How to solve P(0 < z < a) = 0.2 for a

[ms. confused]

How would I solve for a on a question like this: P(0 < z < a)=0.2 ?

I know that for a question like P(z < a)= 0.85 I would find the inverse-norm of 0.85 to solve for a. I've tried the same thing for the first question, but of course it doesn't work and I'm out of ideas as to how else I should try and solve it. Could someone please help?

 

[HallsofIvy]

I assume you are talking about the normal distribution. How is your table of normal distribution set up? Some of them, for example, the one at
http://people.hofstra.edu/faculty/Stefan_Waner/RealWorld/normaltable.html
give the value under the curve from 0 to a which is exactly what you want.

If your table gives from negative infinity to a, then, since the distribution is symmetric about 0, the value from negative infinity to 0 is 1/2 and you just have to subtract 1/2 from the table value.

In general, with either kind of table, to find P(a< z< b), look up the values for a and b separately and subtractP: P(a< z< b)= P(z< b)- P(z< a).

 

[wais]

To solve for a in the equation P(0 < z < a) = 0.2, we need to use the inverse-norm function on a calculator. This function will give us the value of z that corresponds to a given probability. In this case, we want to find the value of z that has a probability of 0.2 between 0 and a.

To do this, we can follow these steps:

1. Start by writing the equation as P(z < a) = 0.2. This is because the probability between 0 and a is the same as the probability of being less than a.

2. Use a calculator or a statistical table to find the inverse-norm of 0.2. This will give you the value of z that corresponds to a probability of 0.2.

3. The inverse-norm of 0.2 is approximately -0.84. This means that the z-value that corresponds to a probability of 0.2 is -0.84.

4. Now we can substitute this value into our equation: P(z < a) = 0.2. This gives us the equation P(-0.84 < a) = 0.2.

5. Since we want to solve for a, we can simply add 0.84 to both sides of the equation, giving us P(a) = 0.2 + 0.84.

6. Finally, we can use a calculator to find the inverse-norm of 1.04, which is approximately 1.04. This means that a has a value of approximately 1.04.

Therefore, the solution to the equation P(0 < z < a) = 0.2 is a = 1.04.

In summary, to solve for a in an equation like this, we need to use the inverse-norm function on a calculator or statistical table to find the corresponding z-value, and then substitute it back into the equation to solve for a.