The Sine Function
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Every confidence interval has associated z value. As confidence interval increases so do the z value associated with it.
The confidence interval can be calculated using following formula:
Where
is the mean value, z is the associated z value, s is the standard deviation and n is the number of samples.
We know that standard deviation is simply a square root of variance:
The confidence interval of 95% has associated z value of <span>1.960.
</span>Now we can calculate the confidence interval for our income:
The confidence interval can be calculated using following formula:
Where
We know that standard deviation is simply a square root of variance:
The confidence interval of 95% has associated z value of <span>1.960.
</span>Now we can calculate the confidence interval for our income:
First notice that the triangle with sides 
and the triangle with sides 
are similar. This is true because the angle between sides 
in the smaller triangle is clearly 
, while the angle between sides 
in the larger triangle is clearly 
. So the triangles are similar with sides corresponding to 
, respectively.
Now both triangles are
, which means there's a convenient ratio between its sides. If the length of the shortest leg is 
, then the length of the longer leg is 
and the hypotenuse has length 
.
Since
is the shortest leg in the larger triangle, it follows that 
, so 
Now both triangles are
Since