Question

A toy manufacturing firms is producing balls with diameter 50 mm and std deviation 1 mm. In that case, out of 10 million balls, what proportion is expected to have dimensions between 47 mm and 48 mm?

Answer 1

In that case out of 10 million balls, what proportion is expected to have dimensions between 47 mm and 48 mm? Q: 2 .

Answer 2

The proportion is expected to have dimensions between 47 mm and 48 mm = 0.0215

Explanation:

Given : A toy manufacturing firms is producing balls with diameter 50 mm and std deviation 1 mm.

i.e. $\mu=50$     and   $\sigma=1$

Let x be the random variable that denotes the diameter of balls.

We assume that diameters of balls are normally distributed.

Sample size = 10 million , extremely large .

Now , the probability that dimensions lies between 47 mm and 48 mm:-

$P(47<x<48)=P(\dfrac{47-50}{1}<\dfrac{x-\mu}{\sigma}<\dfrac{48-50}{1})\\\\=P(-3<z<-2)\ \ [\because\ z=\dfrac{x-\mu}{\sigma}]\\\\ =P(z<-2)-P(z<-3)=0.0228-0.0013\\\\=0.0215$

Hence, the proportion is expected to have dimensions between 47 mm and 48 mm = 0.0215.

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Construction of normal distribution from mean and standard deviation

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