of $1/3$ by $3$ rectangles, the area beneath it is $1$

As a result, the supplied curve satisfies the density curve's two conditions.

Part (b) Step 1. Explanation

We can see from the density curve that one-third of all accidents happen in the first mile: that is, a $1/3$by $1$rectangle circumscribed between $0$and $1$mile. As a result, the first mile of the path accounts for$1/3$of all incidents.

As a result, the first mile of the path accounts for 1/3 of all incidents.

Part (c) Step 1. Explanation

We've told Sue that her property borders the bike route between the $0.8$and $1.1$-mile markers. That is to say, the distance between his homes is$0.3$miles.

According to the definitions, one-third of all accidents occur within $0.3$miles of Sue's home.

As a result, the percentage of accidents that occur in front of Sue's house is$0.1$