Question

Suppose $X~N(–3,1)$. Between what $x$values does $34.14\%$of the data lie?

Short answer

The $34.14\%$of data lies between $(-3.44,-2.56)$.

Step-by-step solution

Given Information

Given in the question that, $X~N(–3,1).$

We have to find between what $x$values does $34.14\%$of the data lie.

Explanation 

The $34.14\%$is lies in the middle of the normal curve,

Therefore the portion will be divided as equal in both side.

$\frac{34.14\mathrm{\%}}{2}=17.07\mathrm{\%}$

So it will be

$50\mathrm{\%}-17.07\mathrm{\%}=32.93\mathrm{\%}$

So the value of the outside will be$32.93\%$

Now, we need to find the z-value for the $32.93\%$

$z0.3293=-0.44(\text{left}-\text{tail})$

$z0.3293=0.44(\text{right}-\text{tail})$

The $z$- score formula is

$z=\frac{x-\mu }{\sigma }$

$-0.44=\frac{x-(-3)}{1}\Rightarrow -0.44\times 1=x+3\Rightarrow x=-3-0.44\Rightarrow x=-3.44$

$0.44=\frac{x-(-3)}{1}\Rightarrow 0.44\times 1=x+3\Rightarrow x=-3+0.44\Rightarrow x=-2.56$