Question

A certain machine makes electrical resistors having a mean resistance of 40 ohms and a standard deviation of 2 ohms. Assuming that the resistance follows a normal distribution and can be measured to any degree of accuracy, what percentage of resistors will have a resistance that exceeds 43 ohms?

Answer 1

$\large\underline{\bf{Solution-}}$

Let X be the random variable which represents the resistance of an electrical resistor.

Given that,

Mean resistance of electrical resistor = 40 ohms

$\rm :\longmapsto\: \mu \: = 40 \: ohms$

Standard deviation of electrical resistor = 2 ohms

$\rm :\longmapsto\: \sigma \: = \: 2 \: ohms$

Let Z be a normal variable corresponding to X = 43, then

$\rm :\longmapsto\:Z = \dfrac{X - \mu}{ \sigma}$

$\rm :\longmapsto\:Z = \dfrac{43 - 40}{2}$

$\rm :\longmapsto\:Z = \dfrac{3}{2}$

$\bf\implies \:Z = 1.5$

Now, we have to find,

$\rm :\longmapsto\:P(X > 43)$

$\: \: \rm = \: \: P(Z > 1.5)$

$\: \: \rm = \: \: 0.5 - P(0 \leqslant Z \leqslant 1.5)$

$\: \: \rm = \: \: 0.5 - 0.4332$

$\: \: \rm = \: \: 0.0668$

Hence,

$\red{ \boxed{ \rm \: The \: required \:\% \: age = 0.0668 × 100 = 6.68 \: \%}}$

Additional Information :-

1. In normal distribution, mean, median and mode are equal.

2. Normal distribution is symmetric around the mean.

3. Area under the normal curve is 1.

Answer 2

Step-by-step explanation:

A certain machine makes electrical resistors having a mean resistance of 40 ohms and a

standard deviation of 17 ohms. Assuming that the resistance follows the normal distribution

and can be measured to any degree of accuracy, evaluate for resistance exceeding 43 ohm.