Suppose Z is a standard normal random variable

Question

Question 1: If P( -z < Z < z ) = 0.7814, find z.

Question 2: Find P( -2.11 < Z < -0.42).

I believe the first question, I have to do: 0.5 + (1/2)(0.7814). But I don't know what to do after.

Answer 1

You are correct. You do want to do that first. I would suggest drawing a picture for this. Since we are looking for the z score that has 0.7814 between it and it's negative, I'd draw a bell curve (it doesn't need to be perfect) and label zero in the middle. z and -z are the same distance from the middle (zero) but on opposite sides. Shade the area in the middle and label it as 0.7814.

What you are doing when you take half of 0.7814 is you are using the fact that the bell curve is symmetrical and finding the part of the area that is above zero (the middle.) Then you add it to 0.5 because you are adding the part that is above zero (the middle) to all the area that is below zero. Now we can either use a z-table or the invNorm( function if you are using a TI graphing calculator.

We need to find the z-score that corresponds to the area from our upper number "z" all the way down to the left. That is the area that you just found (0.5 + (1/2)(0.7814)) = 0.8907. We can find this score on our table, or at least a score that is close to it. If you look at the 1.2 row of a standard normal table, you should be able to find the column that is at or really close to 0.8907. The column label at the top will tell you what the second digit of the z-score is (e.g. if the number at the top of the column is ".08" then the z would be 1.28)

If you are allowed to (and have) a TI graphing calculator, you can use the "invNorm(" function that can be found in the DISTR (for distribution) key. You get to this by pressin [2nd] then [VARS]. Instead of looking for the area, you just need to enter it in and the calculator will give you the z-score. If you round to two decimal places on the number it gives you, the answer should be the same as you would get from the table. It should look like

invNorm(0.8907)

Hint: Later on, when you're dealing with a normal distribution that isn't standard, you can use this function with the mean and standard deviation to answer similar questions. Just add them on to the end, separating all the values with commas and closing the parentheses.

invNorm(0.8907, mean, sd)

For the second question, you are doing the opposite of this. Instead of having an area, they are giving you the values and asking you to find the are between them.

Using the table, you would go to the "-0.4" row and then the ".02" column and that would give you the area fto the left of -0.42. for the other z-score, you would go to the "-2.1" row and the ".01) column. Now this is another place where a picture will help you. -0.42 is a z score (along the "x-axis" if you think of it like most graphs) so we are slightly to the left of zero or the tallest part on the curve. The area that you find on the table is from that number all the way off to the left of the curve (which should go on forever.) Since the question only wants the probability (or area) between -2.11 and -0.42, we need to make sure that we don't count the area that is to the left of -2.11. So, we need to take that area away from the area we found for -0.42 through a process called subtraction. I hope that you're familiar with that. It is important to understand that this is what we are doing to understand what is going on when we are talking about probabilities with these kind of functions, so I suggest you always draw a picture and try to understand what is going on.

If you are using a TI-Graphing Calculator, it's much easier. There is a function in that same DISTR that will help us with this. You want to use the "normalcdf(" function. CDF stands for cumulative distribution function. What it does is finds the are under the curve if you give it two values. You need to give it the lower and upper values (in that order) and it will spit out the area which is the probability that you are looking for. It should look like.

normalcdf(lower, upper)

Hint : Just like the "invNorm(" function, this can be used with non-standard normal distributions by putting in the mean and standard deviation. Just add them on to the end, separating all the values with commas and closing the parentheses.

normalcdf(lower, upper, mean, sd)

If you had a question where it wanted to know all the area to the left of a certain z-score, you can just enter a number that is very far to the left, like -9999.