# Suppose that the current measurements in a strip of wire are assumed to follow a normal distribution with a mean of 10 milliamperes and a standard deviation of 2 milliamperes. What is the probability that a measurement exceeds 13 milliamperes? What is the probability that a current measurement is between 9 and 11 milliamperes.

Question
Measurement
Suppose that the current measurements in a strip of wire are assumed to follow a normal distribution with a mean of 10 milliamperes and a standard deviation of 2 milliamperes.
What is the probability that a measurement exceeds 13 milliamperes? What is the probability that a current measurement is between 9 and 11 milliamperes.

2021-02-15
Step 1
Standard normal distribution:
The standard normal distribution is a special case of normal distribution. The standard normal distribution will have mean 0 and standard deviation 1. If a random variable y follows normal distribution with mean (m) and standard deviation (s), then the standard normal variable z will be as given below:
$$z=\frac{y-E(y)}{S.D(y)}=\frac{y-\mu}{s}-N(0.1)$$
Step 2
Find the probability that the current measurement exceeds 13 milliamperes:
Consider the current measurements in a strip of wire be denoted by y. It is given that the current measurements in a strip of wire are normally distributed.
The mean and standard deviation for current measurements in a strip of wire is y-bar = 10 milliamperes and s(y) = 2 milliamperes, respectively.
The given requirement is that the current measurement in a strip of wire should be greater than 13 milliamperes. That is, y > 13.
The value of $$P(y > 13)$$ is obtained as 0.06681 from the calculation given below:
$$P(y>13)=1-P(y\leq 13)$$
$$=1-P((y-\frac{10}{2})\leq \frac{13-10}{2})$$
$$=1-P((y-E\frac{y}{S}.D(y))\leq (1.5)$$
$$=1-P(Z\leq 1.5)$$
$$\cong1-\phi(1.5),\left[Fromcthe\ table\ values\ of\ standard\ normal\ distribution.\ The\ area\ corresponding\ to\ left\ of\ 1.5\ is\ 0.06681\right]$$
$$\cong1-0.93319\cong0.06681$$
Step 3
Find the probability that the current measurement lies between 9 and 11 milliamperes:
The given requirement is that the current measurement in a strip of wire should lie between 9 and 11 milliamperes. That is, $$9 < y < 11$$.
The value of $$P(9 < y < 11)$$ is obtained as 0.38292 from the calculation given below:
$$P(9 \(=P(-0.5<(y-E\frac{y}{S.D(y)}<(0.5)$$</span>
$$=P(-0.5 \(=\phi(0.5)-\phi(-0.5)$$
$$\cong2\phi(0.5)-1$$
$$\cong(2\times 0.69146)-1,\left[From\ the\ table\ values\ of\ standard\ normal\ distribution.\ The\ area\ corresponding\ to\ left\ of\ 0.5\ is\ 0.69146\right]$$
$$\cong0.38292$$
Step 4
The probability that the current measurement exceeds 13 milliamperes is 0.06681.
The probability that the current measurement lies between 9 and 11 milliamperes is 0.38292.

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