Use the Statistical Applet: Normal Density Curve to answer the question. Set the mean and standard deviation to 0 and 5 , respectively. Note that 0.023 of the area under the curve falls below −10.0 . Now set the mean to 10 and keep the standard deviation at 5 . Under these settings, 0.023 of the area under the curve falls below 0.0 . Proportion?

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