Question

Bridge inspection ratings. According to the National Bridge Inspection Standard (NBIS), public bridges over 20 feet in length must be inspected and rated every 2 years. The NBIS rating scale ranges from 0 (poorest rating) to 9 (highest rating). University of Colorado engineers used a probabilistic model to forecast the inspection ratings of all major bridges in Denver (Journal of Performance of Constructed Facilities, February 2005). For the year 2020, the engineers forecast that 9% of all major Denver bridges will have ratings of 4 or below.

1. Use the forecast to find the probability that in a random sample of 10 major Denver bridges, at least 3 will have an inspection rating of 4 or below in 2020.

2. Suppose that you actually observe 3 or more of the sample of 10 bridges with inspection ratings of 4 or below in 2020. What inference can you make? Why?

Short answer

1. The probability that at least 3 will have an inspection rating of 4 or below in 2020 is 0.05404.

2. The probability that 3 or more of the sample of 10 bridges with inspection ratings of 4 or below in 2020 is 0.9912.

Step-by-step solution

Given information

9% is the probability of success,

The probability of success and the probability of failure are,

$$\begin{gathered} \mathrm{p}=1-\mathrm{q} \\ =1-0.09 \\ =0.91 \end{gathered}$$

(a) Describing the probability

Here, the probability that at least 3 is,

The formula for the probability is,

$p\left(X\ge x\right)={\sum _{}}^n{C}_x{\left(p\right)}^x{\left(1-p\right)}^{n-x}$

Substituting the values we get,

localid="1664385878414" $$\begin{gathered} \mathrm{p}\left(\mathrm{X}\ge 3\right)=\sum _{\mathrm{x}=3}^{10}{}^{10}{\mathrm{C}}_{\mathrm{x}}{\left(0.09\right)}^{\mathrm{x}}{\left(1-0.09\right)}^{10-\mathrm{x}} \\ =0.05404 \end{gathered}$$

(b) Inspection rating

The probability of 3 or more of a sample of 10 is,

$p\left(X\le 3\right)=\sum _{x=0}^{3}P\left(X=x\right)$

Substituting the values, we get

$P\left(x\le 3\right)=0.9912$

There are 3 or more of the sample of 10 bridges with an inspection rating 4 or below in 2020 is 0.9912.