Finding locations Using a Table
Once we have actually the basic idea that the regular Distribution, the next step is come learn how to find areas under the curve. We"ll learn two various ways  making use of a table and also using technology.
You are watching: Find the area under the standard normal curve between z = 1 and z = 2.
Since every normally spread random variable has actually a slightly different distribution shape, the only means to find areas using a table is come standardize the variable  transform our variable so it has a mean of 0 and a traditional deviation of 1. Just how do we carry out that? Use the zscore!
As we noted in ar 7.1, if the arbitrarily variable X has a median μ and also standard deviation σ, climate transforming X making use of the zscore create a arbitrarily variable with typical 0 and also standard deviation 1! through that in mind, we simply need to learn just how to find areas under the standard common curve, which can then be used to any type of normally dispersed random variable.
Finding Area under the typical Normal Curve come the Left
Before us look a couple of examples, we require to first see how the table works. Before we begin the section, you require a copy the the table. You deserve to download a printable copy the this table, or use the table in the back of your textbook. It should look something prefer this:
It"s pretty overwhelming at first, however if you look at the photo at the height (take a minute and check that out), you have the right to see the it is denote the area come the left. That"s the vital  the values in the middle represent locations to the left of the matching zvalue. To recognize which zvalue it"s referring to, us look to the left to acquire the an initial two number and above to the columns to acquire the hundredths value. (Zvalues with an ext accuracy have to be rounded come the percentage percent in bespeak to usage this table.)
Say we"re searching for the area left that 2.84. To do that, we"d begin on the 2.8 row and go across until we obtain to the 0.04 column. (See picture.)
From the picture, we have the right to see the the area left the 2.84 is 0.0023.
Finding areas Using StatCrunch
Click ~ above Stat > Calculators > Normal Enter the mean, typical deviation, x, and also the direction that the inequality. Then press Compute. The image below shows P(Z Example 1 a. Discover the area left the Z = 0.72 < disclose answer > The area left that 0.72 is approximately 0.2358. b. Uncover the area left the Z = 1.90 < reveal answer > The area left of 1.90 is approximately 0.9713. Finding Area under the typical Normal Curve come the RightTo find areas to the right, we have to remember the match rule. Take a minute and also look back at the dominion from ar 5.2. Since we understand the entire area is 1, (Area come the ideal of z0) = 1  (Area come the left that z0) Example 2 a. Uncover the area to the ideal of Z = 0.72 < expose answer >
b. Discover the area come the right of Z = 2.68 < disclose answer >
An alternate idea is to usage the symmetric building of the typical curve. Rather of looking come the right of Z=2.68 in example 2 above, we might have looked at the area left of 2.68. Since the curve is symmetric, those locations are the same. Finding Area under the standard Normal Curve in between Two ValuesTo find the area between two values, we think of it in 2 pieces. Suppose we want to uncover the area between Z = 2.43 and also Z = 1.81. What we perform instead, is uncover the area left of 1.81, and also then subtract the area left that 2.43. Prefer this:
So the area in between 2.43 and 1.81 = 0.9649  0.0075 = 0.9574 Note: StatCrunch is able to calculation the "between" probabilities, so you won"t must perform the calculation over if you"re utilizing StatCrunch. Example 3 a. Find the area in between Z = 0.23 and Z = 1.64. < expose answer > area in between 0.23 and 1.64 = 0.9495  0.5910 = 0.3585 b. Discover the area in between Z = 3.5 and also Z = 3.0. < disclose answer > area in between 3.5 and also 3.0 = 0.0013  0.0002 = 0.0011 Finding areas Under a common Curve utilizing the Tabledraw a sketch of the typical curve and shade the desired area. Calculation the corresponding Zscores. Discover the corresponding area under the traditional normal curve.If girlfriend remember, this is precisely what we experienced happening in the Area of a Normal circulation demonstration. Monitor the link and also explore again the relationship in between the area under the standard normal curve and also a nonstandard normal curve. Finding locations Under a normal Curve making use of StatCrunchEven despite there"s no "standard" in the location here, the directions room actually precisely the same as those from above!
