Answer:
And if we solve for a we got
Step-by-step explanation:
For this case we have the initial case we have a normal distribution given :
Where and
And from the info of the problem we know that:
We can verify this using the z score formula given by:
And if we replace we got:
Now we have another situation for a new random variable X with a new distribution given by:
Where and
For this part we want to find a value a, such that we satisfy this condition:
(a)
(b)
Both conditions are equivalent on this case. We can use the z score again in order to find the value a.
As we can see on the figure attached the z value that satisfy the condition with 0.023 of the area on the left and 0.977 of the area on the right it's z=-1.995. On this case P(Z<-1.995)=0.023 and P(z>-1.995)=0.977
If we use condition (b) from previous we have this:
But we know which value of z satisfy the previous equation so then we can do this:
And if we solve for a we got