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Question

$$X$$ is a Normally distributed variable with mean $$ = 30$$ and standard deviation $$ = 4$$. Find $$P(30 < x<35)$$


A
0.3698
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B
0.3956
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C
0.2134
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D
0.3944
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Solution

The correct option is C $$0.3944$$
Here we need to determine the value of $$ Z=\dfrac{x-\mu}{\sigma} $$
Here, $$\mu=\text{Mean}=30$$ and $$\sigma=\text{Stand Deviation}=4$$
For, $$x=30\Rightarrow Z=\dfrac{30-30}{4}=0$$
For, $$x=35\Rightarrow Z=\dfrac{35-30}{4}=1.25$$
$$\Rightarrow P(30<x<35)=P(0<Z<1.25) $$
$$\Rightarrow P(0<Z<1.25)=P(Z<1.25)-P(Z<0) $$                         $$(\because P(a<Z<b)=P(Z<b)-P(Z<a))$$
From the Normal Distribution table, $$P(Z<1.25)=0.8944$$ and $$P(Z<0)=0.5$$
$$\Rightarrow P(30<x<35)=0.8944-0.5=0.3944$$

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