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Reliability of gas station air guages. Tire and automobile
Chapter 3, Problem 47E(choose chapter or problem)
Reliability of gas station air guages. Tire and automobile manufacturers and consumer safety experts all recommend that drivers maintain proper tire pressure in their cars. Consequently, many gas stations now provide air pumps and air gauges for their customers. In a Research Note (Nov. 2001), the National Highway Traffic Safety Administration studied the reliability of gas station air guages. The next table gives the percentage of gas stations that provide air gauges that overreport the pressure level in the tire.
\(\begin{array}{l|ccc}
\hline \begin{array}{l}
\text { Station } \\
\text { Gauge } \\
\text { Pressure }
\end{array} & \begin{array}{c}
\text { Overreport } \\
\text { by } 4 \text { psi or } \\
\text { More (\%) }
\end{array} & \begin{array}{c}
\text { Overreport } \\
\text { by } 6 \text { psi or } \\
\text { More (\%) }
\end{array} & \begin{array}{c}
\text { Overreport } \\
\text { by } 8 \text { psi or } \\
\text { More (\%) }
\end{array} \\
\hline 25 \mathrm{psi} & 16 & 2 & 0 \\
35 \mathrm{psi} & 19 & 9 & 0 \\
45 \mathrm{psi} & 19 & 14 & 5 \\
55 \mathrm{psi} & 20 & 15 & 9 \\
\hline
\end{array}\)
a. If the gas station air pressure gauge reads 35 psi, what is the probability that the pressure is overreported by 6 psi or more?
b. If the gas station air pressure gauge reads 55 psi, what is the probability that the pressure is overreported by 8 psi or more?
c. If the gas station air pressure gauge reads 25 psi, what is the probability that the pressure is not overreported by 4 psi or more?
d. Are the events A = {overreport by 4 psi or more} and B = {overreport by 6 psi or more} mutually exclusive? Explain.
e. Based on your answer to part d, why do the probabilities in the table not sum to 1?
Questions & Answers
QUESTION:
Reliability of gas station air guages. Tire and automobile manufacturers and consumer safety experts all recommend that drivers maintain proper tire pressure in their cars. Consequently, many gas stations now provide air pumps and air gauges for their customers. In a Research Note (Nov. 2001), the National Highway Traffic Safety Administration studied the reliability of gas station air guages. The next table gives the percentage of gas stations that provide air gauges that overreport the pressure level in the tire.
\(\begin{array}{l|ccc}
\hline \begin{array}{l}
\text { Station } \\
\text { Gauge } \\
\text { Pressure }
\end{array} & \begin{array}{c}
\text { Overreport } \\
\text { by } 4 \text { psi or } \\
\text { More (\%) }
\end{array} & \begin{array}{c}
\text { Overreport } \\
\text { by } 6 \text { psi or } \\
\text { More (\%) }
\end{array} & \begin{array}{c}
\text { Overreport } \\
\text { by } 8 \text { psi or } \\
\text { More (\%) }
\end{array} \\
\hline 25 \mathrm{psi} & 16 & 2 & 0 \\
35 \mathrm{psi} & 19 & 9 & 0 \\
45 \mathrm{psi} & 19 & 14 & 5 \\
55 \mathrm{psi} & 20 & 15 & 9 \\
\hline
\end{array}\)
a. If the gas station air pressure gauge reads 35 psi, what is the probability that the pressure is overreported by 6 psi or more?
b. If the gas station air pressure gauge reads 55 psi, what is the probability that the pressure is overreported by 8 psi or more?
c. If the gas station air pressure gauge reads 25 psi, what is the probability that the pressure is not overreported by 4 psi or more?
d. Are the events A = {overreport by 4 psi or more} and B = {overreport by 6 psi or more} mutually exclusive? Explain.
e. Based on your answer to part d, why do the probabilities in the table not sum to 1?
ANSWER:Step 1 of 5
(a)
Reliability of gas station air gauges.
Given below table gives the percentage of gas stations that provide air gauges that overreport the pressure level in the tire.
\(\begin{array}{|c|c|c|c|}
\hline \begin{array}{c}
\text { Station Gauge } \\
\text { Pressure }
\end{array} & \begin{array}{c}
\text { Overreport by } 4 p s i \\
\text { or more (\%) }
\end{array} & \begin{array}{c}
\text { Overreport by } 6 \mathrm{psi} \\
\text { or more (\%) }
\end{array} & \begin{array}{c}
\text { Overreport by } 8 \mathrm{psi} \\
\text { or more (\%) }
\end{array} \\
\hline 25 \mathrm{psi} & 16 & 2 & 0 \\
\hline 35 \mathrm{psi} & 19 & 9 & 0 \\
\hline 45 p s i & 19 & 14 & 5 \\
\hline 55 p s i & 20 & 15 & 9 \\
\hline
\end{array}\)
If the gas station air pressure gauge reads 35 psi
We are asked to find the probability that the pressure is over-reported by 6 psi or more.
Let A be the event of air pressure is over-reported by 4 psi or more.
Let B be the event of air pressure is over-reported by 6 psi or more.
Let C be the event of air pressure is over-reported by 8 psi or more.
Hence from the table, If the gas station air pressure gauge reads 35 psi then 9% of gas stations over-report the pressure by 6 psi.
Therefore the probability that the pressure is over-reported by 6 psi or more by gas stations is 9 out of 100.
Hence we can write,
\(P(B)=\frac{9}{100}=0.09\)
Hence the probability that the pressure is over-reported by 6 psi or more is 0.09